Optimal approximation using complex-valued neural networks
Complex-valued neural networks (CVNNs) have recently shown promising
empirical success, for instance for increasing the stability of recurrent
neural networks and for improving the performance in tasks with complex-valued
inputs, such as in MRI fingerprinting. While the overwhelming success of Deep
Learning in the real-valued case is supported by a growing mathematical
foundation, such a foundation is still largely lacking in the complex-valued
case. We thus analyze the expressivity of CVNNs by studying their approximation
properties. Our results yield the first quantitative approximation bounds for
CVNNs that apply to a wide class of activation functions including the popular
modReLU and complex cardioid activation functions. Precisely, our results apply
to any activation function that is smooth but not polyharmonic on some
non-empty open set; this is the natural generalization of the class of smooth
and non-polynomial activation functions to the complex setting. Our main result
shows that the error for the approximation of $C^k$-functions scales as
$m^{-k/(2n)}$ for $m \to \infty$ where $m$ is the number of neurons, $k$ the
smoothness of the target function and $n$ is the (complex) input dimension.
Under a natural continuity assumption, we show that this rate is optimal; we
further discuss the optimality when dropping this assumption. Moreover, we
prove that the problem of approximating $C^k$-functions using continuous
approximation methods unavoidably suffers from the curse of dimensionality.
Paul Geuchen, Felix Voigtlaender
# Optimal approximation of \(C^{k}\)-functions using shallow complex-valued neural networks

###### Abstract.

We prove a quantitative result for the approximation of functions of regularity \(C^{k}\) (in the sense of real variables) defined on the complex cube \(\Omega_{n}:=[-1,1]^{n}+i[-1,1]^{n}\subseteq\mathbb{C}^{n}\) using shallow complex-valued neural networks. Precisely, we consider neural networks with a single hidden layer and \(m\) neurons, i.e., networks of the form \(z\mapsto\sum_{j=1}^{m}\sigma_{j}\cdot\phi\big{(}\rho_{j}^{T}z+b_{j}\big{)}\) and show that one can approximate every function in \(C^{k}\left(\Omega_{n};\mathbb{C}\right)\) using a function of that form with error of the order \(m^{-k/(2n)}\) as \(m\to\infty\), provided that the activation function \(\phi:\mathbb{C}\to\mathbb{C}\) is smooth but not polyharmonic on some non-empty open set. Furthermore, we show that the selection of the weights \(\sigma_{j},b_{j}\in\mathbb{C}\) and \(\rho_{j}\in\mathbb{C}^{n}\) is continuous with respect to \(f\) and prove that the derived rate of approximation is optimal under this continuity assumption. We also discuss the optimality of the result for a possibly _discontinuous_ choice of the weights.

Key words and phrases:Complex-valued neural networks, Shallow neural networks, Continuously differentiable functions, approximation rates 2020 Mathematics Subject Classification: 68T07, 41A25, 41A30, 41A63, 31A30, 30E10 Both authors contributed equally to this work.
the network. Here he assumed that the activation function is smooth on an open interval, where at a point of this interval no derivative vanishes. This is equivalent to the activation function being smooth and non-polynomial on that interval [12, p. 53].

We are going to generalize this result to the setting of complex-valued networks, by proving that one can approximate every function in \(C^{k}\left(\Omega_{n};\mathbb{C}\right)\) with an error of the order \(m^{-k/(2n)}\) using shallow complex-valued neural networks with \(m\) neurons in the hidden layer. Here we define the set \(\Omega_{n}\) as \(\Omega_{n}:=[-1,1]^{n}+i[-1,1]^{n}\). The result holds whenever the activation function is smooth and non-polyharmonic on some non-empty open set. This is a very natural condition, since for polyharmonic activation functions there exist \(C^{k}\)-functions that cannot be approximated at all below some error threshold using neural networks with this activation function [29].

Furthermore, the present paper studies in how far the approximation order of \(m^{-k/(2n)}\) is _optimal_, meaning that an order of \(m^{-(k/2n)-\alpha}\) (where \(\alpha>0\)) cannot be achieved. Here it turns out that the derived order of approximation is indeed optimal in the setting that the weight selection is _continuous_, meaning that the map that assigns to a function in \(C^{k}\left(\Omega^{n};\mathbb{C}\right)\) its corresponding network weights is continuous with respect to some norm on \(C^{k}(\Omega_{n};\mathbb{C})\). We are also going to investigate this optimality result further by dropping the continuity assumption and constructing two special smooth and non-polyharmonic activation functions with the first one having the property that the order of approximation can indeed be strictly increased via a discontinuous selection of the related weights. For the second function we are going to show that the order of \(m^{-k/(2n)}\) cannot be improved, even if one allows a discontinuous weight selection. This in particular shows that in the given generality of arbitrary smooth, non-polyharmonic activation functions, the approximation rate \(m^{-k/(2n)}\) cannot be improved.

### Main Results

In this work we are going to deal with so-called shallow complex-valued neural networks, meaning complex-valued neural networks with a single hidden layer. Furthermore, we are only considering networks where the map connecting the hidden with the final layer is not just affine-linear but even linear, and where the output dimension of the network is \(1\). Precisely, this means that we consider functions of the form

\[\mathbb{C}^{n}\ni z\mapsto\sum_{j=1}^{m}\sigma_{j}\phi\left(\rho_{j}^{T}\cdot z +\eta_{j}\right)\in\mathbb{C},\]

with \(\rho_{1},...,\rho_{m}\in\mathbb{C}^{n}\), \(\sigma_{1},...,\sigma_{m},\eta_{1},...,\eta_{m}\in\mathbb{C}\) and an _activation function_\(\phi:\mathbb{C}\rightarrow\mathbb{C}\). Here, \(m\) denotes the number of neurons of the network.

Our first main result is that it is possible to approximate complex polynomials in \(z\) and \(\overline{z}\) using shallow complex-valued neural networks. Precisely, for \(m\in\mathbb{N}\) let

\[\mathcal{P}_{m}:=\left\{\mathbb{C}^{n}\rightarrow\mathbb{C},\ z\mapsto\sum_{ \mathbf{m}\leq m}\sum_{\boldsymbol{\ell}\leq m}a_{\mathbf{m},\boldsymbol{\ell} }z^{\mathbf{m}}\overline{z}^{\boldsymbol{\ell}}:\ a_{\mathbf{m},\boldsymbol{ \ell}}\in\mathbb{C}\right\},\]

where we are summing over all multi-indices \(\mathbf{m},\boldsymbol{\ell}\in\mathbb{N}_{0}^{n}\) with \(\mathbf{m}_{j},\boldsymbol{\ell}_{j}\leq m\) for every \(j\in\{1,...,n\}\). Note that \(\mathcal{P}_{m}\) is a finite-dimensional space, so that it makes sense to talk about bounded subsets of \(\mathcal{P}_{m}\). Then we show the following:

**Theorem 1.1**.: _Let \(m,n\in\mathbb{N}\), \(\varepsilon>0\) and \(\phi:\mathbb{C}\rightarrow\mathbb{C}\) be smooth and non-polyharmonic on an open set \(\varnothing\neq U\subseteq\mathbb{C}\). Let \(\mathcal{P}^{\prime}\subseteq\mathcal{P}_{m}\) be bounded and set \(N:=(4m+1)^{2n}\). Then there exist certain coefficients \(\rho_{1},...,\rho_{N}\in\mathbb{C}^{n}\) and \(b_{m}\in U\) with the following property: For each polynomial \(p\in\mathcal{P}^{\prime}\) there are coefficients \(\sigma_{1},...,\sigma_{N}\in\mathbb{C}\), such that_

\[\left\|p-f\right\|_{L^{\infty}(\Omega_{n};\mathbb{C})}\leq\varepsilon,\]

_where_

\[f:\quad\Omega_{n}\rightarrow\mathbb{C},\quad z\mapsto\sum_{j=1}^{N}\sigma_{ j}\phi\left(\rho_{j}^{T}z+b_{m}\right). \tag{1.1}\]
Note that \(N\) only depends on the degree of the polynomials, not on the desired degree of approximation and that the coefficients \(\rho_{j}\) are chosen independently from the specific polynomial \(p\).

Using this approximation result and the fact that \(C^{k}\)-functions can be approximated at a certain rate by complex polynomials in \(z\) and \(\overline{z}\), we can then study the approximation of arbitrary \(C^{k}\)-functions.

**Theorem 1.2**.: _Let \(n,k\in\mathbb{N}\). Then there exists a constant \(c=c(n,k)>0\) with the following property: For any function \(\phi:\mathbb{C}\rightarrow\mathbb{C}\) that is smooth and non-polyharmonic on an open set \(\varnothing\neq U\subseteq\mathbb{C}\) and for any \(m\in\mathbb{N}\) there are coefficients \(\rho_{1},...,\rho_{m}\in\mathbb{C}^{n}\) and \(b_{m}\in U\) with the following property: For any \(f\in C^{k}\left(\Omega_{n};\mathbb{C}\right)\) there are complex numbers \(\sigma_{1}=\sigma_{1}(f),...,\sigma_{m}=\sigma_{m}(f)\in\mathbb{C}\), such that_

\[\left\|f-g\right\|_{L^{\infty}\left(\Omega_{n};\mathbb{C}\right)}\leq c\cdot m ^{-k/(2n)}\cdot\left\|f\right\|_{C^{k}\left(\Omega_{n};\mathbb{C}\right)},\]

_where \(g\) is defined as_

\[g:\quad\Omega_{n}\rightarrow\mathbb{C},\quad g(z):=\sum_{j=1}^{m}\sigma_{j} \phi\left(\rho_{j}^{T}z+b_{m}\right).\]

_Furthermore, the map \(f\mapsto\sigma_{j}(f)\) is a continuous linear functional with respect to the \(L^{\infty}\)-norm for every \(j\in\{1,...,m\}\)._

Discussing the optimality of this result, based on work by DeVore et al. [10] we prove the following result.

**Theorem 1.3**.: _Let \(n\in\mathbb{N}\) and \(k\in\mathbb{N}\). Then there exists a constant \(c=c(n,k)>0\) with the following property: For any \(m\in\mathbb{N}\), \(\phi\in C(\mathbb{C};\mathbb{C})\) and any map_

\[\eta:\quad C^{k}\left(\Omega_{n};\mathbb{C}\right)\rightarrow(\mathbb{C}^{n}) ^{m}\times\mathbb{C}^{m}\times\mathbb{C}^{m},\quad g\mapsto\left(\eta_{1}(g), \eta_{2}(g),\eta_{3}(g)\right)\]

_which is continuous with respect to some norm on \(C^{k}(\Omega_{n};\mathbb{C})\) there exists \(f\in C^{k}\left(\Omega_{n};\mathbb{C}\right)\) satisfying \(\left\|f\right\|_{C^{k}\left(\Omega_{n};\mathbb{C}\right)}\leq 1\) and_

\[\left\|f-\Phi(f)\right\|_{L^{\infty}\left(\Omega_{n};\mathbb{C}\right)}\geq c \cdot m^{-k/(2n)},\]

_where \(\Phi(f)\in C(\Omega_{n};\mathbb{C})\) is given by_

\[\Phi(f)(z):=\sum_{j=1}^{m}\left(\eta_{3}(f)\right)_{j}\phi\left(\left[\eta_{1 }(f)\right]_{j}^{T}z+\left(\eta_{2}(f)\right)_{j}\right).\]

This result shows that the approximation order of \(m^{-k/(2n)}\) is optimal if one insists on a continuous selection of the weights. For the case of possibly _discontinuous_ weight selection we deduce the following two results:

**Theorem 1.4**.: _There exists a function \(\phi\in C^{\infty}(\mathbb{C};\mathbb{C})\) which is non-polyharmonic with the following additional property: For every \(k\in\mathbb{N}\) and \(n\in\mathbb{N}\) there exists a constant \(c=c(n,k)>0\) such that for any \(m\in\mathbb{N}\) and \(f\in C^{k}\left(\Omega_{n};\mathbb{C}\right)\) there are \(\rho_{1},...,\rho_{m}\in\mathbb{C}^{n}\) and \(\ell_{1},...,\ell_{m}\in\mathbb{N}\), such that_

\[\left\|f(z)-\sum_{j=1}^{m}\phi\left(\rho_{j}^{T}z+3\ell_{j}\right)\right\|_{L^ {\infty}\left(\Omega_{n};\mathbb{C}\right)}\leq c\cdot m^{-k/(2n-1)}\cdot \left\|f\right\|_{C^{k}\left(\Omega_{n};\mathbb{C}\right)}.\]

**Theorem 1.5**.: _Let \(n\in\mathbb{N}\), \(k\in\mathbb{N}\) and_

\[\phi:\quad\mathbb{C}\rightarrow\mathbb{C},\quad\phi(z):=\frac{1}{1+e^{- \operatorname{Re}(z)}}.\]

_In particular, \(\phi\) is smooth but non-polyharmonic. Then there exists a constant \(c=c(n,k)>0\) with the following property: For any \(m\in\mathbb{N}^{\geq 2}\) there is a function \(f\in C^{k}\left(\Omega^{n};\mathbb{C}\right)\) with \(\left\|f\right\|_{C^{k}\left(\Omega_{n};\mathbb{C}\right)}\leq 1\), such that for every choice of coefficients \(\rho_{1},...,\rho_{m}\in\mathbb{C}^{n}\), \(\eta_{1},...,\eta_{m}\in\mathbb{C}\) and \(\sigma_{1},...,\sigma_{m}\in\mathbb{C}\) we have_

\[\left\|f(z)-\sum_{j=1}^{m}\sigma_{j}\cdot\phi\left(\rho_{j}^{T}z+\eta_{j} \right)\right\|_{L^{\infty}\left(\Omega_{n};\mathbb{C}\right)}\geq c\cdot \left(m\cdot\ln(m)\right)^{-k/2n}.\]
The significance of these results is as follows: The approximation rate provided by Theorem 1.2 can be strictly improved for some special activation functions (e.g., for the function \(\phi\) from Theorem 1.4), but this improvement is not generally possible for arbitrary (smooth, non-polyharmonic) activation functions; there are such activation functions for which the order of \(m^{-k/(2n)}\) is optimal (up to logarithmic factors), even if discontinuous weight selection is allowed.

### Related Work

The main inspiration for the present work is the paper [22] by Mhaskar, where the real-valued counterpart of Theorem 1.2 is stated and proven. Many techniques in the present paper are in fact adaptations of what has been done in [22]. In particular noteworthy are the multidimensional divided differences formula used to approximate polynomials and the approximation of \(C^{k}\)-functions using Fourier-analytic results and Chebyshev polynomials.

In addition, the work [21] should be mentioned, which serves as an inspiration for Theorem 1.4. In [21] it was proven that there exists a real activation function such that one can approximate continuous functions on the unit cube arbitrarily well using neural networks of _constant size_ (only depending on the input dimension) which have two hidden layers. Suitable adaptation of the techniques used in [21] in combination with a result from [14] is the basis for our proof of Theorem 1.4.

Another important inspiration for the present work is [31], in which the approximation properties of real-valued networks using the ReLU activation function are considered. In particular, this paper also distinguishes between a continuous and discontinuous selection of network weights, but generally considers networks with more than one hidden layer. In the setting of [31], the differences between continuous and discontinuous weight assignment are significant: while with continuous weight assignment at most a rate of \(W^{-k/n}\) is possible (and in fact achievable), with discontinuous weight assignment an improvement of the rate to \(W^{-2k/n}\) is possible. Here \(k\) is, as usual, the regularity of the functions under consideration, \(n\) the input dimension and \(W\) the number of parameters of the network.

When it comes to general literature about mathematical properties of complex-valued neural networks, surprisingly little work can be found. The most prominent result in the theory of CVNNs is the _Universal Approximation Theorem for Complex-Valued Neural Networks_[29], discussing for which activation functions CVNNs are uniform approximators on compact sets. In particular, it has been shown that shallow CVNNs have this property if and only if the activation function \(\phi\) is not polyharmonic. Thus, the condition assumed in this work is quite natural.

Another result from the field of approximation theory of CVNNs can be found in [24]. Here it is proven that shallow CVNNs with _entire_ functions as activation functions can uniformly approximate exactly those continuous functions on compact sets which can also be uniformly approximated by complex polynomials on compact sets. Note that, unlike in the real setting, this does not correspond to the entirety of continuous functions. For example, it is not possible to uniformly approximate the function \(z\mapsto\overline{z}\) on compact sets with non-empty interior using holomorphic functions (in particular with polynomials).

Regarding quantitative approximation results for CVNNs, the only existing work of which we are aware is [8], analyzing the approximation capabilities of _deep_ CVNNs where the modReLU is used as activation function. Since the modReLU satisfies our condition on an admissible activation function (as shown in Section 5), the present work can be seen as an improvement to [8]. The main differences can be summarized as follows:

* We consider very general activation functions including but not limited to the modReLU.
* We improve the approximation order by a log factor.
* We show that the order of approximation can in fact be achieved using shallow networks instead of the deep networks used in [8].

### Notation

Throughout the paper, multi-indices (i.e. elements of \(\mathbb{N}_{0}^{n}\)) are denoted using boldface. For \(\mathbf{m}\in\mathbb{N}_{0}^{n}\) we have the usual notation \(|\mathbf{m}|=\sum_{j=1}^{n}\mathbf{m}_{j}\). For a number \(m\in\mathbb{N}_{0}\) and another multi-index \(\mathbf{p}\in\mathbb{N}_{0}^{n}\) we write \(\mathbf{m}\leq m\) iff \(\mathbf{m}_{j}\leq m\) for every \(j\in\{1,...,n\}\) and \(\mathbf{m}\leq\mathbf{p}\) iff \(\mathbf{m}_{j}\leq\mathbf{p}_{j}\) for every
\(j\in\{1,...,n\}\). Furthermore we write

\[\begin{pmatrix}\mathbf{P}\\ \mathbf{r}\end{pmatrix}=\prod_{j=1}^{n}\begin{pmatrix}\mathbf{p}_{j}\\ \mathbf{r}_{j}\end{pmatrix}\]

for two multi-indices \(\mathbf{p},\mathbf{r}\in\mathbb{N}_{0}^{n}\) with \(\mathbf{r}\leq\mathbf{p}\). For a complex vector \(z\in\mathbb{C}^{n}\) we write

\[z^{\mathbf{m}}=\prod_{j=1}^{n}z_{j}^{\mathbf{m}_{j}}\quad\text{and}\quad \overline{z}^{\mathbf{m}}=\prod_{j=1}^{n}\overline{z_{j}}^{\mathbf{m}_{j}}.\]

For a point \(x\in\mathbb{R}^{n}\) and \(r>0\) we define

\[B_{r}(z):=\{y\in\mathbb{R}^{n}:\ \left\|x-y\right\|<r\}\]

with \(\left\|\cdot\right\|\) denoting the usual Euclidian distance. This definition is analogously transferred to \(\mathbb{C}^{n}\). Furthermore, for any function \(f:\mathbb{R}^{n}\to\mathbb{R}^{m}\) and a subset \(A\subseteq\mathbb{R}^{n}\) we denote

\[\left\|f\right\|_{L^{\infty}(A;\mathbb{R}^{m})}:=\sup_{x\in A}\left\|f(x) \right\|.\]

Note that this is finite if \(f\) is continuous and \(A\) compact. This definition is also analogously transferred to \(\mathbb{C}^{n}\) and \(\mathbb{C}^{m}\).

For a function \(f:U\to\mathbb{R}^{m}\) with an open set \(U\subseteq\mathbb{R}^{n}\) we write

\[f\in C^{k}\left(U;\mathbb{R}^{m}\right)\ \Leftrightarrow\ \partial^{\mathbf{p}}f\text{ exists and is continuous for every }\mathbf{p}\in\mathbb{N}_{0}^{n}\text{ with }\left|\mathbf{p}\right|\leq k.\]

\(\mathbb{C}^{n}\) is canonically isomorphic to \(\mathbb{R}^{2n}\) by virtue of the isomorphism

\[\varphi_{n}:\quad\mathbb{R}^{2n}\to\mathbb{C}^{n},\quad(x_{1},...,x_{n},y_{1},...,y_{n})\mapsto(x_{1}+iy_{1},...,x_{n}+iy_{n})\,. \tag{1.2}\]

Let \(U\subseteq\mathbb{C}^{n}\) be any subset of \(\mathbb{C}^{n}\) and \(f:U\to\mathbb{C}\). Then the function \(\widetilde{f}:\varphi_{n}^{-1}(U)\to\mathbb{R}^{2}\) associated to \(f\) is defined as

\[\widetilde{f}:=\varphi_{1}^{-1}\circ f\circ\varphi_{n}\big{|}_{\varphi_{n}^{- 1}(U)}.\]

If \(U\) is open, we can study real differentiability of \(\widetilde{f}\). We write

\[f\in C^{k}(U;\mathbb{C}):\Leftrightarrow\widetilde{f}\in C^{k}\left(\varphi_{ n}^{-1}(U);\mathbb{R}^{2}\right).\]

If \(f\in C^{k}(U;\mathbb{C})\) and \(\mathbf{p}\in\mathbb{N}_{0}^{2n}\) is a multi-index with \(\left|\mathbf{p}\right|\leq k\), we denote

\[\partial^{\mathbf{p}}f:=\varphi_{1}\circ\partial^{\mathbf{p}}\widetilde{f} \circ\varphi_{n}^{-1}\big{|}_{U}.\]

In the present work, the goal is to study the approximation capabilities of complex-valued neural networks on the complex cube

\[\Omega_{n}:=\{(z_{1},...,z_{n})\in\mathbb{C}^{n}:\ \operatorname{Re}\left(z_{j} \right),\ \operatorname{Im}\left(z_{j}\right)\in[-1,1],\ j=1,...,n\}\,.\]

Therefore, we need to introduce \(C^{k}\)-functions on this cube: We define

\[C^{k}\left([-1,1]^{n};\mathbb{R}^{m}\right):=\left\{f\in C([-1,1]^{n};\mathbb{ R}^{m}):\begin{array}{c}f\big{|}_{(-1,1)^{n}}\in C^{k}\left((-1,1)^{n}; \mathbb{R}^{m}\right)\text{ and }\\ \partial^{\mathbf{p}}\big{(}f\big{|}_{(-1,1)^{n}}\big{)}\text{ uniformly continuous }\forall\left|\mathbf{p}\right|\leq k \end{array}\right\}.\]

For a function \(f\in C^{k}\left([-1,1]^{n};\mathbb{R}^{m}\right)\) and any multi-index \(\mathbf{p}\in\mathbb{N}_{0}^{n}\) with \(\left|\mathbf{p}\right|\leq k\) we define \(\partial^{\mathbf{p}}f\) as the unique continuous extension of \(\partial^{\mathbf{p}}\big{(}f\big{|}_{(-1,1)^{n}}\big{)}\) to \([-1,1]^{n}\) which exists because of the uniform continuity of \(\partial^{\mathbf{p}}\big{(}f\big{|}_{(-1,1)^{n}}\big{)}\). We then define the \(C^{k}\)-norm as

\[\left\|f\right\|_{C^{k}\left([-1,1]^{n};\mathbb{R}^{m}\right)}:=\max_{\left| \mathbf{p}\right|\leq k}\ \max_{x\in[-1,1]^{n}}\left\|(\partial^{\mathbf{p}}f)\left(x)\right\|_{2}.\]

We translate this definition to the setting of complex-valued functions by letting

\[C^{k}\left(\Omega_{n};\mathbb{C}\right):=\left\{f:\Omega_{n}\to\mathbb{C}:\ \widetilde{f}\in C^{k}\left([-1,1]^{2n};\mathbb{R}^{2}\right)\right\}.\]

Additionally, as in the setting of open sets, we denote

\[\partial^{\mathbf{p}}f:=\varphi_{1}\circ\partial^{\mathbf{p}}\widetilde{f} \circ\varphi_{n}^{-1}\big{|}_{\Omega_{n}}\]
for any \(\mathbf{p}\in\mathbb{N}_{0}^{2n}\) with \(\left|\mathbf{p}\right|\leq k\) and

\[\left\|f\right\|_{C^{k}(\Omega_{n};\mathbb{C})}:=\max_{\left|\mathbf{p}\right| \leq k}\,\max_{z\in\Omega_{n}}\left|\left(\partial^{\mathbf{p}}f\right)(z) \right|.\]

Note that by definition we have

\[\left\|f\right\|_{C^{k}(\Omega_{n};\mathbb{C})}=\|\widetilde{f}\|_{C^{k}([-1, 1]^{2n};\mathbb{R}^{2})}.\]

For \(f\in C^{1}(U;\mathbb{C})\), where \(U\subseteq\mathbb{C}\) is an open subset of \(\mathbb{C}\), we denote

\[\partial_{\mathrm{wirt}}f:=\frac{1}{2}\left(\partial^{(1,0)}f-i\partial^{(0, 1)}f\right),\quad\overline{\partial}_{\mathrm{wirt}}f:=\frac{1}{2}\left( \partial^{(1,0)}f+i\partial^{(0,1)}f\right).\]

These operators are called the _Wirtinger derivatives_. They have the following properties that we are going to use, which can be found for example in [16, E. 1a]:

1. \(\partial_{\mathrm{wirt}}\) and \(\overline{\partial}_{\mathrm{wirt}}\) are both \(\mathbb{C}\)-linear operators on the set \(C^{1}(U;\mathbb{C})\).
2. \(f\) is complex-differentiable in \(z\in U\) iff \(\overline{\partial}_{\mathrm{wirt}}f(z)=0\) and in this case the equality \[\partial_{\mathrm{wirt}}f(z)=f^{\prime}(z)\] holds true, with \(f^{\prime}(z)\) denoting the complex derivative of \(f\) at \(z\).
3. We have the conjugation rules \[\overline{\partial_{\mathrm{wirt}}f}=\overline{\partial}_{\mathrm{wirt}} \overline{f}\text{ and }\overline{\overline{\partial}_{\mathrm{wirt}}f}= \partial_{\mathrm{wirt}}\overline{f}.\]
4. If \(g\in C^{1}(U;\mathbb{C})\), the following product rules for Wirtinger derivatives hold for every \(z\in U\): \[\partial_{\mathrm{wirt}}(f\cdot g)(z) =\partial_{\mathrm{wirt}}f(z)\cdot g(z)+f(z)\cdot\partial_{ \mathrm{wirt}}g(z),\] \[\overline{\partial}_{\mathrm{wirt}}(f\cdot g)(z) =\overline{\partial}_{\mathrm{wirt}}f(z)\cdot g(z)+f(z)\cdot \overline{\partial}_{\mathrm{wirt}}g(z).\]
5. If \(V\subseteq\mathbb{C}\) is an open set and \(g\in C^{1}(V;\mathbb{C})\) with \(g(V)\subseteq U\), then the following chain rules for Wirtinger derivatives hold true: \[\partial_{\mathrm{wirt}}(f\circ g) =\left[\left(\partial_{\mathrm{wirt}}f\right)\circ g\right]\cdot \partial_{\mathrm{wirt}}g+\left[\left(\overline{\partial}_{\mathrm{wirt}}f \right)\circ g\right]\cdot\partial_{\mathrm{wirt}}\overline{g},\] \[\overline{\partial}_{\mathrm{wirt}}(f\circ g) =\left[\left(\partial_{\mathrm{wirt}}f\right)\circ g\right]\cdot \overline{\partial}_{\mathrm{wirt}}g+\left[\left(\overline{\partial}_{\mathrm{ wirt}}f\right)\circ g\right]\cdot\overline{\partial}_{\mathrm{wirt}}\overline{g}.\]
6. If \(f\in C^{2}(U;\mathbb{C})\) then we have \[\Delta f(z)=4\left(\partial_{\mathrm{wirt}}\overline{\partial}_{\mathrm{wirt}} f\right)(z)\] for every \(z\in U\), with \(\Delta\) denoting the usual Laplace-Operator \(\Delta=\partial^{(2,0)}+\partial^{(0,2)}\) (cf. [6, equation (1.7)]).

A function \(f\in C^{\infty}(U;\mathbb{C})\) is called _polyharmonic_ (on \(U\) with an open set \(U\subseteq\mathbb{C}\)) if there is \(k\in\mathbb{N}\) such that \(\Delta^{k}\left(f\right)\equiv 0\).

Furthermore, we can also define partial Wirtinger derivatives of functions in multiple variables, meaning \(f\in C^{1}(U;\mathbb{C})\), where \(U\subseteq\mathbb{C}^{n}\) is an open set. The formal definition of the partial Wirtinger derivative with respect to the \(j\)-th variable is

\[\partial_{\mathrm{wirt}}^{e_{j}}f:=\frac{1}{2}\left(\partial^{e_{j}}f-i\partial ^{e_{n+j}}f\right),\quad\overline{\partial}_{\mathrm{wirt}}^{e_{j}}f:=\frac{1} {2}\left(\partial^{e_{j}}f+i\partial^{e_{n+j}}f\right).\]

It is easy to see that since Wirtinger derivatives are just a complex linear combination of real derivatives, the symmetry of derivatives also holds in the case of Wirtinger derivatives, i.e., all Wirtinger derivatives commute for functions of sufficient regularity. Thus, we introduce the following notation: For an open set \(U\subseteq\mathbb{C}^{n}\), a function \(f\in C^{k}(U;\mathbb{C})\) and a multi-index \(\boldsymbol{\ell}\in\mathbb{N}_{0}^{n}\) with \(\left|\boldsymbol{\ell}\right|\leq k\) we write \(\partial_{\mathrm{wirt}}^{\boldsymbol{\ell}}f\) and \(\overline{\partial}_{\mathrm{wirt}}^{\boldsymbol{\ell}}f\) for the iterated Wirtinger derivatives according to the multi-index \(\boldsymbol{\ell}\).

**Proposition 1.6**.: _Let \(U\subseteq\mathbb{C}^{n}\) be an open set and \(f\in C^{k}(U;\mathbb{C})\). Then for any multi-indices \(\boldsymbol{m},\boldsymbol{\ell}\in\mathbb{N}_{0}^{n}\) with \(\left|\boldsymbol{m}+\boldsymbol{\ell}\right|\leq k\) we have the representation_

\[\partial_{\mathrm{wirt}}^{\boldsymbol{m}}\overline{\partial}_{\mathrm{wirt}}^{ \boldsymbol{\ell}}f(a)=\sum_{\begin{subarray}{c}\boldsymbol{p}=(\boldsymbol{p}^{ \prime},\boldsymbol{p}^{\prime\prime})\in\mathbb{N}_{0}^{2n}\\ \boldsymbol{p}^{\prime}+\boldsymbol{p}^{\prime\prime}=\boldsymbol{m}+\boldsymbol {\ell}\end{subarray}}b_{\boldsymbol{p}}\left(\partial^{\mathbf{p}}f\right)(a) \quad\forall a\in U\]
_with complex numbers \(b_{\boldsymbol{p}}\in\mathbb{C}\) only depending on \(\boldsymbol{p},\boldsymbol{m}\) and \(\boldsymbol{\ell}\) and writing \(\boldsymbol{p}=(\boldsymbol{p}^{\prime},\boldsymbol{p}^{\prime\prime})\) for the concatenation of the multi-indices \(\boldsymbol{p}^{\prime}\) and \(\boldsymbol{p}^{\prime\prime}\in\mathbb{N}_{0}^{n}\). In particular, the coefficients do not depend on \(f\)._

Proof.: The proof is by induction over \(\mathbf{m}\) and \(\boldsymbol{\ell}\). We first assume \(\mathbf{m}=0\) and show the claim for all \(\boldsymbol{\ell}\in\mathbb{N}_{0}^{n}\) with \(|\boldsymbol{\ell}|\leq k\). In the case \(\boldsymbol{\ell}=0\) there is nothing to show, so we assume the claim to be true for fixed \(\boldsymbol{\ell}\in\mathbb{N}_{0}^{n}\) with \(|\boldsymbol{\ell}|<k\) and take \(j\in\{1,...,n\}\) arbitrarily. Then we get

\[\overline{\partial}_{\mathrm{wirt}}^{\boldsymbol{\ell}+e_{j}}f(a) =\overline{\partial}_{\mathrm{wirt}}^{ej}\overline{\partial}_{ \mathrm{wirt}}^{\boldsymbol{\ell}}f(a)\overset{\mathrm{IH}}{=}\sum_{ \begin{subarray}{c}\mathbf{p}=(\boldsymbol{p}^{\prime},\boldsymbol{p}^{\prime \prime})\in\mathbb{N}_{0}^{2n}\\ \mathbf{p}^{\prime}+\boldsymbol{p}^{\prime\prime}=\boldsymbol{\ell}\end{subarray}} b_{\boldsymbol{p}}\overline{\partial}_{\mathrm{wirt}}^{ej}\left(\partial^{ \mathbf{p}}f\right)(a)\] \[=\sum_{\begin{subarray}{c}\mathbf{p}=(\mathbf{p}^{\prime}, \mathbf{p}^{\prime\prime})\in\mathbb{N}_{0}^{2n}\\ \mathbf{p}^{\prime}+\mathbf{p}^{\prime\prime}=\boldsymbol{\ell}\end{subarray}} b_{\boldsymbol{p}}\left(\partial^{(\mathbf{p}^{\prime}+e_{j},\mathbf{p}^{\prime \prime})}f\right)(a)+\frac{ib_{\mathbf{p}}}{2}\left(\partial^{(\mathbf{p}^{ \prime},\mathbf{p}^{\prime\prime}+e_{j})}f\right)(a)\] \[=\sum_{\begin{subarray}{c}\mathbf{p}=(\mathbf{p}^{\prime}, \mathbf{p}^{\prime\prime})\in\mathbb{N}_{0}^{2n}\\ \mathbf{p}^{\prime}+\mathbf{p}^{\prime\prime}=\boldsymbol{\ell}+e_{j}\end{subarray}} b_{\boldsymbol{p}}\left(\partial^{\mathbf{p}}f\right)(a),\]

as was to be shown.

Since we have shown the case \(\mathbf{m}=0\) we may assume the claim to be true for fixed \(\mathbf{m},\boldsymbol{\ell}\in\mathbb{N}_{0}^{s}\) with \(|\mathbf{m}+\boldsymbol{\ell}|<k\). Then we deduce

\[\partial_{\mathrm{wirt}}^{\mathbf{m}+e_{j}}\overline{\partial}_{ \mathrm{wirt}}^{\boldsymbol{\ell}}f(a) =\partial_{\mathrm{wirt}}^{ej}\partial_{\mathrm{wirt}}^{\boldsymbol {\mathsf{m}}}\overline{\partial}_{\mathrm{wirt}}^{\boldsymbol{\ell}}f(a) \overset{\mathrm{IH}}{=}\sum_{\begin{subarray}{c}\mathbf{p}=(\mathbf{p}^{ \prime},\mathbf{p}^{\prime\prime})\in\mathbb{N}_{0}^{2n}\\ \mathbf{p}^{\prime}+\mathbf{p}^{\prime\prime}=\mathbf{m}+\boldsymbol{\ell} \end{subarray}}b_{\boldsymbol{p}}\partial_{\mathrm{wirt}}^{ej}\left( \partial^{\mathbf{p}}f\right)(a)\] \[=\sum_{\begin{subarray}{c}\mathbf{p}=(\mathbf{p}^{\prime}, \mathbf{p}^{\prime\prime})\in\mathbb{N}_{0}^{2n}\\ \mathbf{p}^{\prime}+\mathbf{p}^{\prime\prime}=\mathbf{m}+\boldsymbol{\ell}+e_{j} \end{subarray}}\frac{b_{\boldsymbol{p}}}{2}\left(\partial^{(\mathbf{p}^{\prime }+e_{j},\mathbf{p}^{\prime\prime})}f\right)(a)-\frac{ib_{\mathbf{p}}}{2}\left( \partial^{(\mathbf{p}^{\prime},\mathbf{p}^{\prime\prime}+e_{j})}f\right)(a)\] \[=\sum_{\begin{subarray}{c}\mathbf{p}=(\mathbf{p}^{\prime}, \mathbf{p}^{\prime})^{\prime}\in\mathbb{N}_{0}^{2n}\\ \mathbf{p}^{\prime}+\mathbf{p}^{\prime\prime}=\mathbf{m}+\boldsymbol{\ell}+e_{j} \end{subarray}}b_{\boldsymbol{p}}\left(\partial^{\mathbf{p}}f\right)(a).\]

Hence the claim follows using the principle of induction. 

Proposition 1.6 is technical but crucial: In the course of the paper it will be useful to be able to approximate Wirtinger derivatives of certain functions. In fact, however, we will approximate real derivatives and take advantage of the fact that Wirtinger derivatives are linear combinations of these.

### Structure of the Paper

The work is divided into five main sections (excluding the introduction) and the appendix. In Section 2 we introduce the class of admissible activation functions \(\phi:\mathbb{C}\to\mathbb{C}\). The definition is stated in a very general way but we show that every function which is smooth and non-polyharmonic on a non-empty open subset of \(\mathbb{C}\) is admissible.

Section 3 is devoted to the approximation of complex polynomials in \(z\) and \(\overline{z}\) and in particular to the proof of Theorem 1.1. An extension of the Mean Value Theorem to multivariate functions using divided differences, which is important for proving the theorem, is proven in the appendix.

In Section 4, we consider general \(C^{k}\)-functions on the complex unit cube \(\Omega_{n}\) and prove Theorem 1.2. Essential to the proof is a quantitative approximation statement for \(C^{k}\)-functions using linear combinations of Chebyshev polynomials. This approximation is based on Fourier analytic methods, which are well-known in principle. Nevertheless, we discuss these methods in the appendix in order to make the paper more self-contained.

Furthermore, Section 5 deals with concrete examples of admissible functions and in particular proves the admissibility of the modReLU and the complex cardioid function, which are the two most commonly used activation functions in applications of complex-valued neural networks.

Finally, Section 6 deals with the optimality of the proven approximation statement for \(C^{k}\)-functions and in particular proves Theorems 1.3 - 1.5.
## 2. Admissibility of Activation Functions

In this section we discuss the notion of admissibility of activation functions \(\phi:\mathbb{C}\to\mathbb{C}\), i.e., the sufficient conditions that a function has to fulfill in order to be suitable for the approximation results established in Sections 3 and 4. We are going to define a slightly weaker notion of admissibility than that the function is smooth and non-polyharmonic on a non-empty open subset of \(\mathbb{C}\).

Let us recall that in [22] the sufficient condition on an activation function \(\phi:\mathbb{R}\to\mathbb{R}\) is that it is smooth on an open interval with one point in the interval where no derivative vanishes. In fact, the proofs in [22] show that it is sufficient to assume that for every \(M\in\mathbb{N}\) there is an open set \(\varnothing\neq U\subseteq\mathbb{R}\) such that \(\phi\big{|}_{U}\in C^{M}\left(U;\mathbb{R}\right)\) with a point \(b\in U\) satisfying

\[f^{(n)}(b)\neq 0\text{ for all }n\leq M.\]

In the case of complex-valued neural networks it turns out that the usual real derivative has to be replaced by mixed Wirtinger derivatives of the form \(\partial_{\text{virt}}^{m}\overline{\partial}_{\text{virt}}^{\ell}\). This leads to the following definition.

**Definition 2.1**.: Let \(\phi:\mathbb{C}\to\mathbb{C}\) and \(M\in\mathbb{N}_{0}\). We call \(\phi\)_\(M\)-admissible_ (in \(b\in\mathbb{C}\)) if there is an open set \(U\subseteq\mathbb{C}\) with \(b\in U\) and \(\phi\big{|}_{U}\in C^{2M}(U;\mathbb{C})\) such that

\[\partial_{\text{virt}}^{m}\overline{\partial}_{\text{virt}}^{\ell}\phi(b)\neq 0 \text{ for all }m,\ell\leq M.\]

In order to approximate complex polynomials in \(z\) and \(\overline{z}\) with bounded degree it will be enough to assume \(M\)-admissibility of an activation function for a certain \(M\in\mathbb{N}\). However, if one wants to approximate arbitrary \(C^{k}\)-functions one needs the following definition.

**Definition 2.2**.: A function \(\phi:\mathbb{C}\to\mathbb{C}\) is called _admissible_ if it is \(M\)-admissible for every \(M\in\mathbb{N}_{0}\).

Note that for an admissible function there does not necessarily have to be an open set \(\varnothing\neq U\subseteq\mathbb{C}\) such that the function is smooth on \(U\). However, if we assume smoothness we can derive the following elegant result.

**Theorem 2.3**.: _Let \(\phi:\mathbb{C}\to\mathbb{C}\), \(\varnothing\neq U\subseteq\mathbb{C}\) an open set and \(\phi\big{|}_{U}\in C^{\infty}(U;\mathbb{C})\). Then the following are equivalent:_

1. \(\phi\big{|}_{U}\) _is not polyharmonic._
2. _For every_ \(M\in\mathbb{N}_{0}\) _there exists_ \(z_{M}\in U\) _such that_ \(\phi\) _is_ \(M\)_-admissible at_ \(z_{M}\)_._

_In particular, in both cases \(\phi\) is admissible._

Proof.: (1) \(\Rightarrow\) (2): Let \(M\in\mathbb{N}_{0}\). Since \(\phi\big{|}_{U}\) is not polyharmonic we can pick \(z\in U\) with \(\Delta^{M}\phi(z)\neq 0\). By continuity we can choose \(\delta>0\) with \(B_{\delta}(z)\subseteq U\) and \(\Delta^{M}\phi(w)\neq 0\) for all \(w\in B_{\delta}(z)\). For all \(m,\ell\in\mathbb{N}_{0}\) let

\[A_{m,\ell}:=\left\{w\in B_{\delta}(z):\ \partial_{\text{virt}}^{m}\overline{ \partial}_{\text{virt}}^{\ell}\phi(w)=0\right\}\]

and assume towards a contradiction that

\[\bigcup_{m,\ell\leq M}A_{m,\ell}=B_{\delta}(z).\]

By [2, Corollary 3.35], \(B_{\delta}(z)\) with its usual topology is completely metrizable. By continuity of \(\partial_{\text{virt}}^{m}\overline{\partial}_{\text{virt}}^{\ell}\phi\) the sets \(A_{m,\ell}\) are closed in \(B_{\delta}(z)\). Hence, using the Baire category theorem [2, Theorems 3.46 and 3.47], there are \(m,\ell\in\mathbb{N}_{0}\) with \(m,\ell\leq M\), \(z^{\prime}\in A_{m,\ell}\) and \(\varepsilon>0\) such that

\[B_{\varepsilon}\left(z^{\prime}\right)\subseteq A_{m,\ell}\subseteq B_{\delta} (z).\]

But thanks to \(\Delta^{M}\phi=4^{M}\partial_{\text{virt}}^{M}\overline{\partial}_{\text{virt}} ^{M}\phi=4^{M}\partial_{\text{virt}}^{M-\ell}\overline{\partial}_{\text{virt}} ^{M-m}\partial_{\text{virt}}^{\ell}\overline{\partial}_{\text{virt}}^{m}\phi\) (see (6) on p. 6), this directly implies \(\Delta^{M}\phi\equiv 0\) on \(B_{\varepsilon}\left(z^{\prime}\right)\) in contradiction to the choice of \(B_{\delta}(z)\).

(2) \(\Rightarrow\) (1): If \(\phi\big{|}_{U}\) is polyharmonic there is \(M\in\mathbb{N}_{0}\) with

\[\partial_{\text{virt}}^{M}\overline{\partial}_{\text{virt}}^{M}\phi(z)=0\]

for every \(z\in U\), which contradicts (2).
Theorem 2.3 justifies the formulation of Theorems 1.1 and 1.2, since it shows that every activation function \(\phi\) satisfying the assumptions of the theorem is \(M\)-admissible for all \(M\in\mathbb{N}_{0}\).

_Remark 2.4_.: In the case of a real function it follows that if the function is smooth and non-polynomial on an open interval, there has to be a point at which no derivative vanishes, see [12, p. 53]. It is an interesting and currently unsolved question if a similar statement also holds true in the case of complex functions and Wirtinger derivatives, i.e., if the following holds: Let \(z\in\mathbb{C},\ r>0\) and \(\phi\in C^{\infty}(B_{r}(z);\mathbb{C})\) be non-polyharmonic. Then there exists \(b\in B_{r}(z)\) such that

\[\partial_{\mathrm{wirt}}^{m}\overline{\partial}_{\mathrm{wirt}}^{\ell}\phi(b) \neq 0\ \text{for all}\ m,\ell\in\mathbb{N}_{0}.\]

## 3. Approximation of Polynomials

The following section is dedicated to proving Theorem 1.1. We are going to show that it is possible to approximate complex polynomials in \(z\) and \(\overline{z}\) arbitrarily well on \(\Omega_{n}\) using shallow complex-valued neural networks.

The following theorem can be seen as a generalization of the classical mean-value theorem. Its proof is deferred to Theorem A.5 in the appendix.

**Theorem 3.1**.: _Let \(f:\mathbb{R}^{s}\to\mathbb{R}\) and \(k\in\mathbb{N}_{0},r>0\), such that \(f\big{|}_{(-r,r)^{s}}\in C^{k}\left((-r,r)^{s};\mathbb{R}\right)\). For \(\boldsymbol{p}\in\mathbb{N}_{0}^{s}\) with \(|\boldsymbol{p}|\leq k\) and \(h>0\) let_

\[f_{\boldsymbol{p},h}:=(2h)^{-|\boldsymbol{p}|}\sum_{0\leq r\leq\boldsymbol{p}} (-1)^{|\boldsymbol{p}|-|\boldsymbol{r}|}\binom{\boldsymbol{p}}{\boldsymbol{r} }f\left(h(2\boldsymbol{r}-\boldsymbol{p})\right).\]

_Let \(m:=\max\limits_{j}\ \boldsymbol{p}_{j}\). Then for \(0<h<\frac{r}{\max\{1,m\}}\) there exists \(\xi\in h[-m,m]^{s}\) satisfying_

\[f_{\boldsymbol{p},h}=\partial^{\boldsymbol{p}}f(\xi).\]

The following theorem describes how one can extract monomials of the form \(z^{\boldsymbol{m}}\overline{z}^{\boldsymbol{\ell}}\) by calculating

\[\partial_{\mathrm{wirt}}^{\boldsymbol{m}}\overline{\partial}_{\mathrm{wirt}}^{ \boldsymbol{\ell}}\left(w\mapsto\phi\left(w^{T}z+b\right)\right)\]

for a \(C^{k}\)-function \(\phi:\mathbb{C}\to\mathbb{C}\).

**Theorem 3.2**.: _Let \(\phi:\mathbb{C}\to\mathbb{C}\) and \(\delta>0,\ b\in\mathbb{C},\ k\in\mathbb{N}_{0}\), such that \(\phi\big{|}_{B_{\delta}(b)}\in C^{k}\left(B_{\delta}(b);\mathbb{C}\right)\). For fixed \(z\in\Omega_{n}\), where we recall that \(\Omega_{n}=[-1,1]^{n}+i[-1,1]^{n}\), we consider the map_

\[\phi_{z}:\quad B_{\frac{\delta}{\sqrt{2n}}}(0)\to\mathbb{C},\quad w\mapsto \phi\left(w^{T}z+b\right),\]

_which is in \(C^{k}\) since for \(w\in B_{\frac{\delta}{\sqrt{2n}}}(0)\subseteq\mathbb{C}\) we have_

\[\big{|}w^{T}z\big{|}\leq\|w\|_{2}\cdot\|z\|_{2}<\frac{\delta}{\sqrt{2n}}\cdot \sqrt{2n}=\delta.\]

_For all multi-indices \(\boldsymbol{m},\boldsymbol{\ell}\in\mathbb{N}_{0}^{n}\) with \(|\boldsymbol{m}+\boldsymbol{\ell}|\leq k\) we have_

\[\partial_{\mathrm{wirt}}^{\boldsymbol{m}}\overline{\partial}_{\mathrm{wirt}}^ {\boldsymbol{\ell}}\phi_{z}(w)=z^{\boldsymbol{m}}\overline{z}^{\boldsymbol {\ell}}\cdot\left(\partial_{\mathrm{wirt}}^{|\boldsymbol{m}|}\overline{ \partial}_{\mathrm{wirt}}^{|\boldsymbol{\ell}|}\phi\right)\left(w^{T}z+b\right)\]

_for all \(w\in B_{\frac{\delta}{\sqrt{2n}}}(0)\)._

Proof.: First we prove the statement

\[\overline{\partial}_{\mathrm{wirt}}^{\boldsymbol{\ell}}\phi_{z}(w)=\overline{ z}^{\boldsymbol{\ell}}\cdot(\overline{\partial}_{\mathrm{wirt}}^{| \boldsymbol{\ell}|}\phi)\left(w^{T}z+b\right)\quad\text{for all}\ w\in B_{ \frac{\delta}{\sqrt{2n}}}(0) \tag{3.1}\]

by induction over \(0\leq|\boldsymbol{\ell}|\leq k\). The case \(\boldsymbol{\ell}=0\) is trivial. Assuming that (3.1) holds for fixed \(\boldsymbol{\ell}\in\mathbb{N}_{0}^{n}\) with \(|\boldsymbol{\ell}|<k\), we want to show

\[\overline{\partial}_{\mathrm{wirt}}^{\ell+e_{j}}\phi_{z}(w)=\overline{z}^{ \boldsymbol{\ell}+e_{j}}\cdot\left(\partial_{\mathrm{wirt}}^{|\boldsymbol{ \ell}|+1}\phi\right)\left(w^{T}z+b\right) \tag{3.2}\]
for all \(w\in B_{\frac{\delta}{\sqrt{2n}}}(0)\), where \(j\in\{1,...,n\}\) is chosen arbitrarily. Firstly we get

\[\overline{\partial}_{\text{wirt}}^{\mathbf{\ell}+e_{j}}\phi_{z}(w) =\overline{\partial}_{\text{wirt}}^{e_{j}}\overline{\partial}_{ \text{wirt}}^{\mathbf{\ell}}\phi_{z}(w)\overset{\text{induction}}{=}\overline{ \partial}_{\text{wirt}}^{e_{j}}\left[w\mapsto\overline{z}^{\mathbf{\ell}}\cdot \left(\partial_{\text{wirt}}^{|\mathbf{\ell}|}\phi\right)\left(w^{T}z+b\right)\right]\] \[=\overline{z}^{\mathbf{\ell}}\overline{\partial}_{\text{wirt}}^{e_{j }}\left[w\mapsto\left(\overline{\partial}_{\text{wirt}}^{|\mathbf{\ell}|}\phi \right)\left(w^{T}z+b\right)\right].\]

Applying the chain rule for Wirtinger derivatives and using

\[\overline{\partial}_{\text{wirt}}^{e_{j}}\left[w\mapsto w^{T}z+b\right]=0\]

we see

\[\overline{\partial}_{\text{wirt}}^{e_{j}}\left[w\mapsto\left( \overline{\partial}_{\text{wirt}}^{|\mathbf{\ell}|}\phi\right)\left(w^{T}z+b \right)\right] =\left(\partial_{\text{wirt}}\overline{\partial}_{\text{wirt}}^{| \mathbf{\ell}|}\phi\right)\left(w^{T}z+b\right)\cdot\overline{\partial}_{\text{ wirt}}^{e_{j}}\left[w\mapsto w^{T}z+b\right]\] \[\quad+\left(\overline{\partial}_{\text{wirt}}^{|\mathbf{\ell}|+1} \phi\right)\left(w^{T}z+b\right)\cdot\overline{\partial}_{\text{wirt}}^{e_{j }}\left[w\mapsto\overline{w^{T}z+b}\right]\] \[=\overline{z}^{e_{j}}\cdot\left(\overline{\partial}_{\text{wirt}} ^{|\mathbf{\ell}|+1}\phi\right)\left(w^{T}z+b\right),\]

using the fact that \(w_{j}\mapsto w^{T}z+b\) is holomorphic and hence

\[\overline{\partial}_{\text{wirt}}^{e_{j}}\left[w\mapsto w^{T}z+b\right]=0\quad \text{and}\quad\partial_{\text{wirt}}^{e_{j}}\left[w\mapsto w^{T}z+b\right]=z _{j}.\]

Thus, we have proven (3.2) and induction yields (3.1).

It remains to show the full claim. We use induction over \(|\mathbf{m}|\) and note that the case \(\mathbf{m}=0\) has just been shown. We assume that the claim holds true for fixed \(\mathbf{m}\in\mathbb{N}_{0}^{n}\) with \(|\mathbf{m}+\mathbf{\ell}|<k\) and choose \(j\in\{1,...,n\}\). Thus, we get

\[\partial_{\text{wirt}}^{\mathbf{m}+e_{j}}\overline{\partial}_{ \text{wirt}}^{\mathbf{\ell}}\phi_{z}(w) =\partial_{\text{wirt}}^{e_{j}}\partial_{\text{wirt}}^{\mathbf{\mathrm{ m}}}\overline{\partial}_{\text{wirt}}^{\mathbf{\ell}}\phi_{z}(w)\overset{\text{IH}}{=} \partial_{\text{wirt}}^{e_{j}}\left(w\mapsto z^{\mathbf{m}}\overline{z}^{\mathbf{ \ell}}\cdot\left(\partial_{\text{wirt}}^{|\mathbf{\mathrm{m}}|}\overline{\partial}_ {\text{wirt}}^{|\mathbf{\ell}|}\phi\right)\left(w^{T}z+b\right)\right)\] \[=z^{\mathbf{m}}\overline{z}^{\mathbf{\ell}}\cdot\partial_{\text{wirt }}^{e_{j}}\left[w\mapsto\left(\partial_{\text{wirt}}^{|\mathbf{\mathrm{m}}|} \overline{\partial}_{\text{wirt}}^{|\mathbf{\ell}|}\phi\right)\left(w^{T}z+b\right) \right].\]

Using the chain rule again, we calculate

\[\partial_{\text{wirt}}^{e_{j}}\left[w\mapsto\left(\partial_{ \text{wirt}}^{|\mathbf{m}|}\overline{\partial}_{\text{wirt}}^{|\mathbf{\ell}|} \phi\right)\left(w^{T}z+b\right)\right]\] \[=\left(\partial_{\text{wirt}}^{|\mathbf{m}|+1}\overline{\partial}_ {\text{wirt}}^{|\mathbf{\ell}|}\phi\right)\left(w^{T}z+b\right)\cdot\partial_{ \text{wirt}}^{e_{j}}\left[w\mapsto w^{T}z+b\right]\] \[\quad+\left(\partial_{\text{wirt}}^{|\mathbf{m}|}\overline{ \partial}_{\text{wirt}}^{|\mathbf{\ell}|+1}\phi\right)\left(w^{T}z+b\right)\cdot \partial_{\text{wirt}}^{e_{j}}\left[w\mapsto\overline{w^{T}z+b}\right]\] \[=z^{e_{j}}\cdot\left(\partial_{\text{wirt}}^{|\mathbf{m}|+1} \overline{\partial}_{\text{wirt}}^{|\mathbf{\ell}|}\phi\right)\left(w^{T}z+b \right)\cdot\overline{\partial}_{\text{wirt}}^{e_{j}}\left[w\mapsto w^{T}z+b\right]\] \[=z^{e_{j}}\cdot\left(\partial_{\text{wirt}}^{|\mathbf{m}|+1} \overline{\partial}_{\text{wirt}}^{|\mathbf{\ell}|}\phi\right)\left(w^{T}z+b \right).\]

By induction, this proves the claim. 

The preceding result shows that monomials can be represented by Wirtinger derivatives, which in turn are linear combinations of "usual" partial derivatives, see Proposition 1.6. We now show that these partial derivatives can be approximated by shallow neural networks.

**Lemma 3.3**.: _Let \(\phi:\mathbb{C}\to\mathbb{C}\) and \(\delta>0,\ b\in\mathbb{C},\ k\in\mathbb{N}_{0}\), such that \(\phi\big{|}_{B_{\delta}(b)}\in C^{k}\left(B_{\delta}(b);\mathbb{C}\right)\). Let \(m,n\in\mathbb{N}\) and \(\varepsilon>0\). For \(\mathbf{p}\in\mathbb{N}_{0}^{2n},h>0\) and \(z\in\Omega_{n}\) we write_

\[\Phi_{\mathbf{p},h}(z) :=(2h)^{-|\mathbf{p}|}\sum_{0\leq r\leq\mathbf{p}}(-1)^{|\mathbf{p}|-|r|}\binom{ \mathbf{p}}{\mathbf{r}}\left(\phi_{z}\circ\varphi_{n}\right)\left(h(2\mathbf{r}-\mathbf{p})\right)\] \[=(2h)^{-|\mathbf{p}|}\sum_{0\leq r\leq\mathbf{p}}(-1)^{|\mathbf{p}|-|r|}\binom{ \mathbf{p}}{\mathbf{r}}\phi\left(\left[\varphi_{n}\left(h(2\mathbf{r}-\mathbf{p})\right)\right]^ {T}\cdot z+b\right),\]
_where \(\phi_{z}\) is as introduced in Theorem 3.2 and \(\varphi_{n}\) is as in (1.2) on page 5. Furthermore, let_

\[\phi_{\boldsymbol{p}}:\quad\Omega_{n}\times B_{\frac{\delta}{\sqrt{2n}}}(0) \rightarrow\mathbb{C},\quad(z,w)\mapsto\partial^{\boldsymbol{p}}\phi_{z}(w).\]

_Then there exists \(h^{*}>0\) such that_

\[\|\Phi_{\boldsymbol{p},h}-\phi_{\boldsymbol{p}}(\cdot,0)\|_{L^{\infty}(\Omega_ {n};\mathbb{C})}\leq\varepsilon\]

_for all \(\boldsymbol{p}\in\mathbb{N}_{0}^{2n}\) with \(|\boldsymbol{p}|\leq k\) and \(\boldsymbol{p}_{j}\leq m\) for all \(j\in\{1,...,2n\}\) and \(h\in(0,h^{*})\)._

Proof.: Fix \(\boldsymbol{p}\in\mathbb{N}_{0}^{2n}\) with \(|\boldsymbol{p}|\leq k\) and \(\boldsymbol{p}_{j}\leq m\) for all \(j\in\{1,...,2n\}\). The map

\[B_{\sqrt{2n}}(0)\times B_{\delta/\sqrt{2n}}(0)\rightarrow\mathbb{C},\quad(z, w)\mapsto\phi\left(w^{T}z+b\right)\]

is in \(C^{k}\) since

\[\left|w^{T}z\right|\leq\|w\|\cdot\|z\|<\frac{\delta}{\sqrt{2n}}\cdot\sqrt{2n} =\delta.\]

Therefore, the map

\[B_{\sqrt{2n}}(0)\times B_{\delta/\sqrt{2n}}(0)\rightarrow\mathbb{C},\quad(z, w)\mapsto\partial^{\boldsymbol{p}}\phi_{z}(w)\]

is continuous and in particular uniformly continuous on the compact set

\[\Omega_{n}\times\overline{B_{\delta/2n}}(0)\subseteq B_{\sqrt{2n}}(0)\times B _{\delta/\sqrt{2n}}(0).\]

Hence, there exists \(h_{\boldsymbol{p}}\in(0,\frac{\delta}{2mn})\), such that

\[\left|\phi_{\boldsymbol{p}}(z,\xi)-\phi_{\boldsymbol{p}}(z,0)\right|\leq\frac {\varepsilon}{\sqrt{2}}\]

for all \(\xi\in\varphi_{n}\left(h\cdot[-m,m]^{2n}\right),\ h\in(0,h_{\boldsymbol{p}})\) and \(z\in\Omega_{n}\). Now fix such an \(h\in(0,h_{\boldsymbol{p}})\) and \(z\in\Omega_{n}\). Applying Theorem 3.1 to both components of \(\left(\varphi_{1}^{-1}\circ\Phi_{\boldsymbol{p},h}\right)(z)\) and \(\varphi_{1}^{-1}\circ\phi_{z}\circ\varphi_{n}|_{\left(-\frac{\delta}{2n},\frac {\delta}{2n}\right)^{2n}}\) separately yields the existence of two real vectors \(\xi_{1},\ \xi_{2}\in h\cdot[-m,m]^{2n}\), such that

\[\left(\varphi_{1}^{-1}\circ\Phi_{\boldsymbol{p},h}(z)\right)_{1}=\left[ \partial^{\boldsymbol{p}}\left(\varphi_{1}^{-1}\circ\phi_{z}\circ\varphi_{n} \right)(\xi_{1})\right]_{1},\ \left(\varphi_{1}^{-1}\circ\Phi_{\boldsymbol{p},h}(z)\right)_{2}=\left[ \partial^{\boldsymbol{p}}\left(\varphi_{1}^{-1}\circ\phi_{z}\circ\varphi_{n} \right)(\xi_{2})\right]_{2}.\]

Rewriting this yields

\[\operatorname{Re}\left(\Phi_{\boldsymbol{p},h}(z)\right)=\operatorname{Re} \left(\phi_{\boldsymbol{p}}(z,\varphi_{n}\left(\xi_{1}\right))\right)\quad \text{and}\quad\operatorname{Im}\left(\Phi_{\boldsymbol{p},h}(z)\right)= \operatorname{Im}\left(\phi_{\boldsymbol{p}}(z,\varphi_{n}\left(\xi_{2} \right))\right).\]

Using this property we deduce

\[\left|\operatorname{Re}\left(\Phi_{\boldsymbol{p},h}(z)-\phi_{\boldsymbol{p}} (z,0)\right)\right|=\left|\operatorname{Re}\left(\phi_{\boldsymbol{p}}(z, \varphi_{n}\left(\xi_{1}\right))-\phi_{\boldsymbol{p}}(z,0)\right)\right|\leq \left|\phi_{\boldsymbol{p}}(z,\varphi_{n}\left(\xi_{1}\right))-\phi_{ \boldsymbol{p}}(z,0)\right|\leq\frac{\varepsilon}{\sqrt{2}}\]

and analogously for the imaginary part. Since \(z\in\Omega_{n}\) and \(h\in(0,h_{\boldsymbol{p}})\) have been chosen arbitrarily we get the claim by choosing

\[h^{*}:=\min\left\{h_{\boldsymbol{p}}:\ \boldsymbol{p}\in\mathbb{N}_{0}^{2n}\text{ with }| \boldsymbol{p}|\leq k\text{ and }\max_{j}\ \boldsymbol{p}_{j}\leq m\right\}.\qed\]

**Definition 3.4**.: For \(m\in\mathbb{N}_{0}\) we define

\[\mathcal{P}_{m}:=\left\{\mathbb{C}^{n}\rightarrow\mathbb{C},\ z\mapsto\sum_{ \boldsymbol{m}\leq m}\sum_{\boldsymbol{\ell}\leq m}a_{\boldsymbol{m},\boldsymbol {\ell}}z^{\boldsymbol{m}}\overline{z}^{\boldsymbol{\ell}}:\ a_{\boldsymbol{m}, \boldsymbol{\ell}}\in\mathbb{C}\right\}\]

as the space of functions from \(\mathbb{C}^{n}\) to \(\mathbb{C}\) that are represented by a complex polynomial in \(z\) and \(\overline{z}\) of coordinatewise degree at most \(m\).

We can now state and prove the first major result of this paper, which already appeared in a slightly simplified form in the introduction as Theorem 1.1. Specifically, we are going to show that it is possible to approximate complex polynomials in \(z\) and \(\overline{z}\) of coordinatewise degree of at most \(m\) using shallow complex-valued neural networks with \((4m+1)^{2n}\) neurons in the hidden layer of the network, provided that the activation function is \(mn\)-admissible.
**Theorem 3.5**.: _Let \(m,n\in\mathbb{N}\), \(\varepsilon>0\) and let \(\phi:\mathbb{C}\to\mathbb{C}\) be \(mn\)-admissible in \(b\in\mathbb{C}\). Let \(\mathcal{P}^{\prime}\subseteq\mathcal{P}_{m}\) be bounded and set \(N:=(4m+1)^{2n}\). Then there exist certain coefficients \(\rho_{1},...,\rho_{N}\in\mathbb{C}^{n}\) with the following property: For each polynomial \(p\in\mathcal{P}^{\prime}\) there are coefficients \(\sigma_{1},...,\sigma_{N}\in\mathbb{C}\), such that_

\[\left\|p-f\right\|_{L^{\infty}(\Omega_{n};\mathbb{C})}\leq\varepsilon,\]

_where_

\[f:\quad\Omega_{n}\to\mathbb{C},\quad z\mapsto\sum_{j=1}^{N}\sigma_{j}\phi\left( \rho_{j}^{T}z+b\right). \tag{3.3}\]

Proof.: Let \(p\in\mathcal{P}^{\prime}\). Fix \(\mathbf{m},\boldsymbol{\ell}\in\mathbb{N}_{0}^{n}\) with \(\mathbf{m},\boldsymbol{\ell}\leq m\). For each \(z\in\Omega_{n}\), using Theorem 3.2, we then have

\[z^{\mathbf{m}}\overline{z}^{\boldsymbol{\ell}}=\left[\left(\partial_{\text{ wirt}}^{|\mathbf{m}|}\overline{\partial}_{\text{wirt}}^{|\boldsymbol{\ell}|} \phi\right)(b)\right]^{-1}\partial_{\text{wirt}}^{\boldsymbol{\ell}}\overline {\partial}_{\text{wirt}}^{\boldsymbol{\ell}}\phi_{z}(0)\overset{1.6}{=}\left[ \left(\partial_{\text{wirt}}^{|\mathbf{m}|}\overline{\partial}_{\text{wirt}}^ {|\boldsymbol{\ell}|}\phi\right)(b)\right]^{-1}\cdot\sum_{\begin{subarray}{c }\mathbf{p}^{\prime},\mathbf{p}^{\prime\prime}\in\mathbb{N}_{0}^{n}\\ \mathbf{p}^{\prime}+\mathbf{p}^{\prime\prime}=\mathbf{m}+\boldsymbol{\ell} \end{subarray}}b_{\mathbf{p}^{\prime},\mathbf{p}^{\prime\prime}}\partial^{( \mathbf{p}^{\prime},\mathbf{p}^{\prime\prime})}\phi_{z}(0)\]

with suitably chosen complex coefficients \(b_{\mathbf{p}^{\prime},\mathbf{p}^{\prime\prime}}\in\mathbb{C}\) depending only on \(\mathbf{p}^{\prime},\ \mathbf{p}^{\prime\prime},\ \mathbf{m}\) and \(\boldsymbol{\ell}\). Here we used that \(|\mathbf{m}|,\ |\boldsymbol{\ell}|\leq mn\) and that \(\phi\) is \(mn\)-admissible in \(b\). Since \(\mathcal{P}^{\prime}\) is bounded and \(p\in\mathcal{P}^{\prime}\), we can write

\[p(z)=\sum_{\begin{subarray}{c}\mathbf{m},\boldsymbol{\ell}\in\mathbb{N}_{0}^{ n}\\ \mathbf{m},\boldsymbol{\ell}\leq m\end{subarray}}a_{\mathbf{m},\boldsymbol{ \ell}^{\geq}\mathbf{m}}\overline{z}^{\boldsymbol{\ell}}\]

with \(|a_{\mathbf{m},\boldsymbol{\ell}}|\leq c\) for some constant \(c=c(\mathcal{P}^{\prime})>0\). This easily implies that we can rewrite \(p\) as

\[p(z)=\sum_{\mathbf{p}\leq 2m}c_{\mathbf{p}}(p)\partial^{\mathbf{p}}\phi_{z}(0)\]

with coefficients \(c_{\mathbf{p}}(p)\in\mathbb{C}\) satisfying \(|c_{\mathbf{p}}(p)|\leq c^{\prime}\) for some constant \(c^{\prime}=c^{\prime}(\phi,b,\mathcal{P}^{\prime},m,n)\). By Lemma 3.3, we choose \(h^{*}>0\), such that

\[|\Phi_{\mathbf{p},h^{*}}(z)-\partial^{\mathbf{p}}\phi_{z}(0)|\leq\frac{ \varepsilon}{\sum_{\mathbf{p}\leq 2m}c^{\prime}}\]

for all \(z\in\Omega_{n}\) and \(\mathbf{p}\in\mathbb{N}_{0}^{2n}\) with \(\mathbf{p}\leq 2m\) (note that we then automatically have \(|\mathbf{p}|\leq 2mn\) and that \(\phi\big{|}_{B_{\delta}(b)}\in C^{2mn}(B_{\delta}(b);\mathbb{C})\) for some \(\delta>0\)). Furthermore, we can rewrite each function \(\Phi_{\mathbf{p},h^{*}}\) as

\[\Phi_{\mathbf{p},h^{*}}(z)=\sum_{\begin{subarray}{c}\boldsymbol{\alpha}\in \mathbb{Z}^{2n}\\ |\boldsymbol{\alpha}|_{j}\leq 2m\ \forall j\end{subarray}}\lambda_{\boldsymbol{ \alpha},\mathbf{p}}\phi(\left[\varphi_{n}\left(h^{*}\boldsymbol{\alpha} \right)\right]^{T}\cdot z+b)\]

with suitable coefficients \(\lambda_{\boldsymbol{\alpha},\mathbf{p}}\in\mathbb{C}\). Since the cardinality of the set

\[\left\{\boldsymbol{\alpha}\in\mathbb{Z}^{2n}:\ |\boldsymbol{\alpha}_{j}|\leq 2m\ \forall j\right\}\]

is \((4m+1)^{2n}\), this can be converted to

\[\Phi_{\mathbf{p},h^{*}}(z)=\sum_{j=1}^{N}\lambda_{j,\mathbf{p}}\phi\left(\rho_ {j}^{T}\cdot z+b\right).\]

If we then again take any arbitrary polynomial \(p\in\mathcal{P}^{\prime}\), we define

\[\theta(z):=\sum_{\mathbf{p}\leq 2m}c_{\mathbf{p}}(p)\cdot\Phi_{\mathbf{p},h^{*}}(z)\]

and note

\[|\theta(z)-p(z)|\leq\sum_{\mathbf{p}\leq 2m}|c_{\mathbf{p}}(p)|\cdot|\Phi_{ \mathbf{p},h^{*}}(z)-\partial^{\mathbf{p}}\phi_{z}(0)|\leq\varepsilon.\]

Since the coefficients \(\rho_{j}\) have been chosen independently from the polynomial \(p\), we can rewrite \(\theta\) so it has the form (3.3).
## 4. Approximation of differentiable functions

For any natural number \(\ell\in\mathbb{N}_{0}\), we denote by \(T_{\ell}\) the \(\ell\)-th Chebyshev polynomial, satisfying

\[T_{\ell}\left(\cos(x)\right)=\cos(\ell x),\quad x\in\mathbb{R}.\]

For a multi-index \(\mathbf{k}\in\mathbb{N}_{0}^{s}\) we define

\[T_{\mathbf{k}}(x):=\prod_{j=1}^{s}T_{\mathbf{k}_{j}}\left(x_{j}\right),\quad x \in[-1,1]^{s}.\]

We then get the following approximation result.

**Theorem 4.1**.: _Let \(s,m\in\mathbb{N}\), \(k\in\mathbb{N}_{0}\). Then there is a constant \(c=c(s,k)>0\) with the following property: For any \(f\in C^{k}\left([-1,1]^{s};\mathbb{R}\right)\) the polynomial \(P\) defined as_

\[P(x):=\sum_{0\leq\mathbf{k}\leq 2m-1}\mathcal{V}_{\mathbf{k}}^{m}(f)\cdot T_{ \mathbf{k}}(x)\]

_satisfies_

\[\left\|f-P\right\|_{L^{\infty}\left([-1,1]^{s}\right)}\leq\frac{c}{m^{k}}\cdot \left\|f\right\|_{C^{k}\left([-1,1]^{s};\mathbb{R}\right)}.\]

_Here, the maps_

\[C\left([-1,1]^{s};\mathbb{R}\right)\rightarrow\mathbb{R},\quad f\mapsto \mathcal{V}_{\mathbf{k}}^{m}(f)\]

_are continuous and linear functionals with respect to the \(L^{\infty}\)-norm. Furthermore, there is a constant \(\widetilde{c}=\widetilde{c}(s)>0\), such that the inequality_

\[\sum_{0\leq\mathbf{k}\leq 2m-1}\left|\mathcal{V}_{\mathbf{k}}^{m}(f)\right| \leq\widetilde{c}\cdot m^{s/2}\cdot\left\|f\right\|_{L^{\infty}\left([-1,1]^{ s}\right)}\]

_holds for any \(f\in C\left([-1,1]^{s};\mathbb{R}\right)\)._

The proof of Theorem 4.1 is deferred to the appendix (Theorem B.15). We can now formulate the final result for the approximation of a function \(f\in C^{k}\left(\Omega_{n};\mathbb{C}\right)\), a simplified version of which appeared in the introduction as Theorem 1.2.

**Theorem 4.2**.: _Let \(n,k\in\mathbb{N}\). Then there is a constant \(c=c(n,k)>0\) with the following property: For any admissible activation function \(\phi:\mathbb{C}\rightarrow\mathbb{C}\) and for any \(m\in\mathbb{N}\) there are complex vectors \(\rho_{1},...,\rho_{m}\in\mathbb{C}^{n}\) and \(b_{m}\in\mathbb{C}\) with the following property: For any \(f\in C^{k}\left(\Omega_{n};\mathbb{C}\right)\) there are complex numbers \(\sigma_{1}=\sigma_{1}(f),...,\sigma_{m}=\sigma_{m}(f)\in\mathbb{C}\), such that_

\[\left\|f-g\right\|_{L^{\infty}\left(\Omega_{n}\right)}\leq c\cdot m^{-k/(2n)} \cdot\left\|f\right\|_{C^{k}\left(\Omega_{n};\mathbb{C}\right)},\]

_where \(g\) is defined as_

\[g:\quad\Omega_{n}\rightarrow\mathbb{C},\quad g(z):=\sum_{j=1}^{m}\sigma_{j} \phi\left(\rho_{j}^{T}z+b_{m}\right).\]

_Furthermore, the map \(f\mapsto\sigma_{j}(f)\) is a continuous linear functional with respect to the \(L^{\infty}\)-norm for every \(j\in\{1,...,m\}\)._

Proof.: Choose \(M\in\mathbb{N}\) as the largest integer, such that \((16M-7)^{2n}\leq m\), where we assume without loss of generality that \(9^{2n}\leq m\), which can be done by choosing \(\sigma_{j}=0\) for all \(j\in\{1,...,m\}\) for \(m<9^{2n}\), at the cost of possibly enlarging \(c\). First we note that by the choice of \(M\) the inequality

\[m\leq(16M+9)^{2n}\]

holds true. Since \(16M+9\leq 25M\), we get \(m\leq 25^{2n}\cdot M^{2n}\) or equivalently

\[\frac{m^{1/2n}}{25}\leq M. \tag{4.1}\]
According to Theorem 4.1 we choose a constant \(c_{1}=c_{1}(n,k)\) with the property that for any function \(f\in C^{k}\left([-1,1]^{2n};\mathbb{R}\right)\) there exists a polynomial

\[P=\sum_{0\leq\mathbf{k}\leq 2M-1}\mathcal{V}_{\mathbf{k}}^{M}(f)\cdot T_{ \mathbf{k}}\]

of coordinatewise degree at most \(2M-1\) satisfying

\[\left\|f-P\right\|_{L^{\infty}([-1,1]^{2n})}\leq\frac{c_{1}}{M^{k}}\cdot\left\| f\right\|_{C^{k}([-1,1]^{2n};\mathbb{R})}.\]

Furthermore, according to Theorem 4.1, we choose a constant \(c_{2}=c_{2}(n)\), such that the inequality

\[\sum_{0\leq\mathbf{k}\leq 2M-1}\left|\mathcal{V}_{\mathbf{k}}^{M}(f)\right|\leq c _{2}\cdot M^{n}\left\|f\right\|_{C^{k}([-1,1]^{2n};\mathbb{R})}\]

holds for all \(f\in C^{k}\left([-1,1]^{2n}\right)\). The final constant is then defined to be

\[c=c(n,k):=\sqrt{2}\cdot 25^{k}\cdot(c_{1}+c_{2})\,.\]

Since \(\phi\) is admissible, there exists \(b_{m}\in\mathbb{C}\) such that \(\phi\) is \((4M-2)n\)-admissible in \(b_{m}\). Fix \(\mathbf{k}\leq 2M-1\). Since \(T_{\mathbf{k}}\) is a polynomial of componentwise degree less or equal to \(2M-1\), we have a representation

\[\left(T_{\mathbf{k}}\circ\varphi_{n}^{-1}\right)(z)=\sum_{\begin{subarray}{c }\boldsymbol{\ell}^{1},\boldsymbol{\ell}^{2}\in\mathbb{N}_{0}^{\infty}\\ \boldsymbol{\ell}^{1},\boldsymbol{\ell}^{2}\leq 2M-1\end{subarray}}a_{ \boldsymbol{\ell}^{1},\boldsymbol{\ell}^{2}}^{\mathbf{k}}\prod_{t=1}^{n} \operatorname{Re}\left(z_{t}\right)^{\boldsymbol{\ell}^{1}_{t}}\operatorname{ Im}\left(z_{t}\right)^{\boldsymbol{\ell}^{2}_{t}}\]

with suitably chosen coefficients \(a_{\boldsymbol{\ell}^{1},\boldsymbol{\ell}^{2}}^{\mathbf{k}}\in\mathbb{C}\). By using the identities \(\operatorname{Re}\left(z_{t}\right)=\frac{1}{2}\left(z_{t}+\overline{z_{t}}\right)\) and also \(\operatorname{Im}\left(z_{t}\right)=\frac{1}{2i}\left(z_{t}-\overline{z_{t}}\right)\) we can rewrite \(T_{\mathbf{k}}\circ\varphi_{n}^{-1}\) into a complex polynomial in \(z\) and \(\overline{z}\), i.e.

\[\left(T_{\mathbf{k}}\circ\varphi_{n}^{-1}\right)(z)=\sum_{\begin{subarray}{c }\boldsymbol{\ell}^{1},\boldsymbol{\ell}^{2}\in\mathbb{N}_{0}^{\infty}\\ \boldsymbol{\ell}^{1},\boldsymbol{\ell}^{2}\leq 4M-2\end{subarray}}b_{ \boldsymbol{\ell}^{1},\boldsymbol{\ell}^{2}}^{\mathbf{k}}z^{\boldsymbol{\ell}^ {1}}\overline{z}^{\boldsymbol{\ell}^{2}}\]

with complex coefficients \(b_{\boldsymbol{\ell}^{1},\boldsymbol{\ell}^{2}}^{\mathbf{k}}\in\mathbb{C}\). Using Theorem 3.5, we choose \(\rho_{1},...,\rho_{m}\in\mathbb{C}^{n}\), such that for any polynomial \(P\in\left\{T_{\mathbf{k}}\circ\varphi_{n}^{-1}:\;\mathbf{k}\leq 2M-1\right\} \subseteq\mathcal{P}_{4M-2}\) there are coefficients \(\sigma_{1}(P),...,\sigma_{m}(P)\in\mathbb{C}\), such that

\[\left\|g_{P}-P\right\|_{L^{\infty}(\Omega_{n})}\leq M^{-k-n}, \tag{4.2}\]

where

\[g_{P}:=\sum_{t=1}^{m}\sigma_{h}\phi\left(\rho_{t}^{T}z+b\right).\]

Note that here we implicitly use the bound \((4\cdot(4M-2)+1)^{2n}\leq m\). We are now going to show that the chosen constant and the chosen vectors \(\rho_{t}\) have the desired property.

Let \(f\in C^{k}\left(\Omega_{n};\mathbb{C}\right)\). By splitting \(f\) into real and imaginary part, we write \(f=f_{1}+i\cdot f_{2}\) with \(f_{1},f_{2}\in C^{k}\left(\Omega_{n};\mathbb{R}\right)\). For the following, fix \(j\in\{1,2\}\) and note that \(f_{j}\circ\varphi_{n}\in C^{k}\left([-1,1]^{2n};\mathbb{R}\right)\). By choice of \(c_{1}\), there is a polynomial \(P\) with the property

\[\left\|f_{j}\circ\varphi_{n}-P\right\|_{L^{\infty}([-1,1]^{2n})}\leq\frac{c_{ 1}}{M^{k}}\cdot\left\|f_{j}\circ\varphi_{n}\right\|_{C^{k}([-1,1]^{2n}; \mathbb{R})}\]

or equivalently

\[\left\|f_{j}-P\circ\varphi_{n}^{-1}\right\|_{L^{\infty}(\Omega_{n})}\leq\frac{ c_{1}}{M^{k}}\cdot\left\|f_{j}\circ\varphi_{n}\right\|_{C^{k}([-1,1]^{2n}; \mathbb{R})}, \tag{4.3}\]

where \(P\circ\varphi_{n}^{-1}\) can be written in the form

\[\left(P\circ\varphi_{n}^{-1}\right)(z)=\sum_{0\leq\mathbf{k}\leq 2M-1}\mathcal{V}_{ \mathbf{k}}^{M}\left(f_{j}\circ\varphi_{n}\right)\cdot\left(T_{\mathbf{k}} \circ\varphi_{n}^{-1}\right)(z).\]

We choose the function \(g_{T_{\mathbf{k}}\circ\varphi_{n}^{-1}}\) according to (4.2). Thus, writing

\[g_{j}:=\sum_{0\leq\mathbf{k}\leq 2M-1}\mathcal{V}_{\mathbf{k}}^{M}\left(f_{j} \circ\varphi_{n}\right)\cdot g_{T_{\mathbf{k}}\circ\varphi_{n}^{-1}},\]
we obtain

\[\left\|P\circ\varphi_{n}^{-1}-g_{j}\right\|_{L^{\infty}(\Omega_{n})} \leq\sum_{0\leq\mathbf{k}\leq 2M-1}\left|\mathcal{V}_{\mathbf{k}}^{M} \left(f_{j}\circ\varphi_{n}\right)\right|\cdot\underbrace{\left\|T_{\mathbf{k}} \circ\varphi_{n}^{-1}-g_{T_{\mathbf{k}}\circ\varphi_{n}^{-1}}\right\|_{L^{ \infty}(\Omega_{n})}}_{\leq M^{-k-n}}\] \[\leq M^{-k-n}\cdot\sum_{0\leq\mathbf{k}\leq 2M-1}\left|\mathcal{V}_ {\mathbf{k}}^{M}\left(f_{j}\circ\varphi_{n}\right)\right|\] \[\leq\frac{c_{2}}{M^{k}}\left\|f_{j}\circ\varphi_{n}\right\|_{C^{ k}([-1,1]^{2n};\mathbb{R})}\leq\frac{c_{2}}{M^{k}}\left\|f_{j}\right\|_{C^{k}( \Omega_{n};\mathbb{C})}. \tag{4.4}\]

Combining (4.3) and (4.4) we see

\[\left\|f_{j}-g_{j}\right\|_{L^{\infty}(\Omega_{n})}\leq\frac{c_{1}+c_{2}}{M^ {k}}\cdot\left\|f_{j}\right\|_{C^{k}(\Omega_{n};\mathbb{R})}\leq\frac{c_{1}+c_ {2}}{M^{k}}\cdot\left\|f\right\|_{C^{k}(\Omega_{n};\mathbb{C})}.\]

In the end, define

\[g:=g_{1}+i\cdot g_{2}.\]

Since the vectors \(\rho_{t}\) have been chosen fixed, it is clear that, after rearranging, \(g\) has the desired form. Furthermore, one obtains the bound

\[\left\|f-g\right\|_{L^{\infty}(\Omega_{n})} \leq\sqrt{\left\|f_{1}-g_{1}\right\|_{L^{\infty}(\Omega_{n})}^{2} +\left\|f_{2}-g_{2}\right\|_{L^{\infty}(\Omega_{n})}^{2}}\] \[\leq\frac{c_{1}+c_{2}}{M^{k}}\cdot\sqrt{\left\|f\right\|_{C^{k}( \Omega_{n};\mathbb{C})}^{2}+\left\|f\right\|_{C^{k}(\Omega_{n};\mathbb{C})}^{ 2}}\] \[\leq\frac{\sqrt{2}\cdot(c_{1}+c_{2})}{M^{k}}\cdot\ \left\|f\right\|_{C^{k}( \Omega_{n};\mathbb{C})}.\]

Using (4.1), we see

\[\left\|f-g\right\|_{L^{\infty}(\Omega_{n};\mathbb{C})}\leq\frac{\sqrt{2}\cdot 2 5^{k}\cdot(c_{1}+c_{2})}{m^{k/2n}}\cdot\ \left\|f\right\|_{C^{k}(\Omega_{n};\mathbb{C})},\]

as desired.

The linearity and continuity of the maps \(f\mapsto\sigma_{j}(f)\) follow directly from the fact that the map \(f\mapsto\mathcal{V}_{\mathbf{k}}^{M}(f)\) is a continuous linear functional for every \(0\leq\mathbf{k}\leq 2M-1\). 

## 5. Concrete examples of admissible functions

In this section we want to show the admissibility of concrete activation functions that are commonly used when applying complex-valued neural networks to machine learning. Proposition 5.1 introduces a large class of admissible functions.

**Proposition 5.1**.: _Let \(\rho\in C^{\infty}(\mathbb{R};\mathbb{R})\) be non-polynomial and let \(\psi:\mathbb{C}\to\mathbb{C}\) be defined as_

\[\psi(z):=\rho(\operatorname{Re}(z))\quad\text{resp.}\quad\psi(z):=\rho( \operatorname{Im}(z)).\]

_Then \(\psi\) is admissible._

Proof.: Since \(\psi\) depends only on the real resp. imaginary part of the input, we see directly from the definition of the Wirtinger derivatives that

\[\partial_{\mathrm{wirt}}\psi(z)=\overline{\partial}_{\mathrm{wirt}}\psi(z)= \frac{1}{2}\rho^{\prime}(\operatorname{Re}(z))\quad\text{resp.}\quad\partial_ {\mathrm{wirt}}\psi(z)=-\overline{\partial}_{\mathrm{wirt}}\psi(z)=-\frac{i}{ 2}\rho^{\prime}\operatorname{Im}(z).\]

Hence we see for arbitrary \(m,\ell\in\mathbb{N}_{0}\) that

\[\left|\partial_{\mathrm{wirt}}^{m}\overline{\partial}_{\mathrm{wirt}}^{\ell} \psi(z)\right|=\frac{1}{2^{m+\ell}}\left|\rho^{(m+\ell)}(\operatorname{Re}(z) )\right|\quad\text{resp.}\quad\left|\partial_{\mathrm{wirt}}^{m}\overline{ \partial}_{\mathrm{wirt}}^{\ell}\psi(z)\right|=\frac{1}{2^{m+\ell}}\left|\rho^ {(m+\ell)}(\operatorname{Im}(z))\right|.\]

Since \(\rho\) is non-polynomial we can choose a real number \(x\), such that all derivatives of \(\rho\) in \(x\) do not vanish (cf. for instance [12, p. 53]). Thus, all the Wirtinger derivatives \(\partial_{\mathrm{wirt}}^{m}\overline{\partial}_{\mathrm{wirt}}^{\ell}\psi\) do not vanish for all complex numbers with real, resp. imaginary part \(x\). 

In the following, we consider a special activation function, which has been proposed in [4]. Its representative power has already been discussed to some extent in [8].
**Definition 5.2**.: For \(b\in(-\infty,0)\) we define

\[\sigma_{\mathrm{modReLU},b}:\quad\mathbb{C}\to\mathbb{C},\quad\sigma_{\mathrm{ modReLU},b}(z):=\begin{cases}(|z|+b)\frac{z}{|z|},&|z|+b\geq 0,\\ \qquad 0,&\mathrm{otherwise}.\end{cases}\]

**Theorem 5.3**.: _Let \(b\in(-\infty,0)\). Writing \(\sigma=\sigma_{\mathrm{modReLU},b}\) one has for every \(z\in\mathbb{C}\) with \(|z|+b>0\) the identity_

\[\left(\partial_{\mathrm{wirt}}^{m}\overline{\partial}_{\mathrm{wirt}}^{\ell} \sigma\right)(z)=\begin{cases}z+b\frac{z}{|z|},&m=\ell=0,\\ 1+\frac{b}{2}\cdot\frac{1}{|z|},&m=1,\ell=0,\\ b\cdot q_{m,\ell}\cdot\frac{\frac{\ell-m+1}{|z|^{2m+1}}}{|z|^{2m+1}},&m\leq\ell +1,\ell\geq 1,\\ b\cdot q_{m,\ell}\cdot\frac{\overline{z}-m-\ell-1}{|z|^{2m-1}},&m\geq\ell+1,m \geq 2\end{cases}\]

_for every \(m\in\mathbb{N}_{0}\) and \(\ell\in\mathbb{N}_{0}\). Here, the numbers \(q_{m,\ell}\) are non-zero and rational. Furthermore, note that all cases for choices of \(m\) and \(\ell\) are covered by observing that we can either have the case \(m\geq\ell+1\) (where either \(m\geq 2\) or \(m=1,\ell=0\)) or \(m\leq\ell+1\) where the latter is again split into \(\ell=0\) and \(\ell\geq 1\)._

Proof.: We first calculate certain Wirtinger derivatives for \(z\neq 0\). First note

\[\overline{\partial}_{\mathrm{wirt}}\left(\frac{1}{|z|^{m}}\right) =\frac{1}{2}\left(\partial^{(1,0)}\left(\frac{1}{|z|^{m}}\right) +i\cdot\partial^{(0,1)}\left(\frac{1}{|z|^{m}}\right)\right)\] \[=\frac{1}{2}\left(\left(-\frac{m}{2}\right)\frac{2\operatorname{ Re}(z)+i\cdot 2\operatorname{Im}(z)}{|z|^{m+2}}\right)\] \[=-\frac{m}{2}\cdot\frac{z}{|z|^{m+2}}\]

and similarly

\[\partial_{\mathrm{wirt}}\left(\frac{1}{|z|^{m}}\right)=-\frac{m}{2}\cdot\frac {\overline{z}}{|z|^{m+2}}\]

for any \(m\in\mathbb{N}\). Using the product rule for Wirtinger derivatives, we see

\[\overline{\partial}_{\mathrm{wirt}}\left(\frac{z^{\ell}}{|z|^{m}}\right)= \underbrace{\overline{\partial}_{\mathrm{wirt}}\left(z^{\ell}\right)}_{=0} \cdot\frac{1}{|z|^{m}}+z^{\ell}\cdot\overline{\partial}_{\mathrm{wirt}} \left(\frac{1}{|z|^{m}}\right)=-\frac{m}{2}\cdot\frac{z^{\ell+1}}{|z|^{m+2}} \tag{5.1}\]

for any \(m\in\mathbb{N}\) and \(\ell\in\mathbb{N}_{0}\) and furthermore

\[\partial_{\mathrm{wirt}}\left(\frac{z^{\ell}}{|z|^{m}}\right) =\partial_{\mathrm{wirt}}\left(z^{\ell}\right)\cdot\frac{1}{|z|^ {m}}+z^{\ell}\cdot\partial_{\mathrm{wirt}}\left(\frac{1}{|z|^{m}}\right)\] \[=\ell\cdot z^{\ell-1}\cdot\frac{1}{|z|^{m}}-z^{\ell}\cdot\frac{m} {2}\cdot\frac{\overline{z}}{|z|^{m+2}}\] \[=\left(\ell-\frac{m}{2}\right)\cdot\frac{z^{\ell-1}}{|z|^{m}} \tag{5.2}\]

for \(m,\ell\in\mathbb{N}\), and finally

\[\partial_{\mathrm{wirt}}\left(\frac{\overline{z}^{\ell}}{|z|^{m}}\right)= \underbrace{\partial_{\mathrm{wirt}}\left(\overline{z}^{\ell}\right)}_{=0} \cdot\frac{1}{|z|^{m}}+\overline{z}^{\ell}\cdot\partial_{\mathrm{wirt}} \left(\frac{1}{|z|^{m}}\right)=-\frac{m}{2}\cdot\frac{\overline{z}^{\ell+1}} {|z|^{m+2}} \tag{5.3}\]

for \(m\in\mathbb{N}\) and \(\ell\in\mathbb{N}_{0}\).

Having proven those three identities, we can start with the actual computation. We first fix \(m=0\) and prove the claimed identity by induction over \(\ell\). The case \(\ell=0\) is just the definition of the function and furthermore, we calculate

\[\overline{\partial}_{\mathrm{wirt}}\sigma(z)=\underbrace{\overline{\partial}_ {\mathrm{wirt}}(z)}_{=0}+b\cdot\overline{\partial}_{\mathrm{wirt}}\left(\frac{ z}{|z|}\right)\overset{\eqref{eq:2.1}}{=}b\cdot\left(-\frac{1}{2}\right)\frac{z^{2}}{|z|^{3}},\]
which is the claimed form. Then, using induction, we compute

\[\overline{\partial}_{\rm wirt}^{\ell+1}\sigma(z)=\overline{\partial}_{\rm wirt} \left(b\cdot q_{0,\ell}\cdot\frac{z^{\ell+1}}{|z|^{2\ell+1}}\right)\stackrel{{ \eqref{eq:def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_ \Rightarrow:q_{0,\ell+1}}}{=:q_{0,\ell+1}}\cdot\frac{z^{\ell+2}}{|z|^{2 \ell+3}},\]

so that the case \(m=0\) is complete.

Now we deal with the case \(m\leq\ell+1\). The case \(\ell=0\) is proven by computing

\[\partial_{\rm wirt}\sigma(z)=\partial_{\rm wirt}(z)+b\cdot\partial_{\rm wirt} \left(\frac{z}{|z|}\right)\stackrel{{\eqref{eq:def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def_def__def_def_def_def_def__def_def_def__def_def__def_def_def_def_def__def_def_def__def_def__def_def__def__def_def__def__def_def__def__def__def_def__def__def_def__def__def__def_def__def__def__def_def__def__def__def__def__def__def__def_def__def__def_def__def__def_def__def__def__def__def_def__def__def_def__def__def_def__def_def__def__def__def_def__def_def__def_def__def_def__def_def__def_def__def_def__def_def_def__def_def__def_def_def__def_def__def_def__def_def_def__def_def_def__def_
_Here, the numbers \(a_{m,\ell}\) and \(b_{m,\ell}\) are again non-zero and rational. Furthermore, note that all cases for possible choices of \(m\) and \(\ell\) are covered: The case \(m\leq\ell\) is split into \(\ell=0\) and \(\ell\neq 0\). The case \(m=\ell+1\) is split into \(m=1\) and \(m>1\). Then, the case \(m\geq\ell+2\) remains._

Proof.: For the following we always assume \(z\in\mathbb{C}\) with \(z\neq 0\). Then we can simplify \(\cos(\spherical z)=\frac{\operatorname{Re}(z)}{|z|}\), so we can rewrite

\[\operatorname{card}(z)=\frac{1}{2}\left(1+\frac{\operatorname{Re}(z)}{|z|} \right)z=\frac{1}{2}z+\frac{1}{4}\frac{(z+\overline{z})\,z}{|z|}=\frac{1}{2}z +\frac{1}{4}\frac{z^{2}}{|z|}+\frac{|z|}{4}.\]

First, we compute

\[\overline{\partial}_{\operatorname{wirt}}\left(|z|\right)=\frac{1}{2}\left( \frac{1}{2}\frac{2\operatorname{Re}(z)}{|z|}+\frac{i}{2}\frac{2\operatorname {Im}(z)}{|z|}\right)=\frac{1}{2}\frac{z}{|z|} \tag{5.4}\]

and similarly

\[\partial_{\operatorname{wirt}}\left(|z|\right)=\frac{1}{2}\frac{\overline{z} }{|z|}. \tag{5.5}\]

We deduce

\[\overline{\partial}_{\operatorname{wirt}}\operatorname{card}(z)\overset{\eqref {eq:1.1},\eqref{eq:2.1}}{=}\underbrace{\frac{1}{4}\cdot\left(-\frac{1}{2} \right)}_{=:b_{0,1}}\cdot\frac{z^{3}}{|z|^{3}}+\underbrace{\frac{1}{8}}_{=:a_ {0,1}}\cdot\frac{z}{|z|}.\]

Using induction, we derive

\[\overline{\partial}_{\operatorname{wirt}}^{\ell+1}\operatorname{ card}(z) =a_{0,\ell}\overline{\partial}_{\operatorname{wirt}}\left(\frac{z^{\ell}}{|z|^{2 \ell-1}}\right)+b_{0,\ell}\overline{\partial}_{\operatorname{wirt}}\left( \frac{z^{\ell+2}}{|z|^{2\ell+1}}\right)\] \[\overset{\eqref{eq:1.1}}{=:a_{0,\ell}\cdot\left(-\frac{2\ell-1} {2}\right)}\cdot\frac{z^{\ell+1}}{|z|^{2\ell+1}}+\underbrace{b_{0,\ell}\cdot \left(-\frac{2\ell+1}{2}\right)}_{=:b_{0,\ell+1}}\cdot\frac{z^{\ell+3}}{|z|^{2 \ell+3}},\]

so the claim has been shown if \(m=0\). If we now fix any \(\ell\in\mathbb{N}\) and assume that the claim holds true for some \(m\in\mathbb{N}_{0}\) with \(m<\ell\), we get

\[\partial_{\operatorname{wirt}}^{m+1}\overline{\partial}_{ \operatorname{wirt}}^{\ell}\operatorname{card}(z) =a_{m,\ell}\partial_{\operatorname{wirt}}\left(\frac{z^{\ell-m}}{|z|^{2 \ell-1}}\right)+b_{m,\ell}\partial_{\operatorname{wirt}}\left(\frac{z^{\ell+2- m}}{|z|^{2\ell+1}}\right)\] \[\overset{\eqref{eq:2.1}}{=:a_{m+1,\ell}}\cdot\frac{z^{\ell-m-1}} {|z|^{2\ell-1}}+\underbrace{b_{m,\ell}\cdot\left(\frac{3}{2}-m\right)}_{=:b_{ m+1,\ell}}\cdot\frac{z^{\ell+2-m-1}}{|z|^{2\ell+1}},\]

so the claim holds true if \(m\leq\ell\).

The case \(m=\ell+1\) is split into the case \(m=1\) and \(m>1\). If \(m=1\), then \(\ell=0\) and we compute

\[\partial_{\operatorname{wirt}}\operatorname{card}(z)\overset{\eqref{eq:2.1}, \eqref{eq:2.1}}{=}\frac{1}{2}+\frac{1}{4}\left(2-\frac{1}{2}\right)\cdot\frac {z}{|z|}+\frac{1}{8}\cdot\frac{\overline{z}}{|z|}.\]

If \(m>1\) we get

\[\partial_{\operatorname{wirt}}^{\ell+1}\overline{\partial}_{ \operatorname{wirt}}^{\ell}\operatorname{card}(z) =a_{\ell,\ell}\partial_{\operatorname{wirt}}\left(\frac{1}{|z|^{2\ell-1}} \right)+b_{\ell,\ell}\cdot\partial_{\operatorname{wirt}}\left(\frac{z^{2}}{|z |^{2\ell+1}}\right)\] \[\overset{\eqref{eq:2.1},\eqref{eq:2.1}}{=:a_{\ell+1,\ell}} \cdot\frac{\overline{z}}{|z|^{2\ell+1}}+\underbrace{b_{\ell,\ell}\cdot\left(2 -\frac{2\ell+1}{2}\right)}_{=:b_{\ell+1,\ell}}\cdot\frac{z}{|z|^{2\ell+1}}.\]

Next, we deal with the case \(m=\ell+2\): Here we see

\[\partial_{\operatorname{wirt}}^{\ell+2}\overline{\partial}_{ \operatorname{wirt}}^{\ell}\operatorname{card}(z) =\partial_{\operatorname{wirt}}\left(\frac{1}{2}\delta_{(m,\ell)=(1,0)}+a_{ \ell+1,\ell}\frac{\overline{z}}{|z|^{2\ell+1}}+b_{\ell+1,\ell}\frac{z}{|z|^{2 \ell+1}}\right)\] \[\overset{\eqref{eq:2.1},\eqref{eq:2.1}}{=:a_{0,\ell+1}}\cdot \frac{\overline{z}}{|z|^{2\ell+1}}+\underbrace{b_{\ell,\ell}\cdot\left(2-\frac{ 2\ell+1}{2}\right)}_{=:b_{\ell+1,\ell}}\cdot\frac{z}{|z|^{2\ell+1}}.\]

Next, we deal with the case \(m=\ell+2\): Here we see

\[\partial_{\operatorname{wirt}}^{\ell+2}\overline{\partial}_{ \operatorname{wirt}}^{\ell}\operatorname{card}(z) =\partial_{\operatorname{wirt}}\left(\frac{1}{2}\delta_{(m,\ell)=(1,0)}+a_{ \ell+1,\ell}\frac{\overline{z}}{|z|^{2\ell+1}}+b_{\ell+1,\ell}\frac{z}{|z|^{2 \ell+1}}\right)\] \[\overset{\eqref{eq:2.1},\eqref{eq:2.1}}{=:a_{0,\ell+1}}\cdot \frac{\overline{z}}{|z|^{2\ell+1}}+\underbrace{b_{\ell,\ell}\cdot\left(2-\frac{ 2\ell+1}{2}\right)}_{=:b_{\ell+1,\ell}}\cdot\frac{z}{|z|^{2\ell+1}}.\]

Next, we deal with the case \(m=\ell+2\): Here we see

\[\partial_{\operatorname{wirt}}^{\ell+2}\overline{\partial}_{ \operatorname{wirt}}^{\ell}\operatorname{card}(z) =\partial_{\operatorname{wirt}}\left(\frac{1}{2}\delta_{(m,\ell)=(1,0)}+a_{ \ell+1,\ell}\frac{\overline{z}}{|z|^{2\ell+1}}+b_{\ell+1,\ell}\frac{z}{|z|^{2 \ell+1}}\right)\] \[\overset{\eqref{eq:2.1},\eqref{eq:2.1}}{=:a_{0,\ell+1}}\cdot \frac{\overline{z}}{|z|^{2\ell+1}}+\underbrace{b_{\ell,\ell}\cdot\left(2-\frac{ 2\ell+1}{2}\right)}_{=:b_{\ell+1,\ell}}\cdot\frac{z}{|z|^{2\ell+1}}.\]
\[\overset{\eqref{eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eqeq:eq:eq:eqeq:eq:eq:eq:eqeq:eq:eqeq:eq:eq:eqeq:eqeq:eq:eq:eq:eqeq:eq:eqeq:eq:eqeq:eq:eqeq:eq:eqeq:eq:eq:eqeq:eqeq:eq:eq:eqeq:eqeq:eqeq:eq:eq:eq:eq:eqeq:eqeq:eq:eqeq:eqeq:eqeq:eq:eqeq:eq:eqeq:eqeq:eqeq:eq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeqeq:eqeq:eqeq:eqeq:eqeqeq:eqeq:eqeq:eqeq:eqeqeq:eqeqeq:eqeqeq:eqeqeq:eqeq:eqeqeq:eqeqeq:eqeq:eqeqeq:eqeqeqeq:eqeqeq:eqeqeqeq:eqeqeqeq:eqeqeqeq:eqeqeqeq:eqeqeqeq:eqeqeqeq:eqeqeqeq:eqeqeqeq:eqeqeqeqeq:eqeqeqeq:eqeqeq:eqeqeqeqeq:eqeqeqeq:eqeqeqeqeq:eqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeqeqeqeq:eqeqeqeqeqeqeqeq:eqeqeqeqeqeqeqeq:eqeqeqeqeqeqeqeqeq:eqeqeqeqeqeq:eqeqeqeqeqeqeqeqeq:eqeqeqeqeqeqeqeq:eqeqeqeqeqeqeqeqeqeq:eqeqeqeqeqeqeqeqeq:eqeqeqeqeqeqeqeq:eqeqeqeqeqeqeqeqeq:eqeqeqeqeqeqeqeq:eqeqeqeqeqeqeqeqeqeq:eqeqeqeqeqeqeqeq:eqeqeqeqeqeqeqeqeqeqeqeq:
**Theorem 6.1**.: _Let \(s,k\in\mathbb{N}\). Then there is a constant \(c=c(s,k)>0\) with the following property: For any \(m\in\mathbb{N}\) and any map \(\overline{a}:C^{k}([-1,1]^{s};\mathbb{R})\to\mathbb{R}^{m}\) that is continuous with respect to some norm on \(C^{k}([-1,1]^{s};\mathbb{R})\) and any (possibly discontinuous) map \(M_{m}:\mathbb{R}^{m}\to C([-1,1]^{s};\mathbb{R})\) we have_

\[\sup_{\begin{subarray}{c}f\in C^{k}([-1,1]^{s};\mathbb{R})\\ \|f\|_{C^{k}([-1,1]^{s};\mathbb{R})}\leq 1\end{subarray}}\|f-M_{m}(\overline{a} (f))\|_{L^{\infty}([-1,1]^{s};\mathbb{R})}\geq c\cdot m^{-k/s}.\]

Using this result we can prove the following theorem in the context of CVNNs.

**Theorem 6.2**.: _Let \(n\in\mathbb{N}\) and \(k\in\mathbb{N}\). Then there is a constant \(c=c(n,k)>0\) with the following property: For any \(m\in\mathbb{N}\), \(\phi\in C(\mathbb{C};\mathbb{C})\) and any map_

\[\eta:\quad C^{k}\left(\Omega_{n};\mathbb{C}\right)\to\left(\mathbb{C}^{n} \right)^{m}\times\mathbb{C}^{m}\times\mathbb{C}^{m},\quad g\mapsto\left(\eta _{1}(g),\eta_{2}(g),\eta_{3}(g)\right)\]

_which is continuous with respect to some norm on \(C^{k}(\Omega_{n};\mathbb{C})\), there exists \(f\in C^{k}\left(\Omega_{n};\mathbb{C}\right)\) satisfying \(\|f\|_{C^{k}(\Omega_{n};\mathbb{C})}\leq 1\) and_

\[\|f-\Phi(f)\|_{L^{\infty}(\Omega_{n};\mathbb{C})}\geq c\cdot m^{-k/(2n)},\]

_where \(\Phi(f)\in C(\Omega_{n};\mathbb{C})\) is given by_

\[\Phi(f)(z):=\sum_{j=1}^{m}\left(\eta_{3}(f)\right)_{j}\phi\left(\left[\eta_{1} (f)\right]_{j}^{T}z+\left(\eta_{2}(f)\right)_{j}\right).\]

Proof.: The idea is to apply Theorem 6.1. This theorem yields the existence of a constant \(c_{1}=c_{1}(n,k)\) with the property that for any \(m\in\mathbb{N}\), \(\phi\in C(\mathbb{C};\mathbb{C})\) and any map

\[\widetilde{\eta}:\quad C\left([-1,1]^{2n};\mathbb{R}\right)\to\left(\mathbb{C }^{n}\right)^{m}\times\mathbb{C}^{m}\times\mathbb{C}^{m},\quad g\mapsto \left(\widetilde{\eta}_{1}(g),\widetilde{\eta}_{2}(g),\widetilde{\eta}_{3}(g)\right)\]

which is continuous with respect to some norm on \(C^{k}([-1,1]^{2n};\mathbb{R})\) there exists \(\widetilde{f}\in C^{k}\left([-1,1]^{2n};\mathbb{R}\right)\) with \(\|\widetilde{f}\|_{C^{k}([-1,1]^{2n};\mathbb{R})}\leq 1\) satisfying

\[\left\|\widetilde{f}-\mathrm{Re}\left(\widetilde{\Phi}(\widetilde{f})\right) \right\|_{L^{\infty}([-1,1]^{2n};\mathbb{R})}\geq c_{1}\cdot\left(2m(n+2) \right)^{-k/2n}=c\cdot m^{-k/(2n)} \tag{6.1}\]

with \(c:=c_{1}\cdot(2(n+2))^{-k/2n}\). Here, with \(\varphi_{n}\) as introduced in (1.2) on page 5, we define the map \(\widetilde{\Phi}(\widetilde{f})\in C([-1,1]^{2n};\mathbb{C})\) via

\[\widetilde{\Phi}(\widetilde{f})(x):=\sum_{j=1}^{m}\left(\widetilde{\eta}_{3}( \widetilde{f})\right)_{j}\phi\left(\left[\widetilde{\eta}_{1}(\widetilde{f}) \right]_{j}^{T}\varphi_{n}(x)+\left(\widetilde{\eta}_{2}(\widetilde{f})\right) _{j}\right),\quad x\in[-1,1]^{2n}.\]

If we then have any continuous map

\[\eta:\quad C^{k}\left(\Omega_{n};\mathbb{C}\right)\to\left(\mathbb{C}^{n} \right)^{m}\times\mathbb{C}^{m}\times\mathbb{C}^{m},\quad g\mapsto\left(\eta _{1}(g),\eta_{2}(g),\eta_{3}(g)\right),\]

which is continuous with respect to some norm \(\|\cdot\|_{V}\) on \(C^{k}\left(\Omega_{n};\mathbb{C}\right)\), we can define the corresponding map \(\widetilde{\eta}\) via

\[\widetilde{\eta}:\quad C\left([-1,1]^{2n};\mathbb{R}\right)\to\left(\mathbb{C }^{n}\right)^{m}\times\mathbb{C}^{m}\times\mathbb{C}^{m},\quad g\mapsto \left(\widetilde{\eta}_{1}(g),\widetilde{\eta}_{2}(g),\widetilde{\eta}_{3}(g)\right)\]

with

\[\widetilde{\eta}_{i}(g):=\eta_{i}\left(g\circ\varphi_{n}^{-1}\big{|}_{\Omega_{ n}}\right)\]

for every \(g\in C\left([-1,1]^{2n};\mathbb{R}\right)\). Note that \(\widetilde{\eta}\) is continuous with respect to the norm \(\|\cdot\|_{\widetilde{V}}\) defined on \(C^{k}([-1,1]^{2n};\mathbb{R})\) as

\[\|\widetilde{f}\|_{\widetilde{V}}:=\left\|\widetilde{f}\circ\varphi_{n}^{-1} \big{|}_{\Omega_{n}}\right\|_{V}\]

for every \(\widetilde{f}\in C^{k}([-1,1]^{2n};\mathbb{R})\).
Now take \(\widetilde{f}\in C^{k}\left([-1,1]^{2n};\mathbb{R}\right)\) with \(\|\widetilde{f}\|_{C^{k}\left([-1,1]^{2n};\mathbb{R}\right)}\leq 1\) satisfying (6.1). For the function defined as \(f:=\widetilde{f}\circ\varphi_{n}^{-1}\big{|}_{\Omega_{n}}\) we see that \(f\in C^{k}\left(\Omega_{n};\mathbb{R}\right)\subseteq C^{k}\left(\Omega_{n}; \mathbb{C}\right)\), \(\|f\|_{C^{k}\left(\Omega_{n};\mathbb{C}\right)}\leq 1\) and we furthermore deduce

\[\left\|f-\Phi(f)\right\|_{L^{\infty}\left(\Omega_{n};\mathbb{C}\right)}\geq\left\| f-\operatorname{Re}(\Phi(f))\right\|_{L^{\infty}\left(\Omega_{n};\mathbb{R} \right)}=\left\|\widetilde{f}-\operatorname{Re}\left(\widetilde{\Phi}( \widetilde{f})\right)\right\|_{L^{\infty}\left([-1,1]^{2n};\mathbb{R}\right)} \overset{\eqref{eq:f
We then define \(g_{j}\in C(\Omega_{1};\mathbb{R})\) by \(g_{j}(z):=\widetilde{g}_{j}\left(\mathrm{Re}(z)\right)\) for any \(j\in\{1,...,m\}\). We compute

\[g_{j}\left(\left(b_{j}\right)^{T}z\right) =\widetilde{g}_{j}\left(\mathrm{Re}\left(\sum_{\ell=1}^{n}\left(b_ {j}\right)_{\ell}\cdot z_{\ell}\right)\right)\] \[=\widetilde{g}_{j}\left(\mathrm{Re}\left(\sum_{\ell=1}^{n}\left( \left(a_{j}\right)_{\ell}-i\cdot\left(a_{j}\right)_{n+\ell}\right)\left(\varphi _{n}^{-1}(z)_{\ell}+i\cdot\varphi_{n}^{-1}(z)_{n+\ell}\right)\right)\right)\] \[=\widetilde{g}_{j}\left(\sum_{\ell=1}^{n}\left[\left(a_{j}\right) _{\ell}\varphi_{n}^{-1}(z)_{\ell}+\left(a_{j}\right)_{n+\ell}\varphi_{n}^{-1} (z)_{n+\ell}\right]\right)\] \[=\widetilde{g}_{j}\left(\left(a_{j}\right)^{T}\cdot\varphi_{n}^{- 1}(z)\right) \tag{6.2}\]

and deduce

\[\left\|f(z)-\sum_{j=1}^{m}g_{j}\left(b_{j}^{T}z\right)\right\|_{L ^{\infty}(\Omega_{n};\mathbb{R})} =\left\|(f\circ\varphi_{n})(x)-\sum_{j=1}^{m}g_{j}\left(b_{j}^{T} \cdot\varphi_{n}(x)\right)\right\|_{L^{\infty}([-1,1]^{2n};\mathbb{R})}\] \[\overset{\eqref{eq:f_2}}{=}\left\|(f\circ\varphi_{n})(x)-\sum_{j =1}^{m}\widetilde{g}_{j}\left(a_{j}^{T}\cdot x\right)\right\|_{L^{\infty}([-1,1 ]^{2n};\mathbb{R})}\] \[\leq c_{1}\cdot m^{-k/(2n-1)}\cdot\|f\circ\varphi_{n}\|_{C^{k}([- 1,1]^{2n};\mathbb{R})}\,.\]

For a function \(f\in C^{k}\left(\Omega_{n};\mathbb{C}\right)\) we can pick functions \(g_{1}^{\mathrm{Re}},...,g_{m}^{\mathrm{Re}},g_{1}^{\mathrm{Im}},...,g_{m}^{ \mathrm{Im}}\in C^{k}\left(\Omega;\mathbb{R}\right)\) satisfying

\[\left\|\mathrm{Re}(f(z))-\sum_{j=1}^{m}g_{j}^{\mathrm{Re}}\left( b_{j}^{T}z\right)\right\|_{L^{\infty}(\Omega_{n};\mathbb{R})} \leq c_{1}\cdot m^{-k/(2n-1)}\cdot\left\|\mathrm{Re}\left(f\circ \varphi_{n}\right)\right\|_{C^{k}([-1,1]^{2n};\mathbb{R})},\] \[\left\|\mathrm{Im}(f(z))-\sum_{j=1}^{m}g_{j}^{\mathrm{Im}}\left( b_{j}^{T}z\right)\right\|_{L^{\infty}(\Omega_{n};\mathbb{R})} \leq c_{1}\cdot m^{-k/(2n-1)}\cdot\left\|\mathrm{Im}\left(f\circ \varphi_{n}\right)\right\|_{C^{k}([-1,1]^{2n};\mathbb{R})}.\]

Defining \(g_{j}:=g_{j}^{\mathrm{Re}}+i\cdot g_{j}^{\mathrm{Im}}\) yields

\[\left\|f(z)-\sum_{j=1}^{m}g_{j}\left(b_{j}^{T}z\right)\right\|_{L^{\infty}( \Omega_{n};\mathbb{C})}\leq c_{1}\cdot\sqrt{2}\cdot m^{-k/(2n-1)}\cdot\|f\|_{C ^{k}(\Omega_{n};\mathbb{C})}\,,\]

completing the proof. 

**Lemma 6.5**.: _Let \(\{u_{\ell}\}_{\ell=1}^{\infty}\) be an enumeration of the set of complex polynomials in \(z\) and \(\overline{z}\) with coefficients in \(\mathbb{Q}+i\mathbb{Q}\). Then there exists a function \(\phi\in C^{\infty}\left(\mathbb{C};\mathbb{C}\right)\) with the following properties:_

1. _For every_ \(\ell\in\mathbb{N}\) _and_ \(z\in\Omega_{1}\) _one has_ \[\phi(z+3\ell)=u_{\ell}(z).\]
2. \(\phi\) _is admissible._

Proof.: Let \(\psi\in C^{\infty}\left(\mathbb{C};\mathbb{R}\right)\) with \(0\leq\psi\leq 1\) and

\[\psi\big{|}_{\Omega_{1}}\equiv 1,\qquad\mathrm{supp}(\psi)\subseteq\widetilde{ \Omega},\]

where \(\widetilde{\Omega}:=\left\{z\in\mathbb{C}:\ \left|\mathrm{Re}\left(z\right)\right|, \left|\mathrm{Im}\left(z\right)\right|<\frac{3}{2}\right\}\). We then define

\[\phi:=\sum_{\ell=1}^{\infty}u_{\ell}(\bullet-3\ell)\cdot\psi(\bullet-3\ell)+f \cdot\psi,\]
where \(f(z)=e^{\operatorname{Re}(z)}\). Note that \(\phi\) is smooth since it is a locally finite sum of smooth functions. Furthermore, \(\phi\) is admissible, since the calculation in the proof of Proposition 5.1 shows for \(z\) in the interior of \(\Omega_{1}\) and \(\rho:\mathbb{R}\to\mathbb{R},\ t\mapsto e^{t}\) that

\[\left|\partial_{\operatorname{wit}}^{m}\overline{\partial}_{\operatorname{wit} }^{\ell}\phi(z)\right|=\left|\partial_{\operatorname{wit}}^{m}\overline{ \partial}_{\operatorname{wit}}^{\ell}f(z)\right|=\frac{1}{2^{m+\ell}}\left| \rho^{(m+\ell)}(\operatorname{Re}(z))\right|>0\]

for every \(m,\ell\in\mathbb{N}_{0}\). Finally, property (1) follows directly by construction of \(\phi\) because

\[(\widetilde{\Omega}+3\ell)\cap(\widetilde{\Omega}+3\ell^{\prime})=\varnothing\]

for \(\ell\neq\ell^{\prime}\). 

For the special activation function constructed in Lemma 6.5 we can improve the result from Theorem 4.2.

**Theorem 6.6**.: _Let \(k\in\mathbb{N}\) and \(n\in\mathbb{N}\). Then there is a constant \(c=c(n,k)>0\) such that for any \(m\in\mathbb{N}\) and \(f\in C^{k}\left(\Omega_{n};\mathbb{C}\right)\) there are \(\rho_{1},...,\rho_{m}\in\mathbb{C}^{n}\) and \(\ell_{1},...,\ell_{m}\in\mathbb{N}\), such that_

\[\left\|f(z)-\sum_{j=1}^{m}\phi\left(\rho_{j}^{T}z+3\ell_{j}\right)\right\|_{L^ {\infty}(\Omega_{n})}\leq c\cdot m^{-k/(2n-1)}\cdot\left\|f\right\|_{C^{k}( \Omega_{n};\mathbb{C})}.\]

_Here, \(\phi\) is the function constructed in Lemma 6.5._

Proof.: We choose the constant \(c\) according to Proposition 6.4. Let \(m\in\mathbb{N}\) and \(f\in C^{k}\left(\Omega_{n};\mathbb{C}\right)\). Again, according to Proposition 6.4, we can choose \(\rho_{1},...,\rho_{m}\in\mathbb{C}^{n}\) with \(\left\|\rho_{j}\right\|=1/\sqrt{2n}\) and functions \(g_{1},...,g_{m}\in C\left(\Omega;\mathbb{C}\right)\) with the property

\[\left\|f-\sum_{j=1}^{m}g_{j}\left(\rho_{j}^{T}\cdot\bullet\right)\right\|_{L^ {\infty}(\Omega_{n})}\leq c\cdot m^{-k/(2n-1)}\cdot\left\|f\right\|_{C^{k}( \Omega_{n};\mathbb{C})}.\]

Recall from Lemma 6.5 that \(\{u_{\ell}\}_{\ell=1}^{\infty}\) is an enumeration of the set of complex polynomials in \(z\) and \(\overline{z}\). Hence, using the complex version of the Stone-Weierstrass-Theorem (see for instance [13, Theorem 4.51]), pick \(\ell_{1},...,\ell_{m}\in\mathbb{N}\) such that

\[\left\|g_{j}-u_{\ell_{j}}\right\|_{L^{\infty}(\Omega_{1};\mathbb{C})}\leq m^ {-1-k/(2n-1)}\cdot\left\|f\right\|_{C^{k}(\Omega_{n};\mathbb{C})} \tag{6.3}\]

for every \(j\in\{1,...,m\}\). Since \(\phi\left(\bullet+3\ell\right)=u_{\ell}\) on \(\Omega_{1}\) and \(\rho_{j}^{T}z\in\Omega_{1}\) for \(j\in\{1,...,m\}\), \(\ell\in\mathbb{N}\) and \(z\in\Omega_{n}\) we estimate

\[\left\|f(z)-\sum_{j=1}^{m}\phi\left(\rho_{j}^{T}z+3\ell_{j}\right) \right\|_{L^{\infty}(\Omega_{n};\mathbb{C})}\] \[\leq c\cdot m^{-k/(2n-1)}\cdot\left\|f\right\|_{C^{k}(\Omega_{n}; \mathbb{C})}+\sum_{j=1}^{m}\left\|g_{j}\left(z\right)-u_{\ell_{j}}\left(z \right)\right\|_{L^{\infty}(\Omega_{n};\mathbb{C})}\] \[\stackrel{{\eqref{eq:2}}}{{\leq}}c\cdot m^{-k/(2n-1) }\cdot\left\|f\right\|_{C^{k}(\Omega_{n};\mathbb{C})}+m^{-k/(2n-1)}\cdot\left\| f\right\|_{C^{k}(\Omega_{n};\mathbb{C})}\] \[=(c+1)\cdot m^{-k/(2n-1)}\cdot\left\|f\right\|_{C^{k}(\Omega_{n}; \mathbb{C})}.\qed\]

The parameters of the neural network in Theorem 6.6 do not continuously depend on the function \(f\), which is obvious since \(\ell_{1},...,\ell_{m}\) are natural numbers. Furthermore, if the parameters depended continuously on the function \(f\), this would be a contradiction to the optimality result in Theorem 6.2.

Last but not least, we show that there are certain admissible activation functions, for which one can not improve the result from Theorem 4.2, even if we do not assume a continuous weight selection. To
this end, we first formulate a corresponding result in the real setting. The idea of the proof is based on [30, Theorem 4] but instead of the bound for the VC dimension of ReLU networks used in [30], we will employ a bound for the VC dimension stated in [3, Theorem 8.11]. For a detailed introduction of the concept of the VC dimension and related topics, see for instance [26, Chapter 6].

**Theorem 6.7**.: _Let \(n,k\in\mathbb{N}\) and_

\[\phi:\quad\mathbb{R}\to\mathbb{R},\quad\phi(x):=\frac{1}{1+e^{-x}}\]

_be the sigmoid function. Then there is a constant \(c=c(n,k)>0\) with the following property: If \(\varepsilon\in(0,\frac{1}{2})\) and \(m\in\mathbb{N}\) are such that for every function \(f\in C^{k}\left([-1,1]^{n};\mathbb{R}\right)\) with \(\left\|f\right\|_{C^{k}\left([-1,1]^{n};\mathbb{R}\right)}\leq 1\) there exist coefficients \(\rho_{1},...,\rho_{m}\in\mathbb{R}^{n}\), \(\eta_{1},...,\eta_{m}\in\mathbb{R}\) and \(\sigma_{1},...,\sigma_{m}\in\mathbb{R}\) satisfying_

\[\left\|f(x)-\sum_{j=1}^{m}\sigma_{j}\cdot\phi\left(\rho_{j}^{T}x+\eta_{j} \right)\right\|_{L^{\infty}\left([-1,1]^{n};\mathbb{R}\right)}\leq\varepsilon,\]

_then necessarily_

\[m\geq c\cdot\frac{\varepsilon^{-n/k}}{\ln\left(1/\varepsilon\right)}.\]

Proof.: We first pick a function \(\psi\in C^{\infty}\left(\mathbb{R}^{n};\mathbb{R}\right)\) with the property that \(\psi(0)=1\) and \(\psi(x)=0\) for every \(x\in\mathbb{R}^{n}\) with \(\left|x\right|>\frac{1}{4}\). We then choose

\[c_{1}=c_{1}(n,k):=\left(\left\|\psi\right\|_{C^{k}\left([-1,1]^{n};\mathbb{R} \right)}\right)^{-1}.\]

Now, let \(\varepsilon\in(0,\frac{1}{2})\) and \(m\in\mathbb{N}\) be arbitrary with the property stated in the formulation of the theorem. If \(\varepsilon>\frac{c_{1}}{2}\cdot\frac{1}{6^{k}}\), then \(m\geq c\cdot\frac{\varepsilon^{-n/k}}{\ln(1/\varepsilon)}\) trivially holds (as long as \(c=c(n,k)\) is sufficiently small). Hence, we can assume that \(\varepsilon\leq\frac{c_{1}}{2}\cdot\frac{1}{6^{k}}\). Now, let \(N\) be the smallest integer with \(N\geq 2\), such that

\[\frac{c_{1}}{2^{k+1}}\cdot N^{-k}\leq\varepsilon.\]

Note that this implies

\[N^{k}\geq\frac{c_{1}}{\varepsilon}\cdot\frac{1}{2^{k+1}}\geq\frac{c_{1}}{2^{k +1}}\cdot\frac{2}{c_{1}}\cdot 6^{k}=3^{k}\]

and hence \(N\geq 3\), whence \(N-1\geq 2\). Therefore, since \(\frac{N}{2}\leq N-1\) because of \(N\geq 2\), it follows that

\[\varepsilon<\frac{c_{1}}{2^{k+1}}\cdot(N-1)^{-k}\leq\frac{c_{1}}{2^{k+1}}2^{k }\cdot N^{-k}=\frac{c_{1}}{2}\cdot N^{-k}. \tag{6.4}\]

Now, for every \(\alpha\in\{-N,...,N\}^{n}\) pick \(z_{\alpha}\in\{0,1\}\) arbitrary and let \(y_{\alpha}:=z_{\alpha}c_{1}N^{-k}\). Define the function

\[f(x):=\sum_{\alpha\in\{-N,...,N\}^{n}}y_{\alpha}\psi\left(Nx-\alpha\right), \quad x\in\mathbb{R}^{n}.\]

Clearly, \(f\in C^{\infty}(\mathbb{R}^{n};\mathbb{R})\). Furthermore, for any multi-index \(\mathbf{k}\in\mathbb{N}_{0}^{n}\) with \(\left|\mathbf{k}\right|\leq k\) we have

\[\left\|\partial^{\mathbf{k}}f\right\|_{L^{\infty}\left([-1,1]^{n};\mathbb{R} \right)}\leq N^{\left|\mathbf{k}\right|}\cdot\max_{\alpha}\left|y_{\alpha} \right|\cdot\left\|\partial^{\mathbf{k}}\psi\right\|_{L^{\infty}\left([-1,1]^ {n};\mathbb{R}\right)}\leq N^{k}\cdot\max_{\alpha}\left|y_{\alpha}\right|\cdot \left\|\psi\right\|_{C^{k}\left([-1,1]^{n};\mathbb{R}\right)}\leq 1,\]

so we conclude that \(\left\|f\right\|_{C^{k}\left([-1,1]^{n};\mathbb{R}\right)}\leq 1\). Additionally, for any fixed \(\beta\in\{-N,...,N\}^{n}\) we see

\[f\left(\frac{\beta}{N}\right)=y_{\beta},\]

which follows from the choice of \(\psi\).

By assumption, we can choose suitable coefficients \(\rho_{1},...,\rho_{m}\in\mathbb{R}^{n}\), \(\eta_{1},...,\eta_{m}\in\mathbb{R}\) and \(\sigma_{1},...,\sigma_{m}\in\mathbb{R}\) such that

\[\left\|f-g\right\|_{L^{\infty}\left([-1,1]^{n};\mathbb{R}\right)}\leq\varepsilon\]
with

\[g:=\sum_{j=1}^{m}\sigma_{j}\cdot\phi\left(\rho_{j}^{T}\cdot\bullet+\eta_{j}\right).\]

Letting

\[\widetilde{g}:=g(\bullet/N)=\sum_{j=1}^{m}\sigma_{j}\cdot\phi\left(\frac{\rho_{ j}^{T}}{N}\cdot\bullet+\eta_{j}\right) \tag{6.5}\]

we see for every \(\alpha\in\{-N,...,N\}^{n}\) that

\[\widetilde{g}(\alpha)=g\left(\frac{\alpha}{N}\right)=\begin{cases}\geq y_{ \alpha}-\varepsilon=c_{1}N^{-k}-\varepsilon\overset{\eqref{eq:eq:eq:eq:eq: eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eqeq:eq:eq:eqeq:eq:eq:eq:eqeq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eqeq:eq:eq:eq:eq:eq:eqeq:eq:eq:eqeq:eq:eq:eq:eqeq:eqeq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eqeq:eq:eq:eqeq:eq:eq:eqeq:eq:eq:eqeq:eqeq:eq:eq:eqeq:eq:eq:eqeq:eq:eq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eq:eq:eqeq:eqeq:eqeq:eq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eqeq:eq:eqeqeq:eqeq:eqeq:eqeqeq:eqeqeq:eqeq:eqeqeq:eqeqeq:eqeq:eq:eqeq:eqeqeq:eqeq:eqeqeqeq:eqeqeq:eqeqeq:eqeqeq:eqeqeq:eqeqeqeq:eqeqeq:eqeqeq:eqeqeqeq:eqeqeqeq:eqeqeqeq:eqeqeqeqeq:eqeqeqeqeq:
Proof.: We choose the constant \(c=c(2n,k)\) according to the previous Theorem 6.7 and let \(\varphi_{n}\) as in (1.2). Then, let \(\varepsilon\in(0,\frac{1}{2})\) and \(m\in\mathbb{N}\) with the properties assumed in the statement of the corollary. If we then take an arbitrary function \(f\in C^{k}\left([-1,1]^{2n};\mathbb{R}\right)\) with \(\left\|f\right\|_{C^{k}\left([-1,1]^{2n};\mathbb{R}\right)}\leq 1\), we deduce the existence of \(\rho_{1},...,\rho_{m}\in\mathbb{C}^{n}\), \(\eta_{1},...,\eta_{m}\in\mathbb{C}\) and \(\sigma_{1},...,\sigma_{m}\in\mathbb{C}\), such that

\[\left\|f(x)-\operatorname{Re}\left(\sum_{j=1}^{m}\sigma_{j}\cdot \phi\left(\rho_{j}^{T}\cdot\varphi_{n}(x)+\eta_{j}\right)\right)\right\|_{L^{ \infty}\left([-1,1]^{2n};\mathbb{R}\right)}\] \[\leq \left\|(f\circ\varphi_{n}^{-1})(z)-\sum_{j=1}^{m}\sigma_{j}\cdot \phi\left(\rho_{j}^{T}z+\eta_{j}\right)\right\|_{L^{\infty}\left(\Omega_{n}; \mathbb{C}\right)}\leq\varepsilon.\]

In the next step, we want to show that

\[x\mapsto\operatorname{Re}\left(\sum_{j=1}^{m}\sigma_{j}\cdot\phi\left(\rho_{j} ^{T}\cdot\varphi_{n}(x)+\eta_{j}\right)\right)\]

is a real-valued shallow neural network with \(m\) neurons in the hidden layer and the real sigmoid function as activation function. Then the claim follows using Theorem 6.7.

For every \(j\in\{1,...,m\}\) we pick a matrix \(\widetilde{\rho_{j}}\in\mathbb{R}^{2n\times 2}\) with the property that one has

\[\widetilde{\rho_{j}}^{T}\cdot\varphi_{n}^{-1}(z)=\varphi_{1}^{-1}\left(\rho_{ j}^{T}\cdot z\right)\]

for every \(z\in\mathbb{C}^{n}\). This is possible, since this is equivalent to

\[\widetilde{\rho_{j}}^{T}v=\varphi_{1}^{-1}(\rho_{j}^{T}\varphi_{n}(v))\]

for all \(v\in\mathbb{R}^{2n}\), where the right-hand side is an \(\mathbb{R}\)-linear map \(\mathbb{R}^{2n}\to\mathbb{R}^{2}\). Denoting the first column of \(\widetilde{\rho_{j}}\) with \(\widehat{\rho_{j}}\), we get

\[\widehat{\rho_{j}}^{T}\cdot\varphi_{n}^{-1}(z)=\operatorname{Re}\left(\rho_{ j}^{T}\cdot z\right).\]

Writing \(\gamma\) for the classical real-valued sigmoid function (i.e. \(\gamma(x)=\frac{1}{1+e^{-x}}\)), we deduce

\[\operatorname{Re}\left(\sum_{j=1}^{m}\sigma_{j}\cdot\phi\left( \rho_{j}^{T}\cdot\varphi_{n}(x)+\eta_{j}\right)\right) =\operatorname{Re}\left(\sum_{j=1}^{m}\sigma_{j}\cdot\gamma\left( \operatorname{Re}\left(\rho_{j}^{T}\cdot\varphi_{n}(x)+\eta_{j}\right)\right)\right)\] \[=\operatorname{Re}\left(\sum_{j=1}^{m}\sigma_{j}\cdot\gamma\left( \widehat{\rho_{j}}^{T}x+\operatorname{Re}\left(\eta_{j}\right)\right)\right)\] \[=\sum_{j=1}^{m}\operatorname{Re}\left(\sigma_{j}\right)\cdot \gamma\left(\widehat{\rho_{j}}^{T}x+\operatorname{Re}\left(\eta_{j}\right) \right),\]

where in the last step we used that \(\gamma\) is real-valued. As noted above, this completes the proof. 

As a further corollary, we can formulate the result in the context of our approximation order derived in Theorem 4.2.

**Corollary 6.9**.: _Let \(n\in\mathbb{N}\), \(k\in\mathbb{N}\) and_

\[\phi:\quad\mathbb{C}\to\mathbb{C},\quad\phi(z):=\frac{1}{1+e^{-\operatorname{ Re}(z)}}.\]

_Then there exists a constant \(c=c(n,k)>0\) with the following property: For any \(m\in\mathbb{N}^{\geq 2}\) there exists a function \(f\in C^{k}\left(\Omega_{n};\mathbb{C}\right)\) with \(\left\|f\right\|_{C^{k}\left(\Omega_{n};\mathbb{C}\right)}\leq 1\) such that for every choice of coefficients \(\rho_{1},...,\rho_{m}\in\mathbb{C}^{n}\), \(\eta_{1},...,\eta_{m}\in\mathbb{C}\) and \(\sigma_{1},...,\sigma_{m}\in\mathbb{C}\), we have_

\[\left\|f(z)-\sum_{j=1}^{m}\sigma_{j}\cdot\phi\left(\rho_{j}^{T}z+\eta_{j} \right)\right\|_{L^{\infty}\left(\Omega_{n};\mathbb{C}\right)}\geq c\cdot\left( m\cdot\ln(m)\right)^{-k/2n}.\]
Proof.: Choose \(c=c(n,k)>0\) to be exactly determined later, where we may assume

\[c\cdot(2\cdot\ln(2))^{-k/2n}<\frac{1}{2}\]

and hence \(c\cdot(m\cdot\ln(m))^{-k/2n}<\frac{1}{2}\) for all \(m\in\mathbb{N}^{\geq 2}\). Assume towards a contradiction that for every \(f\in C^{k}\left(\Omega_{n};\mathbb{C}\right)\) with \(\|f\|_{C^{k}(\Omega_{n};\mathbb{C})}\leq 1\) there are coefficients \(\rho_{1},...,\rho_{m}\in\mathbb{C}^{n},\sigma_{1},...,\sigma_{m}\in\mathbb{C}\) and \(\eta_{1},...,\eta_{m}\in\mathbb{C}\) such that

\[\left\|f(z)-\sum_{j=1}^{m}\sigma_{j}\cdot\phi\left(\rho_{j}^{T}z+\eta_{j} \right)\right\|_{L^{\infty}\left(\Omega_{n};\mathbb{C}\right)}<c\cdot\left(m \cdot\ln(m)\right)^{-k/2n}.\]

Applying Corollary 6.8 we then get

\[m\geq c_{1}\cdot\frac{\varepsilon^{-2n/k}}{\ln\left(1/\varepsilon\right)}\]

with \(\varepsilon=c\cdot\left(m\cdot\ln m\right)^{-k/2n}\in\left(0,\frac{1}{2}\right)\) and \(c_{1}=c_{1}(n,k)>0\). Let \(\alpha:=2n/k\) and \(c_{2}=c_{2}(\alpha)=c_{2}(n,k)>0\) such that \(\ln x\leq c_{2}x^{\alpha/2}\) for every \(x\geq 1\). Therefore we observe

\[m\geq c_{1}\cdot\frac{\varepsilon^{-\alpha}}{\ln\left(1/\varepsilon\right)} \geq\frac{c_{1}}{c_{2}}\varepsilon^{-\alpha/2},\]

which implies

\[\ln(m)\geq\ln\left(\frac{c_{1}}{c_{2}}\right)+\frac{\alpha}{2}\cdot\ln(1/ \varepsilon)\geq c_{3}\cdot\ln(1/\varepsilon)\]

for a suitable constant \(c_{3}=c_{3}(n,k)>0\); here we used that \(\varepsilon<\frac{1}{2}\), so that \(\ln\left(1/\varepsilon\right)\geq\ln(2)\geq 0\). Overall we then get

\[m\geq c_{1}\cdot\frac{\varepsilon^{-\alpha}}{\ln(1/\varepsilon)}=c_{1}\cdot c ^{-\alpha}\cdot\frac{m\cdot\ln(m)}{\ln(1/\varepsilon)}\geq c_{1}\cdot c_{3} \cdot c^{-\alpha}\cdot\frac{m\cdot\ln(m)}{\ln(m)}=c_{1}\cdot c_{3}\cdot c^{- \alpha}\cdot m.\]

Thus, choosing \(c>0\) in a way such that \(c^{\alpha}<c_{1}\cdot c_{3}\) yields the desired contradiction. Since \(\alpha,c_{1}\) and \(c_{3}\) all only depend on \(n\) and \(k\), this is possible for a suitable \(c=c(n,k)>0\). 

The previous corollary shows that, using the activation function

\[\phi:\quad\mathbb{C}\to\mathbb{C},\quad\phi(z):=\frac{1}{1+e^{-\operatorname{ Re}(z)}},\]

we cannot expect an order of approximation which is higher than \(m^{-k/(2n)}\) (up to a log factor), even if we allow discontinuous weight selection. Note also that \(\phi\) is admissible because of Proposition 5.1. In particular, we see that the approximation order of \(m^{-k/(2n-1)}\) derived for the special activation function in Theorem 6.6 cannot be achieved for arbitrary admissible activation functions.

## Appendix A Divided Differences

Divided differences are well-known in numerical mathematics as they are for example used to calculate the coefficients of an interpolation polynomial in its Newton representation. In our case, we are interested in divided differences, since they can be used to obtain a generalization of the classical mean-value theorem for differentiable functions: Given an interval \(I\subseteq\mathbb{R}\) and \(n+1\) pairwise distinct data points \(x_{0},...,x_{n}\in I\) as well as an \(n\)-times differentiable real-valued function \(f:I\to\mathbb{R}\), there exists \(\xi\in\left(\min\left\{x_{0},...,x_{n}\right\},\max\left\{x_{0},...,x_{n} \right\}\right)\), such that

\[f\left[x_{0},...,x_{n}\right]=\frac{f^{(n)}(\xi)}{n!},\]

where the left-hand side is a divided difference of \(f\) (defined below). The classical mean-value theorem is obtained in the case \(n=1\). Our goal in this section is to generalize this result to a multivariate setting by considering divided differences in multiple variables. Such a generalization is probably well-known, but since we could not locate a convenient reference and to make the paper more self-contained, we provide a proof.
Let us first define divided differences formally. Given \(n+1\) data points \(\left(x_{0},y_{0}\right),...,\left(x_{n},y_{n}\right)\in\mathbb{R}\times\mathbb{R}\) with pairwise distinct \(x_{k}\), we define the associated divided differences recursively via

\[\left[y_{k}\right] :=y_{k},\ k\in\{0,...,n\},\] \[\left[y_{k},...,y_{k+j}\right] :=\frac{\left[y_{k+1},...,y_{k+j}\right]-\left[y_{k},...,y_{k+j-1} \right]}{x_{k+j}-x_{k}},\ j\in\{1,...,n\},\ k\in\{0,...,n-j\}.\]

If the data points are defined by a function \(f\) (i.e. \(y_{k}=f\left(x_{k}\right)\) for all \(k\in\{0,...,n\}\)), we write

\[\left[x_{k},...,x_{k+j}\right]f:=\left[y_{k},...,y_{k+j}\right].\]

We first consider an alternative representation of divided differences, the so-called _Hermite-Genocchi-Formula_. To state it, we introduce the notation \(\Sigma^{k}\) for a certain \(k\)-dimensional simplex.

**Definition A.1**.: Let \(s\in\mathbb{N}\). Then we define

\[\Sigma^{s}:=\left\{x\in\mathbb{R}^{s}:\ x_{\ell}\geq 0\ \text{for all }\ell\text{ and}\sum_{\ell=1}^{s}x_{\ell}\leq 1\right\}.\]

The identity \(\lambda^{s}\left(\Sigma^{s}\right)=\frac{1}{s!}\) holds true.

A proof for the fact that the volume of \(\Sigma^{s}\) is indeed \(\frac{1}{s!}\) can be found for example in [27]. We can now consider the alternative representation of divided differences.

**Lemma A.2** (Hermite-Genocchi-Formula).: _For real numbers \(a,b\in\mathbb{R}\), a function \(f\in C^{k}([a,b];\mathbb{R})\) and pairwise distinct \(x_{0},...,x_{k}\in[a,b]\), the divided difference of \(f\) at the data points \(x_{0},...,x_{k}\) is given as_

\[\left[x_{0},...,x_{k}\right]f=\int_{\Sigma^{k}}f^{(k)}\left(x_{0}+\sum_{\ell =1}^{k}s_{\ell}\left(x_{\ell}-x_{0}\right)\right)ds.\] (A.1)

Proof.: See [5, Theorem 3.3]. 

We need the following easy generalization of the mean-value theorem for integrals.

**Lemma A.3**.: _Let \(D\subseteq\mathbb{R}^{s}\) be a connected and compact set with positive Lesbesgue measure and furthermore \(f:D\rightarrow\mathbb{R}\) a continuous function. Then there exists \(\xi\in D\) such that_

\[f(\xi)=\frac{1}{\lambda(D)}\cdot\int_{D}f(x)dx.\]

Proof.: Since \(D\) is compact and \(f\) continuous, there exist \(x_{\min}\in D\) and \(x_{\max}\in D\) satisfying

\[f\left(x_{\min}\right)\leq f(x)\leq f\left(x_{\max}\right)\]

for all \(x\in D\). Thus, one gets

\[f\left(x_{\min}\right)\leq\frac{1}{\lambda(D)}\int_{D}f(x)dx\leq f\left(x_{ \max}\right)\]

so the claim follows using the fact that \(f(D)\subseteq\mathbb{R}\) is connected, i.e., an interval. 

We also get a convenient representation of divided differences if the data points are equidistant.

**Lemma A.4**.: _Let \(f:\mathbb{R}\rightarrow\mathbb{R}\), \(x_{0}\in\mathbb{R}\) and \(h>0\). We consider the case of equidistant data points, meaning \(x_{j}:=x_{0}+jh\) for all \(j=1,...,k\) for a fixed \(h>0\). In this case, we have the formula_

\[\left[x_{0},...,x_{k}\right]f=\frac{1}{k!h^{k}}\cdot\sum_{r=0}^{k}(-1)^{k-r} \binom{k}{r}f\left(x_{r}\right).\] (A.2)
Proof.: We prove the result via induction over the number \(j\) of considered data points, meaning the following: For all \(j\in\{0,...,k\}\) we have

\[\left[x_{\ell},...,x_{\ell+j}\right]f=\frac{1}{j!h^{j}}\cdot\sum_{r=0}^{j}(-1)^{j -r}\binom{j}{r}f\left(x_{\ell+r}\right)\]

for all \(\ell\in\{0,...,k\}\) such that \(\ell+j\leq k\). The case \(j=0\) is trivial. Therefore, we assume the claim to be true for a fixed \(j\in\{0,...,k-1\}\) and choose an arbitrary \(\ell\in\{0,...,k\}\) such that \(\ell+j+1\leq k\). We then get

\[\left[x_{\ell},...,x_{\ell+j+1}\right]f =\frac{\left[x_{\ell+1},...,x_{\ell+j+1}\right]f-\left[x_{\ell},...,x_{\ell+j}\right]f}{x_{\ell+j+1}-x_{\ell}}\] \[=\frac{1}{j!h^{j}}\cdot\frac{\sum_{r=0}^{j}(-1)^{j-r}\binom{j}{r }\left(f\left(x_{\ell+r+1}\right)-f\left(x_{\ell+r}\right)\right)}{(j+1)h}\] \[=\frac{1}{(j+1)!h^{j+1}}\sum_{r=0}^{j}(-1)^{j-r}\binom{j}{r} \left(f\left(x_{\ell+r+1}\right)-f\left(x_{\ell+r}\right)\right).\]

Using an index shift we deduce

\[\sum_{r=0}^{j}(-1)^{j-r}\binom{j}{r}f\left(x_{\ell+r+1}\right)- \sum_{r=0}^{j}(-1)^{j-r}\binom{j}{r}f\left(x_{\ell+r}\right)\] \[=\sum_{r=1}^{j+1}(-1)^{j+1-r}\binom{j}{r-1}f\left(x_{\ell+r} \right)+\sum_{r=0}^{j}(-1)^{j+1-r}\binom{j}{r}f\left(x_{\ell+r}\right)\] \[=(-1)^{j+1}f\left(x_{\ell}\right)+\sum_{r=1}^{j}\left((-1)^{j+1- r}f\left(x_{\ell+r}\right)\left[\binom{j}{r-1}+\binom{j}{r}\right]\right)+f \left(x_{\ell+j+1}\right)\] \[=\sum_{r=0}^{j+1}(-1)^{j+1-r}\binom{j+1}{r}f\left(x_{\ell+r}\right)\]

which yields the claim. 

The final result for divided differences is stated as follows:

**Theorem A.5**.: _Let \(f:\mathbb{R}^{s}\to\mathbb{R}\) and \(k\in\mathbb{N}_{0},r>0\), such that \(f\big{|}_{(-r,r)^{s}}\in C^{k}\left((-r,r)^{s};\mathbb{R}\right)\). For \(\boldsymbol{p}\in\mathbb{N}_{0}^{s}\) with \(|\boldsymbol{p}|\leq k\) and \(h>0\) let_

\[f_{\boldsymbol{p},h}:=(2h)^{-|\boldsymbol{p}|}\sum_{0\leq r\leq\boldsymbol{p}} (-1)^{|\boldsymbol{p}|-|\boldsymbol{r}|}\binom{\boldsymbol{p}}{\boldsymbol{r} }f\left(h(2\boldsymbol{r}-\boldsymbol{p})\right).\]

_Let \(m:=\max_{j}\,\boldsymbol{p}_{j}\). Then, for \(0<h<\frac{r}{\max\{1,m\}}\) there exists \(\xi\in h[-m,m]^{s}\) satisfying_

\[f_{\boldsymbol{p},h}=\partial^{\boldsymbol{p}}f(\xi).\]

Proof.: We may assume \(m>0\), since \(\boldsymbol{p}=0\) implies \(f_{\boldsymbol{p},h}=f(0)\) and hence, the claim follows with \(\xi=0\).

We prove via induction over \(s\in\mathbb{N}\) that the formula

\[f_{\boldsymbol{p},h}=\boldsymbol{p}!\int_{\Sigma^{\boldsymbol{p}_{s}}}\int_{ \Sigma^{\boldsymbol{p}_{s-1}}}\cdots\int_{\Sigma^{\boldsymbol{p}_{1}}} \partial^{\boldsymbol{p}}f\left(-h\boldsymbol{p}_{1}+2h\sum_{\ell=1}^{ \boldsymbol{p}_{1}}\ell\sigma_{\ell}^{(1)},...,-h\boldsymbol{p}_{s}+2h\sum_{i =1}^{\boldsymbol{p}_{s}}\ell\sigma_{\ell}^{(s)}\right)d\sigma^{(1)}\cdots d \sigma^{(s)}\] (A.3)

holds for all \(\boldsymbol{p}\in\mathbb{N}_{0}^{s}\) with \(1\leq|\boldsymbol{p}|\leq k\) and all \(0<h<\frac{r}{m}\). The case \(s=1\) is exactly the Hermite-Genocchi-Formula (A.1) combined with (A.2) applied to the data points \(-hp,-hp+2h,...,hp-2h,hp\).

By induction, assume that the claim holds for some \(s\in\mathbb{N}\). For a fixed point \(y\in(-r,r)\), let

\[f_{y}:\quad(-r,r)^{s}\to\mathbb{R},\quad x\mapsto f(x,y).\]
For \(\mathbf{p}\in\mathbb{N}_{0}^{s+1}\) with \(|\mathbf{p}|\leq k\) and \(\mathbf{p}^{\prime}:=(p_{1},...,p_{s})\) we define

\[\Gamma:\quad(-r,r)\rightarrow\mathbb{R},\quad y\mapsto(f_{y})_{\mathbf{p}^{ \prime},h}=(2h)^{-|\mathbf{p}^{\prime}|}\sum_{0\leq\mathbf{r}^{\prime}\leq \mathbf{p}^{\prime}}(-1)^{|\mathbf{p}^{\prime}|-|\mathbf{r}^{\prime}|}\binom{ \mathbf{p}^{\prime}}{\mathbf{r}^{\prime}}f\left(h(2\mathbf{r}^{\prime}- \mathbf{p}^{\prime}),y\right).\]

Using the induction hypothesis, we get

\[\Gamma(y)\] \[= \mathbf{p}^{\prime}!\int\limits_{\Sigma^{\mathbf{p}_{s}}}\int \limits_{\Sigma^{\mathbf{p}_{s}-1}}\cdots\int\limits_{\Sigma^{\mathbf{p}_{1}}} \partial^{(\mathbf{p}^{\prime},0)}f\left(-h\mathbf{p}_{1}+2h\sum_{i=1}^{ \mathbf{p}_{1}}i\sigma_{i}^{(1)},...,-h\mathbf{p}_{s}+2h\sum_{i=1}^{\mathbf{ p}_{s}}i\sigma_{i}^{(s)},y\right)d\sigma^{(1)}\cdots d\sigma^{(s)}\]

for all \(y\in(-r,r)\). Furthermore, we calculate

\[\mathbf{p}_{s+1}!\cdot[-h\cdot\mathbf{p}_{s+1},-h\cdot\mathbf{p}_ {s+1}+2h,...,h\cdot\mathbf{p}_{s+1}]\Gamma\] \[\overset{(\ref{eq:p})}{=}(2h)^{-\mathbf{p}_{s+1}}\sum_{r^{\prime }=0}^{\mathbf{p}_{s+1}}(-1)^{\mathbf{p}_{s+1}-r^{\prime}}\binom{\mathbf{p}_{s+1 }}{r^{\prime}}\Gamma\left(h\left(2r^{\prime}-\mathbf{p}_{s+1}\right)\right)\] \[=(2h)^{-|\mathbf{p}|}\sum_{0\leq\mathbf{r}\leq\mathbf{p}}(-1)^{| \mathbf{p}|-|\mathbf{r}|}\binom{\mathbf{p}}{\mathbf{r}}f\left(h(2\mathbf{r}- \mathbf{p})\right)\] \[=f_{\mathbf{p},h}.\]

On the other hand, we get

\[[-h\cdot\mathbf{p}_{s+1},-h\cdot\mathbf{p}_{s+1}+2h,...,h\cdot \mathbf{p}_{s+1}]\Gamma\] \[\overset{(\ref{eq:p})}{=}\int_{\Sigma^{\mathbf{p}_{s+1}}} \Gamma^{(\mathbf{p}_{s+1})}\left(-h\mathbf{p}_{s+1}+2h\sum_{\ell=1}^{\mathbf{ p}_{s+1}}\ell\ell_{t}\right)dt\] \[=\mathbf{p}^{\prime}!\int\limits_{\Sigma^{\mathbf{p}_{s+1}}} \cdots\int\limits_{\Sigma^{\mathbf{p}_{1}}}\partial^{\mathbf{p}}f\left(-h \mathbf{p}_{1}+2h\sum_{\ell=1}^{\mathbf{p}_{1}}\ell\sigma_{\ell}^{(1)},...,-h \mathbf{p}_{s+1}+2h\sum_{\ell=1}^{\mathbf{p}_{s+1}}\ell\sigma_{\ell}^{(s+1)} \right)d\sigma^{(1)}\cdots d\sigma^{(s+1)}.\]

Changing the order of integration and derivative is possible, since we integrate on compact sets and only consider continuously differentiable functions.

We have thus proven (A.3) using the principle of induction. The claim of the theorem then follows directly using the mean-value theorem for integrals and by the fact that all the simplices \(\Sigma^{\mathbf{p}_{\ell}}\) are compact and connected (in fact convex). 

## Appendix B Prerequisites from Fourier Analysis

This section is dedicated to reviewing some notations and results from Fourier Analysis. In the end, a quantitative result for the approximation of \(C^{k}\left([-1,1]^{s};\mathbb{R}\right)\)-functions using linear combinations of multivariate Chebyshev polynomials is derived.

**Definition B.1**.: Let \(s\in\mathbb{N}\) and \(k\in\mathbb{N}_{0}\). We define

\[C^{k}_{2\pi}\left(\mathbb{R}^{s};\mathbb{C}\right):=\left\{f\in C^{k}\left( \mathbb{R}^{s};\mathbb{C}\right):\ \forall\mathbf{p}\in\mathbb{Z}^{s}\ \forall x\in\mathbb{R}^{s}:\ f(x+2\pi \mathbf{p})=f(x)\right\}.\]

For a function \(f\in C^{k}_{2\pi}\left(\mathbb{R}^{s};\mathbb{C}\right)\) we write

\[\left\|f\right\|_{C^{k}\left([-\pi,\pi]^{s};\mathbb{C}\right)} :=\max_{|\mathbf{k}|\leq k}\left\|\partial^{\mathbf{k}}f\right\|_ {L^{\infty}\left([-\pi,\pi];\mathbb{C}\right)}\text{ and }\] \[\left\|f\right\|_{L^{p}\left([-\pi,\pi]^{s};\mathbb{C}\right)} :=\left(\frac{1}{(2\pi)^{s}}\cdot\int_{[-\pi,\pi]^{s}}\left|f(x) \right|^{p}dx\right)^{1/p}\text{ for }p\in[1,\infty).\]
**Definition B.2**.: For any \(s\in\mathbb{N}\) and \(\mathbf{k}\in\mathbb{Z}^{s}\) we write

\[e_{\mathbf{k}}:\quad\mathbb{R}^{s}\to\mathbb{C},\quad e_{\mathbf{k}}(x)=e^{i \langle\mathbf{k},x\rangle}.\]

For any \(f\in C_{2\pi}\left(\mathbb{R}^{s};\mathbb{C}\right)\) we define the \(\mathbf{k}\)-th Fourier coefficient of \(f\) to be

\[\widehat{f}(\mathbf{k}):=\frac{1}{(2\pi)^{s}}\int_{[-\pi,\pi]^{s}}f(x)\overline {e_{\mathbf{k}}(x)}dx.\]

**Definition B.3**.: For two functions \(f,g\in C_{2\pi}\left(\mathbb{R}^{s};\mathbb{C}\right)\) we define their convolution as

\[f*g:\quad\mathbb{R}^{s}\to\mathbb{C},\quad(f*g)(x):=\frac{1}{(2\pi)^{s}}\int_{ [-\pi,\pi]^{s}}f(t)g(x-t)dt.\]

**Definition B.4**.: In the following we define several so-called kernels. Let \(m\in\mathbb{N}_{0}\) be arbitrary.

1. The \(m\)-th one-dimensional _Dirichlet-kernel_ is defined as \[D_{m}:=\sum_{h=-m}^{m}e_{h}.\]
2. The \(m\)-th one-dimensional _Fejer-kernel_ is defined as \[F_{m}:=\frac{1}{m}\sum_{h=0}^{m-1}D_{h}.\]
3. The \(m\)-th one-dimensional _de-la-Vallee-Poussin-kernel_ is defined as \[V_{m}:=(1+e_{m}+e_{-m})\cdot F_{m}.\]
4. Let \(s\in\mathbb{N}\). We extend the above definitions to dimension \(s\) by letting \[D_{m}^{s}\left(x_{1},...,x_{s}\right) :=\prod_{p=1}^{s}D_{m}\left(x_{p}\right),\] \[F_{m}^{s}\left(x_{1},...,x_{s}\right) :=\prod_{p=1}^{s}F_{m}\left(x_{p}\right),\] \[V_{m}^{s}\left(x_{1},...,x_{s}\right) :=\prod_{p=1}^{s}V_{m}\left(x_{p}\right).\]

**Proposition B.5**.: _Let \(m,s\in\mathbb{N}\). Then one has \(\|V_{m}^{s}\|_{L^{1}\left([-\pi,\pi]^{s};\mathbb{C}\right)}\leq 3^{s}\)._

Proof.: From [23, Exercise 1.3 and Lemma 1.4] it follows \(\|F_{m}\|_{L^{1}\left([-\pi,\pi];\mathbb{C}\right)}=1\) and hence using the triangle inequality \(\|V_{m}\|_{L^{1}\left([-\pi,\pi];\mathbb{C}\right)}\leq 3\). The claim then follows using Tonelli's theorem. 

**Definition B.6**.: For any \(s\in\mathbb{N},m\in\mathbb{N}_{0}\) we call a function of the form

\[\mathbb{R}^{s}\to\mathbb{C},\quad x\mapsto\sum_{-m\leq\mathbf{k}\leq m}a_{ \mathbf{k}}e^{i\langle\mathbf{k},x\rangle}\]

with coefficients \(a_{\mathbf{k}}\in\mathbb{C}\) a _trigonometric polynomial of coordinatewise degree at most \(m\)_ and denote the space of all those functions with \(\mathbb{H}_{m}^{s}\). Here, we again consider the sum over all \(\mathbf{k}\in\mathbb{Z}^{s}\) with \(-m\leq\mathbf{k}_{j}\leq m\) for all \(j\in\{1,...,s\}\). We then write

\[E_{m}^{s}(f):=\min_{T\in\mathbb{H}_{m}^{s}}\|f-T\|_{L^{\infty}\left(\mathbb{R }^{s}\right)}\]

for any function \(f\in C_{2\pi}\left(\mathbb{R}^{s};\mathbb{C}\right)\).
**Proposition B.7**.: _Let \(s,m\in\mathbb{N}\) and \(k\in\mathbb{N}_{0}\). The map_

\[v_{m}:\quad C_{2\pi}\left(\mathbb{R}^{s};\mathbb{C}\right)\to\mathbb{H}_{2m-1}^{ s},\quad f\mapsto f*V_{m}^{s}\]

_satisfies_

\[v_{m}(T)=T\] (B.1)

_for all \(T\in\mathbb{H}_{m}^{s}\). Furthermore, there is a constant \(c=c(s)>0\) (independent of \(m\)), such that_

\[\left\|v_{m}(f)\right\|_{C^{k}(\mathbb{R}^{s};\mathbb{C})}\leq c \cdot\left\|f\right\|_{C^{k}(\mathbb{R}^{s};\mathbb{C})}\ \forall f\in C_{2\pi}^{k}\left(\mathbb{R}^{s};\mathbb{C}\right),\] \[\left\|v_{m}(f)\right\|_{L^{\infty}(\mathbb{R}^{s};\mathbb{C})} \leq c\cdot\left\|f\right\|_{L^{\infty}(\mathbb{R}^{s};\mathbb{C})}\ \forall f\in C_{2\pi}\left(\mathbb{R}^{s};\mathbb{C}\right).\] (B.2)

Proof.: The map is well-defined since \(V_{m}\) is a trigonometric polynomial of coordinatewise degree at most \(2m-1\).

The operator is bounded with norm at most \(c=3^{s}\) as follows from Young's inequality [23, Lemma 1.1 (ii)], Proposition B.5 and the fact that one has for all \(\mathbf{k}\in\mathbb{N}_{0}^{s}\) with \(\left|\mathbf{k}\right|\leq k\) the identity

\[\partial^{\mathbf{k}}\left(f*V_{m}^{s}\right)=\left(\partial^{\mathbf{k}}f \right)*V_{m}^{s}\quad\text{for }f\in C_{2\pi}^{k}(\mathbb{R}^{s};\mathbb{C}).\]

It remains to show that \(v_{m}\) is the identity on \(\mathbb{H}_{m}^{s}\). We first prove that

\[e_{k}*V_{m}=e_{k}\] (B.3)

holds for all \(k\in\mathbb{Z}\) with \(\left|k\right|\leq m\). First note that

\[e_{k}*V_{m}=e_{k}*F_{m}+e_{k}*\left(e_{m}\cdot F_{m}\right)+e_{k}*\left(e_{-m} \cdot F_{m}\right).\]

We then compute

\[e_{k}*F_{m}=\frac{1}{m}\sum_{\ell=0}^{m-1}D_{\ell}*e_{k}=\frac{1}{m}\sum_{\ell =0}^{m-1}\sum_{h=-\ell}^{\ell}\underbrace{e_{h}*e_{k}}_{=\delta_{k,h+\cdot e_{ k}}}=\frac{1}{m}\sum_{\ell=\left|k\right|}^{m-1}e_{k}=\frac{m-\left|k\right|}{m} \cdot e_{k}.\]

Similarly, we get

\[e_{k}*\left(e_{m}\cdot F_{m}\right)=\frac{1}{m}\sum_{\ell=0}^{m-1}\left(e_{m} D_{\ell}\right)*e_{k}=\frac{1}{m}\sum_{\ell=0}^{m-1}\sum_{h=-\ell}^{\ell} \underbrace{e_{h+m}*e_{k}}_{=\delta_{k,h+m\cdot e_{k}}}=\frac{1}{m}\sum_{ \begin{subarray}{c}0\leq\ell\leq m-1\\ \ell\geq m-k\end{subarray}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!
**Proposition B.8**.: _Let \(s\in\mathbb{N}\), \(k\in\mathbb{N}_{0}\). Then there is a constant \(c=c(s,k)>0\), such that_

\[E_{m}^{s}(f)\leq\frac{c}{m^{k}}\cdot\|f\|_{C^{k}\left([-\pi,\pi]^{s};\mathbb{R }\right)}\]

_for all \(m\in\mathbb{N}\) and \(f\in C_{2\pi}^{k}\left(\mathbb{R}^{s};\mathbb{C}\right)\)._

Proof.: We apply [19, p. 87, Theorem 6] with \(n_{i}=m\) and \(p_{i}=k\) which yields the existence of a constant \(c_{1}=c_{1}(s,k)>0\), such that

\[E_{m}^{s}(f)\leq c_{1}\cdot\sum_{\ell=1}^{s}\frac{1}{m^{k}}\cdot\omega_{\ell} \left(\frac{1}{m}\right)\]

for all \(m\in\mathbb{N}\) and \(f\in C_{2\pi}^{k}(\mathbb{R}^{s};\mathbb{R})\), where \(\omega_{\ell}\) denotes the modulus of continuity of \(\frac{\partial^{k}f}{\partial x_{\ell}^{k}}\), where we have the trivial bound

\[\omega_{\ell}\left(\frac{1}{m}\right)\leq 2\cdot\|f\|_{C^{k}\left([-\pi,\pi]^{s}; \mathbb{R}\right)}\,.\]

Hence, we get

\[E_{m}^{s}(f)\leq c_{1}\cdot s\cdot 2\cdot\|f\|_{C^{k}\left([-\pi,\pi]^{s}; \mathbb{R}\right)}\,\frac{1}{m^{k}}.\]

so the claim follows by choosing \(c:=2s\cdot c_{1}\). 

**Theorem B.9**.: _Let \(s\in\mathbb{N}\). Then there is a constant \(c=c(s)>0\), such that the operator \(v_{m}\) from Proposition B.7 satisfies_

\[\left\|f-v_{m}(f)\right\|_{L^{\infty}\left(\mathbb{R}^{s}\right)}\leq c\cdot E _{m}^{s}(f)\]

_for any \(m\in\mathbb{N}\) and \(f\in C_{2\pi}\left(\mathbb{R}^{s};\mathbb{C}\right)\)._

Proof.: For any \(T\in\mathbb{H}_{m}^{s}\) one has

\[\left\|f-v_{m}(f)\right\|_{L^{\infty}\left(\mathbb{R}^{s}\right)}\overset{ \eqref{eq:C2}}{\leq}\left\|f-T\right\|_{L^{\infty}\left(\mathbb{R}^{s}\right) }+\left\|v_{m}(T)-v_{m}(f)\right\|_{L^{\infty}\left(\mathbb{R}^{s}\right)} \overset{\eqref{eq:C2}}{\leq}\left(c+1\right)\left\|f-T\right\|_{L^{\infty} \left(\mathbb{R}^{s}\right)}.\qed\]

By combining Proposition B.8 and Theorem B.9, we get the following bound.

**Corollary B.10**.: _Let \(s\in\mathbb{N},k\in\mathbb{N}_{0}\). Then there is a constant \(c=c(s,k)>0\), such that_

\[\left\|f-v_{m}(f)\right\|_{L^{\infty}\left(\mathbb{R}^{s}\right)}\leq\frac{c} {m^{k}}\cdot\|f\|_{C^{k}\left(\mathbb{R}^{s};\mathbb{R}\right)}\]

_for every \(m\in\mathbb{N}\) and \(f\in C_{2\pi}^{k}\left(\mathbb{R}^{s};\mathbb{R}\right)\)._

**Lemma B.11**.: _Let \(k\in\mathbb{N}_{0}\) and \(s\in\mathbb{N}\). For any function \(f\in C^{k}\left([-1,1]^{s};\mathbb{C}\right)\) we define the corresponding periodic function via_

\[f^{*}:\quad\mathbb{R}^{s}\to\mathbb{C},\quad f^{*}\left(x_{1},...,x_{s}\right) =f(\cos\left(x_{1}\right),...,\cos\left(x_{s}\right))\]

_and note \(f^{*}\in C_{2\pi}^{k}\left(\mathbb{R}^{s};\mathbb{C}\right)\). The map_

\[C^{k}\left([-1,1]^{s};\mathbb{C}\right)\to C_{2\pi}^{k}\left(\mathbb{R}^{s}; \mathbb{C}\right),\quad f\mapsto f^{*}\]

_is a continuous linear operator with respect to the \(C^{k}\)-norms on \(C^{k}\left([-1,1]^{s};\mathbb{C}\right)\) and \(C_{2\pi}^{k}\left(\mathbb{R}^{s};\mathbb{C}\right)\)._

Proof.: The map is well-defined since \(\cos\) is a smooth function and \(2\pi\)-periodic. The linearity of the operator follows directly so it remains to show its continuity.

The goal is to apply the closed graph theorem [13, Theorem 5.12]. By definition, we have the equality \(\|f\|_{L^{\infty}\left([-1,1]^{s};\mathbb{C}\right)}=\|f^{*}\|_{L^{\infty}\left( [-\pi,\pi]^{s};\mathbb{C}\right)}\). Let then \(\left(f_{n}\right)_{n\in\mathbb{N}}\) be a sequence of functions \(f_{n}\in C^{k}\left([-1,1]^{s};\mathbb{C}\right)\) and \(g^{*}\in C_{2\pi}^{k}\left(\mathbb{R}^{s};\mathbb{C}\right)\) such that \(f_{n}\to f\) in \(C^{k}\left([-1,1]^{s};\mathbb{C}\right)\) and \(f_{n}^{*}\to g^{*}\) in \(C_{2\pi}^{k}\left(\mathbb{R}^{s};\mathbb{C}\right)\). We then have

\[\|f^{*}-g^{*}\|_{L^{\infty}\left([-\pi,\pi]^{s}\right)} \leq\|f^{*}-f_{n}^{*}\|_{L^{\infty}\left([-\pi,\pi]^{s}\right)}+\| f_{n}^{*}-g^{*}\|_{L^{\infty}\left([-\pi,\pi]^{s}\right)}\] \[\leq\|f-f_{n}\|_{L^{\infty}\left([-1,1]^{s};\mathbb{C}\right)}+\| f_{n}^{*}-g^{*}\|_{L^{\infty}\left([-\pi,\pi]^{s}\right)}\] \[\leq\|f-f_{n}\|_{C^{k}\left([-1,1]^{s};\mathbb{C}\right)}+\|f_{n}^ {*}-g^{*}\|_{C^{k}\left(\mathbb{R}^{s};\mathbb{C}\right)}\to 0\,\,(n\to\infty).\]

It follows \(f^{*}=g^{*}\) and the closed graph theorem yields the desired continuity.
The following identity is a generalization of the well-known product-to-sum formula for \(\cos\).

**Lemma B.12**.: _Let \(s\in\mathbb{N}\). Then it holds for any \(x\in\mathbb{R}^{s}\) that_

\[\prod_{j=1}^{s}\cos(x_{j})=\frac{1}{2^{s}}\sum_{\sigma\in\{-1,1\}^{s}}\cos( \langle\sigma,x\rangle).\]

Proof.: This is an inductive generalization of the product-to-sum formula

\[2\cos(x)\cos(y)=\cos(x-y)+\cos(x+y)\] (B.4)

for \(x,y\in\mathbb{R}\), which can be found for instance in [1, p. 4.3.32]. The case \(s=1\) follows since \(\cos\) is an even function. Assume that the claim holds for a fixed \(s\in\mathbb{N}\) and take \(x\in\mathbb{R}^{s+1}\). Writing \(x^{\prime}=(x_{1},...,x_{s})\) we derive

\[\prod_{j=1}^{s+1}\cos(x_{j}) =\left(\frac{1}{2^{s}}\sum_{\sigma\in\{-1,1\}^{s}}\cos(\langle \sigma,x^{\prime}\rangle)\right)\cdot\cos(x_{s+1})\] \[=\frac{1}{2^{s}}\sum_{\sigma\in\{-1,1\}^{s}}\cos(\langle\sigma,x ^{\prime}\rangle)\cos(x_{s+1})\] \[\stackrel{{\eqref{eq:1}}}{{=}}\frac{1}{2^{s+1}} \sum_{\sigma\in\{-1,1\}^{s}}\left[\cos(\langle\sigma,x^{\prime}\rangle+x_{s+1 })+\cos(\langle\sigma,x^{\prime}\rangle-x_{s+1})\right]\] \[=\frac{1}{2^{s+1}}\sum_{\sigma\in\{-1,1\}^{s+1}}\cos(\langle \sigma,x\rangle),\]

as was to be shown. 

**Proposition B.13**.: _For any \(f\in C^{k}\left([-1,1]^{s};\mathbb{C}\right)\) and \(m\in\mathbb{N}\) the de-la-Vallee-Poussin operator given as \(f\mapsto v_{m}\left(f^{*}\right)\) has a representation_

\[v_{m}\left(f^{*}\right)(x_{1},...,x_{s})=\sum_{\begin{subarray}{c}\mathbf{k}\in \mathbb{N}_{0}^{s}\\ \mathbf{k}\leq 2m-1\end{subarray}}\mathcal{V}_{\mathbf{k}}^{m}(f)\prod_{j=1}^{s}\cos \left(\mathbf{k}_{j}x_{j}\right)\]

_for continuous linear functionals_

\[\mathcal{V}_{\mathbf{k}}^{m}:\ C^{k}\left([-1,1]^{s};\mathbb{C}\right)\to\mathbb{ C}.\]

_Furthermore, if \(f\in C^{k}([-1,1]^{s};\mathbb{R})\) we have \(\mathcal{V}_{\mathbf{k}}^{m}(f)\in\mathbb{R}\) for every \(\mathbf{k}\in\mathbb{N}_{0}^{s}\) with \(\mathbf{k}\leq 2m-1\)._

Proof.: First of all, it is easy to see that \(v_{m}\left(f^{*}\right)\) is even in every variable, which follows directly from the fact that \(f^{*}\) and \(V_{m}^{s}\) are both even in each variable. Furthermore, if we write

\[V_{m}^{s}=\sum_{\begin{subarray}{c}\mathbf{k}\in\mathbb{Z}^{s}\\ -(2m-1)\leq\mathbf{k}\leq 2m-1\end{subarray}}a_{\mathbf{k}}^{m}e_{\mathbf{k}}\]

with appropriately chosen coefficients \(a_{\mathbf{k}}^{m}\in\mathbb{R}\), we easily see

\[v_{m}\left(f^{*}\right)=\sum_{\begin{subarray}{c}\mathbf{k}\in\mathbb{Z}^{s}\\ -(2m-1)\leq\mathbf{k}\leq 2m-1\end{subarray}}a_{\mathbf{k}}^{m}\widehat{f^{*}}(\mathbf{k})e_{ \mathbf{k}}.\]

Using Euler's identity and the fact that \(v_{m}\left(f^{*}\right)\) is an even function, we get the representation

\[v_{m}\left(f^{*}\right)(x)=\sum_{\left|\mathbf{k}\right|\leq 2m-1}a_{\mathbf{k}}^{m} \widehat{f^{*}}(\mathbf{k})\text{cos}(\langle\mathbf{k},x\rangle)\]
for any \(x\in\mathbb{R}^{s}\). Using \(\cdot\) to denote the componentwise product of two vectors of the same size, we see since \(v_{m}\left(f^{*}\right)\) is even in every variable that

\[v_{m}\left(f^{*}\right)\left(x\right) =\frac{1}{2^{s}}\cdot\sum_{\sigma\in\{-1,1\}^{s}}v_{m}\left(f^{*} \right)\left(\sigma\cdot x\right)\] \[=\sum_{\begin{subarray}{c}\mathbf{k}\in\mathbb{Z}^{s}\\ -(2m-1)\leq\mathbf{k}\leq 2m-1\end{subarray}}\left(a_{\mathbf{k}}^{m} \widehat{f^{*}}(\mathbf{k})\frac{1}{2^{s}}\sum_{\sigma\in\{-1,1\}^{s}}\cos( \left\langle\mathbf{k},\sigma\cdot x\right\rangle)\right)\] \[\stackrel{{\text{Lemma B.12}}}{{=}}\sum_{ \begin{subarray}{c}\mathbf{k}\in\mathbb{Z}^{s}\\ -(2m-1)\leq\mathbf{k}\leq 2m-1\end{subarray}}a_{\mathbf{k}}^{m} \widehat{f^{*}}(\mathbf{k})\prod_{j=1}^{s}\cos\left(\mathbf{k}_{j}x_{j}\right)\] \[=\sum_{\begin{subarray}{c}\mathbf{k}\in\mathbb{N}_{0}^{s}\\ \mathbf{k}\leq 2m-1\end{subarray}}2^{\left\|\mathbf{k}\right\|_{0}}a_{ \mathbf{k}}^{m}\widehat{f^{*}}(\mathbf{k})\prod_{j=1}^{s}\cos\left(\mathbf{k} _{j}x_{j}\right)\]

with

\[\left\|\mathbf{k}\right\|_{0}:=\#\big{\{}j\in\{1,...,s\}:\ \mathbf{k}_{j} \neq 0\big{\}}.\]

In the last step we again used that \(\cos\) is an even function and that

\[\widehat{f^{*}}(\mathbf{k})=\widehat{f^{*}}(\sigma\cdot\mathbf{k})\]

for all \(\sigma\in\{-1,1\}^{s}\), which also follows easily since \(f^{*}\) is even in every component. Letting

\[\mathcal{V}_{\mathbf{k}}^{m}(f):=2^{\left\|\mathbf{k}\right\|_{0}}a_{ \mathbf{k}}^{m}\widehat{f^{*}}(\mathbf{k})\]

we have the desired form. The fact that \(\mathcal{V}_{\mathbf{k}}^{m}\) is a continuous linear functional on \(C_{2\pi}^{k}\left([-1,1]^{s};\mathbb{C}\right)\) follows directly since \(f\mapsto\widehat{f^{*}}(\mathbf{k})\) is a continuous linear functional for every \(\mathbf{k}\). If \(f\) is real-valued, so is \(\widehat{f^{*}}(\mathbf{k})\) for every \(\mathbf{k}\in\mathbb{N}_{0}^{s}\) with \(\mathbf{k}\leq 2m-1\), since \(f^{*}\) is real-valued and even in every component. 

**Lemma B.14**.: _Let \(s\in\mathbb{N}\). There is a constant \(c=c(s)>0\), such that the inequality_

\[\sum_{\begin{subarray}{c}\mathbf{k}\in\mathbb{N}_{0}^{s}\\ \mathbf{k}\leq 2m-1\end{subarray}}\left|\mathcal{V}_{\mathbf{k}}^{m}(f) \right|\leq c\cdot m^{s/2}\cdot\left\|f\right\|_{L^{\infty}\left([-1,1]^{s}; \mathbb{C}\right)}\]

_holds for any \(m\in\mathbb{N}\) and \(f\in C\left([-1,1]^{s};\mathbb{C}\right)\), where \(\mathcal{V}_{\mathbf{k}}^{m}\) is as in Proposition B.13._

Proof.: Let \(f\in C\left([-1,1]^{s};\mathbb{C}\right)\) and \(m\in\mathbb{N}\). For any multi-index \(\boldsymbol{\ell}\in\mathbb{N}_{0}^{s}\) one has

\[\widehat{v_{m}\left(f^{*}\right)}(\boldsymbol{\ell})=\sum_{ \begin{subarray}{c}\mathbf{k}\in\mathbb{N}_{0}^{s}\\ \mathbf{k}\leq 2m-1\end{subarray}}\mathcal{V}_{\mathbf{k}}^{m}(f)\widehat{g_{ \mathbf{k}}}(\boldsymbol{\ell}),\]

with

\[g_{\mathbf{k}}:\quad\mathbb{R}^{s}\rightarrow\mathbb{R},\quad \left(x_{1},...,x_{s}\right)\mapsto\prod_{j=1}^{s}\cos\left(\mathbf{k}_{j}x_{j }\right).\]

Now, a calculation using Fubini's theorem and using \(g_{k}=\frac{1}{2}\left(e_{k}+e_{-k}\right)\) for any number \(k\in\mathbb{N}_{0}\) shows

\[\widehat{g_{\mathbf{k}}}(\boldsymbol{\ell})=\begin{cases}\frac{1}{2^{\left\| \mathbf{k}\right\|_{0}}},&\mathbf{k}=\boldsymbol{\ell},\\ 0,&\text{otherwise}\end{cases}\quad\text{ for }\mathbf{k},\boldsymbol{\ell}\in \mathbb{N}_{0}^{s}.\]
Therefore, we have the bound \(|\mathcal{V}_{\mathbf{\ell}}^{m}(f)|\leq 2^{s}\cdot\left|\widehat{v_{m}\left(f^{s} \right)}(\mathbf{\ell})\right|\) for \(\mathbf{\ell}\in\mathbb{N}_{0}^{s}\) with \(|\mathbf{\ell}|\leq 2m-1\). Using Cauchy-Schwarz' and Parseval's inequality one sees

\[\sum_{\begin{subarray}{c}\mathbf{k}\in\mathbb{N}_{0}^{s}\\ \mathbf{k}\leq 2m-1\end{subarray}}|\mathcal{V}_{\mathbf{k}}^{m}(f)| \leq 2^{s}\cdot\sum_{\begin{subarray}{c}\mathbf{k}\in\mathbb{N}_{0}^{s} \\ \mathbf{k}\leq 2m-1\end{subarray}}\left|\widehat{v_{m}\left(f^{s}\right)}(\mathbf{k} )\right|\overset{\mathrm{CS}}{\leq 2}2^{s}\cdot(2m)^{s/2}\cdot\left(\sum_{ \begin{subarray}{c}\mathbf{k}\in\mathbb{N}_{0}^{s}\\ \mathbf{k}\leq 2m-1\end{subarray}}\left|\widehat{v_{m}\left(f^{s}\right)}(\mathbf{k} )\right|^{2}\right)^{1/2}\] \[\overset{\mathrm{Parseval}}{\leq 2^{s}\cdot 2^{s/2}\cdot m^{s/2}\cdot\|v_{m }\left(f^{s}\right)\|_{L^{2}\left(\left[-\pi,\pi\right]^{s};\mathbb{C}\right)}}\] \[\leq \underbrace{2^{s}\cdot 2^{s/2}}_{=:c_{1}(s)}\cdot m^{s/2} \cdot\|v_{m}\left(f^{s}\right)\|_{L^{\infty}\left(\left[-\pi,\pi\right]^{s}; \mathbb{C}\right)}\,.\]

Using Proposition B.7 we get

\[\sum_{\begin{subarray}{c}\mathbf{k}\in\mathbb{N}_{0}^{s}\\ \mathbf{k}\leq 2m-1\end{subarray}}|\mathcal{V}_{\mathbf{k}}^{m}(f)|\leq c_{1}(s) \cdot c_{2}(s)\cdot m^{s/2}\cdot\|f^{*}\|_{L^{\infty}\left(\left[-\pi,\pi \right]^{s}\right)}=c(s)\cdot m^{s/2}\cdot\|f\|_{L^{\infty}\left(\left[-1,1 \right]^{s};\mathbb{C}\right)}\,,\]

as claimed. 

For any natural number \(\ell\in\mathbb{N}_{0}\), we denote by \(T_{\ell}\) the \(\ell\)-th _Chebyshev polynomial_, satisfying

\[T_{\ell}\left(\cos(x)\right)=\cos(\ell x),\quad x\in\mathbb{R}.\]

For a multi-index \(\mathbf{k}\in\mathbb{N}_{0}^{s}\), we define

\[T_{\mathbf{k}}(x):=\prod_{j=1}^{s}T_{\mathbf{k}_{j}}\left(x_{j}\right),\quad x\in[-1, 1]^{s}.\]

We then get the following approximation result.

**Theorem B.15**.: _Let \(s,m\in\mathbb{N}\), \(k\in\mathbb{N}_{0}\). Then there is a constant \(c=c(s,k)>0\) with the following property: For any \(f\in C^{k}\left([-1,1]^{s};\mathbb{R}\right)\) the polynomial \(P\) defined as_

\[P(x):=\sum_{\begin{subarray}{c}\mathbf{k}\in\mathbb{N}_{0}^{s}\\ \mathbf{k}\leq 2m-1\end{subarray}}\mathcal{V}_{\mathbf{k}}^{m}(f)\cdot T_{\mathbf{k}}(x)\]

_with \(\mathcal{V}_{\mathbf{k}}^{m}\) as in Proposition B.13 satisfies_

\[\left\|f-P\right\|_{L^{\infty}\left([-1,1]^{s};\mathbb{R}\right)}\leq\frac{c} {m^{k}}\cdot\left\|f\right\|_{C^{k}\left([-1,1]^{s};\mathbb{R}\right)}.\]

_Here, the maps_

\[C\left([-1,1]^{s};\mathbb{R}\right)\to\mathbb{R},\quad f\mapsto\mathcal{V}_{ \mathbf{k}}^{m}(f)\]

_are continuous and linear functionals with respect to the \(L^{\infty}\)-norm. Furthermore, there is a constant \(\widetilde{c}=\widetilde{c}(s)>0\), such that the inequality_

\[\sum_{\begin{subarray}{c}\mathbf{k}\in\mathbb{N}_{0}^{s}\\ \mathbf{k}\leq 2m-1\end{subarray}}|\mathcal{V}_{\mathbf{k}}^{m}(f)|\leq\widetilde{c} \cdot m^{s/2}\cdot\left\|f\right\|_{L^{\infty}\left([-1,1]^{s}\right)}\]

_holds for any \(f\in C\left([-1,1]^{s};\mathbb{R}\right)\)._

Proof.: We choose the constant \(c_{0}=c_{0}(s,k)\) according to Corollary B.10. Let \(f\in C^{k}\left([-1,1]^{s};\mathbb{R}\right)\) be arbitrary. Then we define the corresponding function \(f^{*}\in C_{2\pi}^{k}\left(\mathbb{R}^{s};\mathbb{R}\right)\) as above. Let \(P\) be defined as in the statement of the theorem. Then the property

\[P^{*}(x)=v_{m}\left(f^{*}\right)(x)\]
is satisfied, where \(P^{*}\) is the corresponding function to \(P\) defined similarly to \(f^{*}\). Overall we get the bound

\[\left\|f-P\right\|_{L^{\infty}([-1,1]^{s};\mathbb{R})}=\left\|f^{*}-P^{*}\right\| _{L^{\infty}([-\pi,\pi]^{s};\mathbb{R})}\stackrel{{\text{Cor. B.10}}}{{\leq}}\frac{c_{0}}{m^{k}}\cdot\left\|f^{*} \right\|_{C^{k}(\mathbb{R}^{s};\mathbb{R})}.\]

The first claim then follows using the continuity of the map \(f\mapsto f^{*}\) as proven in Lemma B.11. The second part of the Theorem has already been proven in Lemma B.14. 

## Appendix C Prerequisites for the optimality result

In this subsection we prove a very general lower bound for the approximation of functions in \(C^{k}([-1,1]^{s};\mathbb{R})\) using a subset of \(C([-1,1]^{s};\mathbb{R})\) that can be parametrized using a certain amount of parameters. Precisely, we prove a lower bound of \(m^{-k/s}\) where \(m\) is the number of parameters that are used, provided that the selection of the parameters, i.e., the map that maps a function in \(C^{k}([-1,1]^{s};\mathbb{R})\) to the parameters of the approximating function, is continuous with respect to some norm on \(C^{k}([-1,1]^{s};\mathbb{R})\). The proofs are in fact almost identical to what is done in [10]. However, we decided to include a detailed proof in this paper, since [10] considers Sobolev functions and not \(C^{k}\)-functions and since the continuity assumption in [10] is not completely clear.

**Proposition C.1** ([10, Theorem 3.1]).: _Let \((X,\|\cdot\|_{X})\) be a normed space, \(\varnothing\neq K\subseteq X\) a subset and \(V\subseteq X\) a linear, not necessarily closed subspace of \(X\) containing K. Let \(m\in\mathbb{N}\), \(\overline{a}:K\to\mathbb{R}^{m}\) be a map which is continuous with respect to some norm \(\|\cdot\|_{V}\) on \(V\) and \(M_{m}:\mathbb{R}^{m}\to X\) some arbitrary map. Let_

\[b_{m}(K)_{X}\mathrel{\mathop{:}}=\sup_{X_{m+1}}\sup\left\{\varrho\geq 0:\ U_{ \varrho}(X_{m+1})\subseteq K\right\},\]

_where the first supremum is taken over all \((m+1)\)-dimensional linear subspaces \(X_{m+1}\) of \(X\) and_

\[U_{\varrho}(X_{m+1})\mathrel{\mathop{:}}=\{y\in X_{m+1}:\ \|y\|_{X}\leq \varrho\}.\]

_Then it holds_

\[\sup_{x\in K}\|x-M_{m}(\overline{a}(x))\|_{X}\geq b_{m}(K)_{X}.\]

Proof.: The claim is trivial if \(b_{m}(K)_{X}=0\). Thus, assume \(b_{m}(K)_{X}>0\). Let \(0<\varrho\leq b_{m}(K)_{X}\) be any number such that there exists an \((m+1)\)-dimensional subspace \(X_{m+1}\) of \(X\) with \(U_{\varrho}(X_{m+1})\subseteq K\). It follows \(U_{\varrho}(X_{m+1})\subseteq V\), hence \(X_{m+1}\subseteq V\), so \(\|\cdot\|_{V}\) defines a norm on \(X_{m+1}\). Thus, the restriction of \(\overline{a}\) to \(\partial U_{\varrho}(X_{m+1})\) is a continuous mapping to \(\mathbb{R}^{m}\) with respect to \(\|\cdot\|_{V}\). Since all norms are equivalent on the finite-dimensional space \(X_{m+1}\), the Borsuk-Ulam-Theorem [9, Corollary 4.2] yields the existence of a point \(x_{0}\in\partial U_{\varrho}(X_{m+1})\) with \(\overline{a}(x_{0})=\overline{a}(-x_{0})\). We then see

\[2\varrho =2\|x_{0}\|_{X}=\|x_{0}-M_{m}(\overline{a}(x_{0}))\|_{X}+\|x_{0} +M_{m}(\overline{a}(-x_{0}))\|_{X}\] \[\leq\|x_{0}-M_{m}(\overline{a}(x_{0}))\|_{X}+\|-x_{0}-M_{m}( \overline{a}(-x_{0}))\|_{X},\]

and hence, at least one of the two summands on the right has to be larger than or equal to \(\varrho\). 

Using this very general result we can deduce our lower bound in the context of \(C^{k}\)-spaces.

**Theorem C.2**.: _Let \(s,k\in\mathbb{N}\). Then there exists a constant \(c=c(s,k)>0\) with the following property: For any \(m\in\mathbb{N}\) and any map \(\overline{a}:C^{k}([-1,1]^{s};\mathbb{R})\to\mathbb{R}^{m}\) that is continuous with respect to some norm on \(C^{k}([-1,1]^{s};\mathbb{R})\) and any map \(M_{m}:\mathbb{R}^{m}\to C([-1,1]^{s};\mathbb{R})\) we have_

\[\sup_{\begin{subarray}{c}f\in C^{k}([-1,1]^{s};\mathbb{R})\\ \|f\|_{C^{k}([-1,1]^{s};\mathbb{R})}\leq 1\end{subarray}}\|f-M_{m}( \overline{a}(f))\|_{L^{\infty}([-1,1]^{s};\mathbb{R})}\geq c\cdot m^{-k/s}.\]

Proof.: The idea is to apply Proposition C.1 to the spaces \(X\mathrel{\mathop{:}}=C([-1,1]^{s};\mathbb{R})\), \(V\mathrel{\mathop{:}}=C^{k}([-1,1]^{s};\mathbb{R})\) and the set \(K\mathrel{\mathop{:}}=\{f\in C^{k}([-1,1]^{s};\mathbb{R}):\ \|f\|_{C^{k}([-1,1]^{s};\mathbb{R})}\leq 1\}\).

Assume in the beginning that \(m=n^{s}\) with an integer \(n>1\). Pick \(\phi\in C^{\infty}(\mathbb{R}^{s})\) with \(\phi\equiv 1\) on \([-3/4,3/4]^{s}\) and \(\phi\equiv 0\) outside of \([-1,1]^{s}\). Fix \(c_{0}=c_{0}(s,k)>0\) with

\[1\leq\|\phi\|_{C^{k}([-1,1]^{s};\mathbb{R})}\leq c_{0}.\]
Let \(Q_{1},...,Q_{m}\) be the partition (disjoint up to null-sets) of \([-1,1]^{s}\) into closed cubes of sidelength \(2/n\). For every \(j\in\{1,...,m\}\) write \(Q_{j}=\mathsf{\times}_{\ell=1}^{s}[a_{\ell}-1/n,a_{\ell}+1/n]\) with some vector \(a=(a_{1},...,a_{s})\in[-1,1]^{s}\) and let

\[\phi_{j}(x):=\phi(nx-na)\text{ for }x\in\mathbb{R}^{s}.\]

By choice of \(\phi\), the maps \(\phi_{j}\) are supported on a proper subset of \(Q_{j}\) for every \(j\in\{1,...,m\}\) and an inductive argument shows

\[\partial^{\mathbf{k}}\phi_{j}(x)=n^{|\mathbf{k}|}\cdot\partial^{\mathbf{k}} \phi(nx-na)\text{ for every }\mathbf{k}\in\mathbb{N}_{0}^{s}\text{ and }x\in\mathbb{R}^{s}\]

and hence in particular

\[\|\phi_{j}\|_{C^{\mathbf{k}}([-1,1]^{s};\mathbb{R})}\leq n^{|\mathbf{k}|} \cdot c_{0}.\] (C.1)

Let \(X_{m}:=\operatorname{span}\{\phi_{1},...,\phi_{m}\}\) and \(S\in U(X_{m})\). Then we can write \(S\) in the form \(S=\sum_{j=1}^{m}c_{j}\phi_{j}\) with real numbers \(c_{1},...,c_{m}\in\mathbb{R}\). Suppose there is \(j^{*}\in\{1,...,m\}\) with \(|c_{j^{*}}|>1\). Then we have

\[\|S\|_{L^{\infty}([-1,1]^{s};\mathbb{R})}\geq\underbrace{|c_{j^{*}}|}_{>1} \cdot\underbrace{\|\phi_{j^{*}}\|_{L^{\infty}([-1,1]^{s};\mathbb{R})}}_{=1}>1,\]

since the functions \(\phi_{j}\) have disjoint support. This is a contradiction to \(S\in U(X_{m})\) and we can thus infer that \(\max\limits_{j}|c_{j}|\leq 1\). Furthermore, we see again because the functions \(\phi_{j}\) have disjoint support that

\[\|\partial^{\mathbf{k}}S\|_{L^{\infty}([-1,1]^{s};\mathbb{R})}=\max\limits_{j }\ |c_{j}|\cdot\|\partial^{\mathbf{k}}\phi_{j}\|_{L^{\infty}([-1,1]^{s}; \mathbb{R})}\stackrel{{\eqref{eq:c_1}}}{{\leq}}n^{|\mathbf{k}|} \cdot c_{0}\leq c_{0}\cdot n^{k}=c_{0}\cdot m^{k/s}\]

for every \(\mathbf{k}\in\mathbb{N}_{0}^{s}\) with \(|\mathbf{k}|\leq k\) and hence

\[\|S\|_{C^{k}([-1,1]^{s};\mathbb{R})}\leq c_{0}\cdot m^{k/s}.\]

Thus, letting \(\varrho:=c_{0}^{-1}\cdot m^{-k/s}\) yields \(U_{\varrho}(X_{m})\subseteq K\), so we see by Proposition C.1 that

\[\sup\limits_{f\in K}\|f-M_{m-1}(\overline{a}(f))\|_{L^{\infty}([-1,1]^{s}; \mathbb{R})}\geq\varrho=c_{1}\cdot m^{-k/s}\]

with \(c_{1}=c_{0}^{-1}\) for every map \(\overline{a}:X\to\mathbb{R}^{m-1}\) which is continuous with respect to some norm on \(V\) and any map \(M_{m-1}:\mathbb{R}^{m-1}\to X\). Using the inequality \(m\leq 2(m-1)\) (note \(m>1\)) we get

\[\sup\limits_{f\in K}\|f-M_{m-1}(\overline{a}(f))\|_{L^{\infty}([-1,1]^{s}; \mathbb{R})}\geq c_{1}\cdot m^{-k/s}\geq c_{1}\cdot(2(m-1))^{-k/s}\geq c_{2} \cdot(m-1)^{-k/s}\]

with \(c_{2}=c_{1}\cdot 2^{-k/s}\). Hence, the claim has been shown for all numbers \(m\) of the form \(n^{s}-1\) with an integer \(n>1\).

In the end, let \(m\in\mathbb{N}\) be arbitrary and pick \(n\in\mathbb{N}\) with \(n^{s}\leq m<(n+1)^{s}\). For given maps \(\overline{a}:V\to\mathbb{R}^{m}\) and \(M_{m}:\mathbb{R}^{m}\to X\) with \(\overline{a}\) continuous with respect to some norm on \(V\), let

\[\widetilde{a}:\quad V\to\mathbb{R}^{(n+1)^{s}-1},\quad f\mapsto(\overline{a} (f),0)\quad\text{and}\quad M_{(n+1)^{s}-1}:\quad\mathbb{R}^{(n+1)^{s}-1}\to X,\quad(x,y)\mapsto M_{m}(x),\]

where \(x\in\mathbb{R}^{m},\ y\in\mathbb{R}^{(n+1)^{s}-1-m}\). Then we get

\[\sup\limits_{f\in K}\|f-M_{m}(\overline{a}(f))\|_{L^{\infty}([-1,1 ]^{s};\mathbb{R})} =\sup\limits_{f\in K}\|f-M_{(n+1)^{s}-1}(\widetilde{a}(f))\|_{L^{ \infty}([-1,1]^{s};\mathbb{R})}\geq c_{2}\cdot((n+1)^{s}-1)^{-k/s}\] \[\geq c_{2}\cdot(2^{s}n^{s})^{-k/s}\geq c_{3}\cdot m^{-k/s}\]

with \(c_{3}=c_{2}\cdot 2^{-k}\). Here we used the bound \((n+1)^{s}-1\leq(2n)^{s}\). This proves the full claim.
## Appendix D Approximation using Ridge Functions

In this section we prove for \(s\in\mathbb{N}^{\geq 2}\) that every function in \(C^{k}([-1,1]^{s};\mathbb{R})\) can be uniformly approximated with an error of the order \(m^{-k/(s-1)}\) using a linear combination of \(m\) so-called _ridge functions_. In fact, we only consider ridge _polynomials_, meaning functions of the form

\[\mathbb{R}^{s}\to\mathbb{R},\quad x\mapsto p(a^{T}x)\]

for a fixed vector \(a\in\mathbb{R}^{s}\) and a polynomial \(p:\mathbb{R}\to\mathbb{R}\). Note that this result has already been obtained in a slightly different form in [20, Theorem 1.1]; namely, it is shown there that the rate of approximation \(m^{-k/(s-1)}\) can be achieved by functions of the form \(\sum_{j=1}^{m}f_{j}(a_{j}^{T}x)\) with \(a_{j}\in\mathbb{R}^{s}\) and \(f_{j}\in L^{1}_{\text{loc}}(\mathbb{R}^{s})\). We will need the fact that the \(f_{j}\) can actually be chosen as polynomials and that the vectors \(a_{1},...,a_{m}\) can be chosen independently from the particular function \(f\). This is shown in the proof of [20], but not stated explicitly. For this reason, and in order to clarify the proof itself and to make the paper more self-contained, we decided to present the proof in this appendix.

**Lemma D.1**.: _Let \(m,s\in\mathbb{N}\). Then we denote_

\[P_{m}^{s}:=\left\{\mathbb{R}^{s}\to\mathbb{R},\quad x\mapsto\sum_{ \begin{subarray}{c}\mathbf{k}\in\mathbb{N}_{0}^{s}\\ |\mathbf{k}|\leq m\end{subarray}}a_{\mathbf{k}}x^{\mathbf{k}}:\ a_{\mathbf{k}}\in\mathbb{R}\right\}\]

_as the set of real polynomials of degree at most \(m\). The subset of homogeneous polynomials of degree \(m\) is defined as_

\[H_{m}^{s}:=\left\{\mathbb{R}^{s}\to\mathbb{R},\quad x\mapsto\sum_{ \begin{subarray}{c}\mathbf{k}\in\mathbb{N}_{0}^{s}\\ |\mathbf{k}|=m\end{subarray}}a_{\mathbf{k}}x^{\mathbf{k}}:\ a_{\mathbf{k}}\in\mathbb{R}\right\}.\]

_There is a constant \(c=c(s)>0\) satisfying_

\[\dim(H_{m}^{s})\leq c\cdot m^{s-1}\quad\forall m\in\mathbb{N}.\]

Proof.: It is immediate that the set

\[\left\{\mathbb{R}^{s}\to\mathbb{R},\ x\mapsto x^{\mathbf{k}}:\ \mathbf{k}\in\mathbb{N}_{0}^{s},\ |\mathbf{k}|=m\right\}\]

forms a basis of \(H_{m}^{s}\), hence

\[\dim(H_{m}^{s})=\#\left\{\mathbf{k}\in\mathbb{N}_{0}^{s}:\ |\mathbf{k}|=m\right\}.\]

This quantity clearly equals the number of possibilities for drawing \(m\) times from a set with \(s\) elements with replacement. Hence, we see

\[\dim(H_{m}^{s})=\binom{s+m-1}{m},\]

see for instance [7, Identity 143]. A further estimation shows

\[\binom{s+m-1}{m}=\prod_{j=1}^{s-1}\frac{m+j}{j}=\prod_{j=1}^{s-1}\left(1+\frac {m}{j}\right)\leq(1+m)^{s-1}\leq 2^{s-1}\cdot m^{s-1}.\]

Hence, the claim follows with \(c(s)=2^{s-1}\). 

A combination of results from [25] together with the fact that it is possible to approximate \(C^{k}\)-functions using polynomials of degree at most \(m\) with an error of the order \(m^{-k}\), as shown in Theorem B.15, yields the desired result.
**Theorem D.2**.: _Let \(s,k\in\mathbb{N}\) with \(s\geq 2\) and \(r>0\). Then there exists a constant \(c=c(s,k)>0\) with the following property: For every \(m\in\mathbb{N}\) there are \(a_{1},...,a_{m}\in\mathbb{R}^{s}\setminus\{0\}\) with \(\|a_{j}\|=r\), such that for every function \(f\in C^{k}([-1,1]^{s};\mathbb{R})\) there are polynomials \(p_{1},...,p_{m}\in P_{m}^{1}\) satisfying_

\[\left\|f(x)-\sum_{j=1}^{m}p_{j}(a_{j}^{T}x)\right\|_{L^{\infty}([-1,1]^{s}; \mathbb{R})}\leq c\cdot m^{-k/(s-1)}\cdot\|f\|_{C^{k}([-1,1]^{s};\mathbb{R})}.\]

Proof.: We first pick the constant \(c_{1}=c_{1}(s)\) according to Lemma D.1. Then we define the constant \(c_{2}=c_{2}(s):=(2s)^{s-1}\cdot c_{1}(s)\) and let \(M\in\mathbb{N}\) be the largest integer satisfying

\[c_{2}\cdot M^{s-1}\leq m.\]

Here, we assume without loss of generality that \(m\geq c_{2}\), which can be justified by choosing \(p_{j}=0\) for every \(j\in\{1,...,m\}\) if \(m<c_{2}\), at the cost of possibly enlarging \(c\). Note that the choice of \(M\) implies \(c_{2}\cdot(2M)^{s-1}\geq c_{2}\cdot(M+1)^{s-1}>m\) and thus

\[M\geq\frac{1}{2}\cdot c_{2}^{-1/(s-1)}\cdot m^{1/(s-1)}=c_{3}\cdot m^{1/(s-1)}\] (D.1)

with \(c_{3}=c_{3}(s):=1/2\cdot c_{2}^{-1/(s-1)}\).

Using [25, Proposition 5.9] and Lemma D.1 we can pick \(a_{1},...,a_{m}\in\mathbb{R}^{s}\setminus\{0\}\) satisfying

\[H_{s(2M-1)}^{s}=\operatorname{span}\left\{x\mapsto(a_{j}^{T}x)^{s(2M-1)}:\ j \in\{1,...,m\}\right\},\] (D.2)

where we used that

\[c_{1}\cdot(s(2M-1))^{s-1}\leq c_{1}\cdot(2s)^{s-1}\cdot M^{s-1}=c_{2}\cdot M^{ s-1}\leq m.\]

Here we can assume \(\|a_{j}\|=r\) for every \(j\in\{1,...,m\}\) since multiplying each \(a_{j}\) with a positive constant does not change the span in (D.2). From [25, Corollary 5.12] we infer that

\[P_{s(2M-1)}^{s}=\operatorname{span}\left\{x\mapsto(a_{j}^{T}x)^{r}:\ j\in\{1,...,m\},\ 0\leq r\leq s(2M-1)\right\}.\] (D.3)

Let \(f\in C^{k}([-1,1]^{s};\mathbb{R})\). Then, according to Theorem B.15, there is a polynomial \(P:\mathbb{R}^{s}\to\mathbb{R}\) of _coordinatewise_ degree at most \(2M-1\) satisfying

\[\|f-P\|_{L^{\infty}([-1,1]^{s};\mathbb{R})}\leq c_{4}\cdot M^{-k}\cdot\|f\|_{ C^{k}([-1,1]^{s};\mathbb{R})},\]

where \(c_{4}=c_{4}(s,k)>0\). Note that by construction it holds \(P\in P_{s(2M-1)}^{s}\). Using (D.3) we deduce the existence of polynomials \(p_{1},...,p_{m}:\mathbb{R}\to\mathbb{R}\) such that

\[P(x)=\sum_{j=1}^{m}p_{j}(a_{j}^{T}x)\quad\text{for all }x\in\mathbb{R}^{s}.\]

Combining the previously shown bounds we get

\[\left\|f(x)-\sum_{j=1}^{m}p_{j}(a_{j}^{T}x)\right\|_{L^{\infty}([-1,1]^{s}; \mathbb{R})} =\|f(x)-P(x)\|_{L^{\infty}([-1,1]^{s};\mathbb{R})}\leq c_{4}\cdot M ^{-k}\cdot\|f\|_{C^{k}([-1,1]^{s};\mathbb{R})}\] \[\stackrel{{\eqref{eq:P_1}}}{{\leq}}c\cdot m^{-k/(s-1 )}\cdot\|f\|_{C^{k}([-1,1]^{s};\mathbb{R})},\]

as desired. Here, we define \(c=c(s,k):=c_{4}\cdot c_{3}^{-k}\). 

**Acknowledgements.** PG acknowledges support by the German Science Foundation (DFG) in the context of the Emmy Noether junior research group VO 2594/1-1. FV is grateful to Hrushikesh Mhaskar for several helpful and interesting discussions.