Universal approximation with complex-valued deep narrow neural networks
We study the universality of complex-valued neural networks with bounded
widths and arbitrary depths. Under mild assumptions, we give a full description
of those activation functions $\varrho:\mathbb{C}\to \mathbb{C}$ that have the
property that their associated networks are universal, i.e., are capable of
approximating continuous functions to arbitrary accuracy on compact domains.
Precisely, we show that deep narrow complex-valued networks are universal if
and only if their activation function is neither holomorphic, nor
antiholomorphic, nor $\mathbb{R}$-affine. This is a much larger class of
functions than in the dual setting of arbitrary width and fixed depth. Unlike
in the real case, the sufficient width differs significantly depending on the
considered activation function. We show that a width of $2n+2m+5$ is always
sufficient and that in general a width of $\max\{2n,2m\}$ is necessary. We
prove, however, that a width of $n+m+4$ suffices for a rich subclass of the
admissible activation functions. Here, $n$ and $m$ denote the input and output
dimensions of the considered networks.
Paul Geuchen, Thomas Jahn, Hannes Matt
# Universal approximation with complex-valued deep narrow neural networks

###### Abstract.

We study the universality of complex-valued neural networks with bounded widths and arbitrary depths. Under mild assumptions, we give a full description of those activation functions \(\varrho:\mathbb{C}\to\mathbb{C}\) that have the property that their associated networks are universal, i.e., are capable of approximating continuous functions to arbitrary accuracy on compact domains. Precisely, we show that deep narrow complex-valued networks are universal if and only if their activation function is neither holomorphic, nor antiholomorphic, nor \(\mathbb{R}\)-affine. This is a much larger class of functions than in the dual setting of arbitrary width and fixed depth. Unlike in the real case, the sufficient width differs significantly depending on the considered activation function. We show that a width of \(2n+2m+5\) is always sufficient and that in general a width of \(\max\{2n,2m\}\) is necessary. We prove, however, that a width of \(n+m+4\) suffices for a rich subclass of the admissible activation functions. Here, \(n\) and \(m\) denote the input and output dimensions of the considered networks.

Key words and phrases:complex-valued neural networks, holomorphic function, polyharmonic function, uniform approximation, universality 2020 Mathematics Subject Classification: 68T07,41A30,41A63,31A30,30E10 All authors contributed equally to this work.

## 1. Introduction

This paper addresses the universality of deep narrow complex-valued neural networks (CVNNs), i.e., the density of neural networks with arbitrarily large depths but bounded widths, in spaces of continuous functions over compact domains with respect to the uniform norm. Our main theorem is as follows.

**Theorem 1.1**.: _Let \(n,m\in\mathbb{N}\), and \(\varrho:\mathbb{C}\to\mathbb{C}\) be a continuous function which at some point is real differentiable with non-vanishing derivative. Then \(\mathcal{N}\mathcal{N}^{\varrho}_{n,m,2n+2m+5}\) is universal if and only if \(\varrho\) is neither holomorphic, nor antiholomorphic, nor \(\mathbb{R}\)-affine._

Here \(\mathcal{N}\mathcal{N}^{\varrho}_{n,m,W}\) denotes the set of complex-valued neural networks with input dimension \(n\), output dimension \(m\), activation function \(\varrho\), and \(W\) neurons per hidden layer. These neural networks are alternating compositions

\[V_{L}\circ\varrho^{\times W}\circ\ldots\circ\varrho^{\times W}\circ V_{1}: \quad\mathbb{C}^{n}\to\mathbb{C}^{m} \tag{1.1}\]

of affine maps \(V_{1}:\mathbb{C}^{n}\to\mathbb{C}^{W}\), \(V_{2},\ldots,V_{L-1}:\mathbb{C}^{W}\to\mathbb{C}^{W}\), \(V_{L}:\mathbb{C}^{W}\to\mathbb{C}^{m}\), and componentwise applications of the activation function \(\varrho\), see Section 2 for a detailed definition.

Studying the expressivity of neural networks is an important part of the mathematical analysis of deep learning. We contribute a qualitative result in that direction. Such qualitative results naturally precede the investigation of approximation rates, i.e., the decay of approximation errors as the class of approximants increases, and the design of numerical algorithms that output near-to-optimal approximants.

### Related work

In the neural network context, universal approximation theorems date back to the 1980s and 1990s [7, 24], where it was shown that real-valued shallow neural networks with output dimension \(1\) and a fixed continuous activation function are universal if and only if the activation function is not a polynomial. Modifications of the setting in which universal approximation is studied appear in the neural network literature over the past decades. These variants of the problem refer to, e.g., the input and the output dimension, the target space (typically \(L_{p}\) for \(1\leq p\leq\infty\), continuous
Introduction

### Background

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\(n\), output dimension \(m\), and \(2n+2m+5\) or even \(n+m+4\) neurons per hidden layer are universal, see Theorem 5.3. This is done by approximating polynomials in \(z\) and \(\overline{z}\) uniformly on compact sets and invoking the Stone-Weierstrass theorem.

### Organization of our paper

In Section 2 we fix our notation and recall some basics from complex analysis, the theory of neural networks, and functional analysis. Section 3 introduces the register model and shows how the identity map on \(\mathbb{C}\) and complex conjugation can be approximated using CVNNs. For non-polyharmonic functions, the proof of Theorem 1.1 can be found in Section 4. In Section 5, we present the proof of universality claimed in Theorem 1.1 in the case of polyharmonic activation functions which are neither holomorphic, nor antiholomorphic, nor \(\mathbb{R}\)-affine. In Section 6, we show that CVNNs whose activation function is holomorphic or antiholomorphic or \(\mathbb{R}\)-affine are never universal, regardless of the number of neurons per hidden layer, and derive a lower bound on the minimum width necessary for universality. In the appendix, we provide basics on the relationship between local uniform convergence and universal approximation, and on Taylor approximations in terms of Wirtinger derivatives.

## 2. Preliminaries

In this section, we recall facts from complex analysis, functional analysis, and the theory of neural networks behind the phrases in Theorem 1.1. The presentation is loosely based on [25, Chapter 7], [26, Chapter 11], and [11, Section 1].

### Complex analysis

We use the symbols \(\mathbb{N}\), \(\mathbb{R}\), and \(\mathbb{C}\) to denote the natural, real, and complex numbers, respectively. By \(\operatorname{Re}(z)\), \(\operatorname{Im}(z)\), and \(\overline{z}\), we denote the componentwise real part, imaginary part, and complex conjugate of a vector \(z\in\mathbb{C}^{n}\), respectively. We call the function \(\varrho:\mathbb{C}\to\mathbb{C}\)_partially differentiable_ at \(z_{0}\), if the _partial derivatives_

\[\frac{\partial\varrho}{\partial x}(z_{0}) :=\lim_{\mathbb{R}\setminus\{0\}\ni h\to 0}\frac{\varrho(z_{0}+h)- \varrho(z_{0})}{h}\] \[\text{and}\]

exist. Higher-order partial derivates are defined in the standard manner. We write \(\varrho\in C^{k}(\mathbb{C};\mathbb{C})\) if \(\varrho\) admits partial derivatives up to order \(k\) at each point of \(\mathbb{C}\) and the \(k\)th-order partial derivatives are

Figure 1. Our results in a nutshell.
continuous functions \(\mathbb{C}\to\mathbb{C}\). Likewise, we write \(\varrho\in C^{\infty}(\mathbb{C};\mathbb{C})\) if \(\varrho\in C^{k}(\mathbb{C};\mathbb{C})\) for all \(k\in\mathbb{N}\). If \(\frac{\partial\varrho}{\partial x}(z_{0})\) and \(\frac{\partial\varrho}{\partial y}(z_{0})\) exist and the identity

\[\lim_{\mathbb{C}\setminus\{0\}\ni h\to 0}\frac{\varrho(z_{0}+h)-\varrho(z_{0})- \frac{\partial\varrho}{\partial x}(z_{0})\operatorname{Re}(h)-\frac{\partial \varrho}{\partial y}(z_{0})\operatorname{Im}(h)}{h}=0 \tag{2.1}\]

holds true, then \(\varrho\) is called _real differentiable in \(z_{0}\)_ with derivative \((\frac{\partial\varrho}{\partial x}(z_{0}),\frac{\partial\varrho}{\partial y}( z_{0}))\). Similarly, \(\varrho\) is called _complex differentiable in \(z_{0}\)_ if

\[\lim_{\mathbb{C}\setminus\{0\}\ni h\to 0}\frac{\varrho(z_{0}+h)-\varrho(z_{0})- ch}{h}=0\]

for some number \(c\in\mathbb{C}\), which in that case is given by

\[\partial_{\operatorname{wirt}}\varrho(z_{0}):=\frac{1}{2}\bigg{(}\frac{ \partial\varrho}{\partial x}(z_{0})-\mathrm{i}\frac{\partial\varrho}{\partial y }(z_{0})\bigg{)}\,.\]

Complex differentiability of \(\varrho\) in \(z_{0}\) can be equivalently stated as

\[\overline{\partial}_{\operatorname{wirt}}\varrho(z_{0}):=\frac{1}{2}\bigg{(} \frac{\partial\varrho}{\partial x}(z_{0})+\mathrm{i}\frac{\partial\varrho}{ \partial y}(z_{0})\bigg{)}=0.\]

The differential operators \(\partial_{\operatorname{wirt}}\) and \(\overline{\partial}_{\operatorname{wirt}}\) are called _Wirtinger derivatives_. If \(\overline{\partial}_{\operatorname{wirt}}\varrho(z)=0\) for all \(z\in\mathbb{C}\), then \(\varrho\) is a _holomorphic_ function. The function \(\varrho\) is called _antiholomorphic_ if the function \(\overline{\varrho}:\mathbb{C}\to\mathbb{C}\), \(\overline{\varrho}(z):=\operatorname{Re}(\varrho(z))-\mathrm{i}\operatorname {Im}(\varrho(z))\) is holomorphic or, equivalently, \(\partial_{\operatorname{wirt}}\varrho(z)=0\) for all \(z\in\mathbb{C}\). As the linear operator that maps the partial derivatives onto the Wirtinger derivatives is invertible, it follows that \((\frac{\partial\varrho}{\partial x}(z_{0}),\frac{\partial\varrho}{\partial y }(z_{0}))=(0,0)\) if and only if \((\partial_{\operatorname{wirt}}\varrho(z_{0}),\overline{\partial}_{ \operatorname{wirt}}\varrho(z_{0}))=(0,0)\). Furthermore, the symmetry of mixed partial derivatives implies for \(\varrho\in C^{2}(\mathbb{C};\mathbb{C})\) that

\[4\partial_{\operatorname{wirt}}\overline{\partial}_{\operatorname{wirt}} \varrho=4\overline{\partial}_{\operatorname{wirt}}\partial_{\operatorname{wirt }}\varrho=\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{ \partial y^{2}}\right)\varrho=:\Delta\varrho. \tag{2.2}\]

If \(\varrho\in C^{\infty}(\mathbb{C};\mathbb{C})\) and \(\Delta^{m}\varrho=0\) for some \(m\in\mathbb{N}\), then \(\varrho\) is called _polyharmonic of order \(m\)_. Because of (2.2), holomorphic and antiholomorphic functions are _harmonic_, i.e., polyharmonic of order \(1\).

The following lemma generalizes the classical real-valued Taylor expansion to the complex-valued setting; see Lemma B.1 for a proof.

**Lemma 2.1**.: _Let \(\varrho\in C(\mathbb{C};\mathbb{C})\) and \(z,z_{0}\in\mathbb{C}\). If \(\varrho\) is real differentiable in \(z_{0}\), then_

\[\varrho(z+z_{0})=\varrho(z_{0})+\partial_{\operatorname{wirt}}\varrho(z_{0})z+ \overline{\partial}_{\operatorname{wirt}}\varrho(z_{0})\overline{z}+\Theta_{1} (z) \tag{2.3}\]

_for a function \(\Theta_{1}:\mathbb{C}\to\mathbb{C}\) with \(\lim_{\mathbb{C}\setminus\{0\}\ni z\to 0}\frac{\Theta_{1}(z)}{z}=0\). If \(\varrho\in C^{2}(\mathbb{C};\mathbb{C})\), then_

\[\varrho(z+z_{0})=\varrho(z_{0})+\partial_{\operatorname{wirt}}\varrho(z_{0})z+ \overline{\partial}_{\operatorname{wirt}}\varrho(z_{0})\overline{z}+\frac{1}{2 }\partial_{\operatorname{wirt}}^{2}\varrho(z_{0})z^{2}+\partial_{\operatorname{ wirt}}\overline{\partial}_{\operatorname{wirt}}\varrho(z_{0})z\overline{z}+\frac{1}{2} \overline{\partial}_{\operatorname{wirt}}^{2}\varrho(z_{0})\overline{z}^{2}+ \Theta_{2}(z) \tag{2.4}\]

_for a function \(\Theta_{2}:\mathbb{C}\to\mathbb{C}\) with \(\lim_{\mathbb{C}\setminus\{0\}\ni z\to 0}\frac{\Theta_{2}(z)}{z^{2}}=0\)._

### Neural networks

A (fully connected feed-forward) _complex-valued neural network_ (CVNN) is a function

\[V_{L}\circ\varrho^{\times N_{L-1}}\ldots\circ\varrho^{\times N_{1}}\circ V_{1}: \quad\mathbb{C}^{N_{0}}\to\mathbb{C}^{N_{L}}\]

where

* \(L\in\mathbb{N}_{\geq 2}\) is called the _depth_ of the CVNN,
* \(N_{j}\in\mathbb{N}\) is the _width_ of the \(j\)th layer,
* \(\max\,\{N_{0},\ldots,N_{L}\}\) is the _width_ of the CVNN,
* \(V_{j}:\mathbb{C}^{N_{j-1}}\to\mathbb{C}^{N_{j}}\) is a \(\mathbb{C}\)-affine map, abbreviated as \(V_{j}\in\operatorname{Aff}(\mathbb{C}^{N_{j-1}};\mathbb{C}^{N_{j}})\), i.e., there exist \(A_{j}\in\mathbb{C}^{N_{j}\times N_{j-1}}\) and \(b_{j}\in\mathbb{C}^{N_{j}}\) such that \(V_{j}(z)=A_{j}z+b_{j}\) for all \(z\in\mathbb{C}^{\mathbb{N}_{j-1}}\),
* \(\varrho^{\times N_{j}}(z_{1},\ldots,z_{N_{j}})=(\varrho(z_{1}),\ldots,\varrho(z_{ N_{j}}))\) is the componentwise application of a (potentially non-affine) map \(\varrho:\mathbb{C}\to\mathbb{C}\) called the _activation function_.
We refer to the numbers \(N_{0}\) and \(N_{L}\) as the _input dimension_ and _output dimension_, respectively. The layers \(1,\ldots,L-1\) are the _hidden layers_ of the CVNN. Since it is always possible to pad matrices and vectors by additional zero rows and columnns, we may and will assume without loss of generality that \(N_{1}=N_{2}=\ldots=N_{L-1}\).

We introduce a short-hand notation for the CVNNs that arise this way.

**Definition 2.2**.: Let \(n,m,W\in\mathbb{N}\) and \(\varrho:\mathbb{C}\to\mathbb{C}\). We denote by \(\mathcal{N}\mathcal{N}_{n,m,W}^{\varrho}\) the set of CVNNs with arbitrary depth \(L\), input dimension \(n\), output dimension \(m\), and \(N_{j}=W\) for \(j\in\{1,\ldots,L-1\}\).

The elements of \(\mathcal{N}\mathcal{N}_{n,m,W}^{\varrho}\) are thus alternating compositions

\[V_{L}\circ\varrho^{\times W}\circ\ldots\circ\varrho^{\times W}\circ V_{1}: \quad\mathbb{C}^{n}\to\mathbb{C}^{m} \tag{2.5}\]

of \(\mathbb{C}\)-affine maps \(V_{j}\) and the componentwise applications of the activation function \(\varrho\). In the subsequent sections Sections 4 and 5, we have \(W\geq\max\left\{n,m\right\}\), such that \(W\) turns out to be the width of the neural networks under consideration.

A typical way of thinking about neural networks is viewing the component functions of \(\varrho^{\times N_{j}}\circ V_{j}\) as building blocks called _neurons_. Each neuron performs a computation of the form

\[z\mapsto\varrho(w^{\top}z+b)\]

where \(z\) is the output of the previous layer, \(w\) a vector of _weights_, and \(b\) a number called _bias_.

Since the composition of affine maps is affine, it is also possible to think about neural networks as maps

\[\big{(}\Psi_{L}\circ\varrho^{\times W}\circ\Phi_{L}\big{)}\circ\big{(}\Psi_{L- 1}\circ\varrho^{\times W}\circ\Phi_{L-1}\big{)}\cdots\circ\big{(}\Psi_{1}\circ \varrho^{\times W}\circ\Phi_{1}\big{)},\]

where each of the maps \(\Phi_{k},\Psi_{k}\) is affine. This allows to perceive shallow networks, see Definition 2.3, as building blocks for neural networks. This point of view is similar to the notion of _enhanced neurons_ in [15].

For a fixed activation function \(\varrho\), different choices of the \(\mathbb{C}\)-affine functions \(V_{j}\) may lead to the same composite function (2.5). In view of this, both depth and width of a CVNN are not properties of the function (2.5) but of the tuple \((V_{1},\ldots,V_{L})\). For this reason, a different terminology is sometimes used in the literature where \((V_{1},\ldots,V_{L})\) is called the _neural network_ and (2.5) is its _realization_, cf. [11, Section 1].

Apart from the choice of the activation function \(\varrho\), restrictions on the depth or the width are common ingredients in the analysis of (fully connected feed-forward) neural networks. A CVNN is called _shallow_ if its depth equals \(2\), and _deep_ otherwise.

**Definition 2.3**.: Let \(n,m\in\mathbb{N}\) and \(\varrho:\mathbb{C}\to\mathbb{C}\). We denote by \(\mathcal{N}\mathcal{N}_{n,m}^{\varrho}\) the set of _shallow CVNNs_, i.e., CVNNs with depth \(L=2\), input dimension \(n\), and output dimension \(m\).

In contrast to shallowness, _narrowness_ is not an individual property of CVNNs but a class property. A set \(\mathcal{F}\) of CVNNs is said to be _narrow_ if it does not contain CVNNs of arbitrarily large widths, i.e., if \(\mathcal{F}\subseteq\mathcal{N}\mathcal{N}_{n,m,W}^{\varrho}\) for suitable \(n,m,W\in\mathbb{N}\) and \(\varrho:\mathbb{C}\to\mathbb{C}\).

### Functional analysis

On \(\mathbb{C}^{m}\), consider the topology induced by the Euclidean norm

\[\left\|(z_{1},\ldots,z_{m})\right\|_{\mathbb{C}^{m}}= \left(\sum_{j=1}^{m}\left|z_{j}\right|^{2}\right)^{\frac{1}{2}}.\]

We denote by

\[B_{\delta}(z_{0}):=\{z\in\mathbb{C}^{m}\;:\;\left\|z-z_{0}\right\|_{\mathbb{C} ^{m}}<\delta\}\]

and

\[\overline{B}_{\delta}(z_{0}):=\{z\in\mathbb{C}^{m}\;:\;\left\|z-z_{0}\right\|_ {\mathbb{C}^{m}}\leq\delta\}\]
the open and the closed ball with center \(z_{0}\in\mathbb{C}^{m}\) and radius \(\delta>0\), respectively. For \(K\subseteq\mathbb{C}^{n}\), we denote the vector space of continuous functions \(K\to\mathbb{C}^{m}\) by \(C(K;\mathbb{C}^{m})\). When \(K\subseteq\mathbb{C}^{n}\) is compact, the expression

\[\left\|f\right\|_{C(K;\mathbb{C}^{m})}:=\sup_{z\in K}\left\|f(z)\right\|_{ \mathbb{C}^{m}}\]

defines a norm on \(C(K;\mathbb{C}^{m})\), called the _uniform norm_, which renders \(C(K;\mathbb{C}^{m})\) a Banach space. The convergence of a sequence \((f_{j})_{j\in\mathbb{N}}\) of elements \(f_{j}\in C(K;\mathbb{C}^{m})\) to a limit \(f\in C(K;\mathbb{C}^{m})\) is written as \(f_{j}\to f\)_as \(j\to\infty\), uniformly on \(K\)_. Similarly, a sequence \((f_{j})_{j\in\mathbb{N}}\) of functions \(f_{j}\in C(\mathbb{C}^{n};\mathbb{C}^{m})\) is said to converge _locally uniformly_ to a function \(f\in C(\mathbb{C}^{n};\mathbb{C}^{m})\) if it converges uniformly to \(f\) on every compact subset \(K\subseteq\mathbb{C}^{n}\). Since compositions of continuous functions are continuous, we have \(\mathcal{N}\mathcal{N}^{\varrho}_{n,m,W}\subseteq C(\mathbb{C}^{n};\mathbb{C }^{m})\) when \(\varrho\in C(\mathbb{C};\mathbb{C})\). The main objective of the paper at hand is to show that under certain additional assumptions on \(\varrho\) and \(W\), the elements of \(\mathcal{N}\mathcal{N}^{\varrho}_{n,m,W}\) are arbitrarily close to the elements of \(C(\mathbb{C}^{n};\mathbb{C}^{m})\) in the following sense.

**Definition 2.4**.: We say that a function class \(\mathcal{F}\subseteq C(\mathbb{C}^{n};\mathbb{C}^{m})\) has the _universal approximation property_ or is _universal_) if for every function \(g\in C(\mathbb{C}^{n};\mathbb{C}^{m})\), every compact subset \(K\subseteq\mathbb{C}^{n}\) and every \(\varepsilon>0\) there is a function \(f\in\mathcal{F}\) such that

\[\sup_{z\in K}\left\|f(z)-g(z)\right\|_{\mathbb{C}^{m}}<\varepsilon.\]

The universal approximation property is equivalent to saying that for every function \(g\in C(\mathbb{C}^{n};\mathbb{C}^{m})\) there is a sequence \((f_{j})_{j\in\mathbb{N}}\) with \(f_{j}\in\mathcal{F}\) for every \(j\in\mathbb{N}\) that converges locally uniformly to \(g\). We elaborate this equivalence in Appendix A.

## 3. Building blocks and register model

In this section we introduce various _building blocks_ for complex-valued networks, i.e., small neural network blocks that are able to represent elementary functions (e.g., the complex identity \(\mathrm{id}_{\mathbb{C}}\) or complex conjugation \(\overline{\mathrm{id}_{\mathbb{C}}}\)) up to an arbitrarily small approximation error. These building blocks are used in Sections 4 and 5 to construct the deep narrow networks that we use to approximate a given continuous function. Throughout the chapter, we assume that the used activation function \(\varrho:\mathbb{C}\to\mathbb{C}\) is differentiable (in the real sense) at one point with non-vanishing derivative at that point. In fact, the strategy for constructing these building blocks is always going to be similar: By using the first- and second-order Taylor expansion of the activation function \(\varrho\) as introduced in Lemma 2.1 one can localize the activation function around its point of differentiability where it behaves like a complex polynomial in \(z\) and \(\overline{z}\) of degree \(1\) and \(2\), respectively. This enables us to extract elementary functions from that Taylor expansion.

Proposition 3.1 is fundamental for the universality results introduced in Sections 4 and 5. It shows that it is possible to uniformly approximate the complex identity or complex conjugation on compact sets using neural networks with a single hidden layer and width at most \(2\). If the activation function (or its complex conjugate) is not just real but even complex differentiable, the width can be reduced to \(1\). See Figure 2 for an illustration of the building blocks.

**Proposition 3.1**.: _Let \(\varrho\in C(\mathbb{C};\mathbb{C})\) be real differentiable in \(z_{0}\in\mathbb{C}\) with \((\partial_{\mathrm{wirt}}\varrho(z_{0}),\overline{\partial}_{\mathrm{wirt}} \varrho(z_{0}))\neq(0,0)\). Furthermore, let \(K\subseteq\mathbb{C}\) be compact and \(\varepsilon>0\)._

_(i) If_ \(\partial_{\mathrm{wirt}}\varrho(z_{0})\neq 0\) _and_ \(\overline{\partial}_{\mathrm{wirt}}\varrho(z_{0})=0\) _there are_ \(\phi,\psi\in\mathrm{Aff}(\mathbb{C};\mathbb{C})\) _such that_

\[\sup_{z\in K}\left|(\psi\circ\varrho\circ\phi)(z)-z\right|<\varepsilon.\]

_(ii) If_ \(\partial_{\mathrm{wirt}}\varrho(z_{0})=0\) _and_ \(\overline{\partial}_{\mathrm{wirt}}\varrho(z_{0})\neq 0\) _there are_ \(\phi,\psi\in\mathrm{Aff}(\mathbb{C};\mathbb{C})\) _such that_

\[\sup_{z\in K}\left|(\psi\circ\varrho\circ\phi)(z)-\overline{z}\right|<\varepsilon.\]

_(iii) If_ \(\partial_{\mathrm{wirt}}\varrho(z_{0})\neq 0\neq\overline{\partial}_{\mathrm{wirt}} \varrho(z_{0})\) _there are_ \(\phi\in\mathrm{Aff}(\mathbb{C};\mathbb{C}^{2})\) _and_ \(\psi\in\mathrm{Aff}(\mathbb{C}^{2};\mathbb{C}^{2})\) _such that_

\[\sup_{z\in K}\left\|(\psi\circ\varrho^{\times 2}\circ\phi)(z)-(z,\overline{z}) \right\|_{\mathbb{C}^{2}}<\varepsilon\]
Proof.: The key idea of the proof is to use the affine maps to localize the activation function around its point of differentiability, where it behaves like its derivative, i.e., like a linear map in \(z\) and \(\bar{z}\), and then to invoke Lemma 2.1. Recall that Lemma 2.1 yields the existence of a function \(\Theta:\mathbb{C}\to\mathbb{C}\) satisfying \(\lim\limits_{z\to 0}\frac{\Theta(z)}{z}=0\) and

\[\varrho(z+z_{0})=\varrho(z_{0})+\partial_{\text{wirt}}\varrho(z_{0})z+ \overline{\partial}_{\text{wirt}}\varrho(z_{0})\overline{z}+\Theta(z)\]

for every \(z\in\mathbb{C}\).

If \(\overline{\partial}_{\text{wirt}}\varrho(z_{0})=0\) and \(\partial_{\text{wirt}}\varrho(z_{0})\neq 0\) we see for every \(h>0\) that

\[\frac{\varrho(z_{0}+hz)-\varrho(z_{0})}{\partial_{\text{wirt}}\varrho(z_{0})h }=z+\frac{\Theta(hz)}{\partial_{\text{wirt}}\varrho(z_{0})h}\quad\text{for all }z\in K. \tag{3.1}\]

Similarly, if \(\overline{\partial}_{\text{wirt}}\varrho(z_{0})\neq 0\) and \(\partial_{\text{wirt}}\varrho(z_{0})=0\) we get

\[\frac{\varrho(z_{0}+hz)-\varrho(z_{0})}{\overline{\partial}_{\text{wirt}} \varrho(z_{0})h}=\overline{z}+\frac{\Theta(hz)}{\overline{\partial}_{\text{ wirt}}\varrho(z_{0})h}\quad\text{for all }z\in K. \tag{3.2}\]

If \(\partial_{\text{wirt}}\varrho(z_{0})\neq 0\neq\overline{\partial}_{\text{wirt}} \varrho(z_{0})\) consider

\[\frac{\mathrm{i}\varrho(z_{0}+hz)+\varrho(z_{0}+\mathrm{i}hz)-(1+\mathrm{i}) \varrho(z_{0})}{2\mathrm{i}h\partial_{\text{wirt}}\varrho(z_{0})}=z+\frac{ \mathrm{i}\Theta(hz)+\Theta(\mathrm{i}hz)}{2\mathrm{i}h\partial_{\text{wirt}} \varrho(z_{0})} \tag{3.3}\]

as well as

\[\frac{-\mathrm{i}\varrho(z_{0}+hz)+\varrho(z_{0}+\mathrm{i}hz)-(1-\mathrm{i}) \varrho(z_{0})}{-2\mathrm{i}h\overline{\partial}_{\text{wirt}}\varrho(z_{0})} =\overline{z}+\frac{-\mathrm{i}\Theta(hz)+\Theta(\mathrm{i}hz)}{-2\mathrm{i} h\overline{\partial}_{\text{wirt}}\varrho(z_{0})}. \tag{3.4}\]

Since \(K\) is compact there exists \(L>0\) satisfying \(|z|\leq L\) for all \(z\in K\). Let \(\varepsilon^{\prime}>0\) be arbitrary and take \(\delta>0\) such that

\[\left|\frac{\Theta(w)}{w}\right|<\frac{\varepsilon^{\prime}}{L}\]

for every \(w\in\mathbb{C}\setminus\{0\}\) with \(|w|<\delta\). Let \(h\in(0,\delta/L)\) and \(z\in K\). Since \(|hz|<\delta\) we see for every \(z\in K\setminus\{0\}\) that

\[\left|\frac{\Theta(hz)}{h}\right|\leq L\cdot\left|\frac{\Theta(hz)}{hz}\right| <\varepsilon^{\prime}\]

and since \(\varepsilon^{\prime}\) has been taken arbitrarily

\[\lim\limits_{h\downarrow 0}\sup\limits_{z\in K}\frac{\Theta(hz)}{h}=0.\]

Here, we used \(\Theta(0)=0\). Thus, Equations (3.1) to (3.4) yield the claim by taking \(h\) sufficiently small. 

Proposition 3.2 is important for the case of polyharmonic activation functions which is considered in Section 5. It essentially states that, given an activation function which is not \(\mathbb{R}\)-affine, one can approximate one of the functions \(z\mapsto z\overline{z},\;\;z\mapsto z^{2}\) or \(z\mapsto\overline{z}^{2}\) by using a shallow neural network of width \(4\), see Figure 3 for an illustration.

Figure 2. Illustration of the neural network building blocks from Proposition 3.1. Neurons in the input and output layers are depicted in filled dots at the top and bottom, respectively. Applications of the activation function \(\varrho\) are shown as circles.
**Proposition 3.2**.: _Let \(\varrho\in C^{2}(\mathbb{C};\mathbb{C})\) be not \(\mathbb{R}\)-affine, let \(K\subseteq\mathbb{C}\) compact and \(\varepsilon>0\). Then there exist \(\phi\in\mathrm{Aff}(\mathbb{C};\mathbb{C}^{4}),\psi\in\mathrm{Aff}(\mathbb{C}^{ 4};\mathbb{C})\) such that at least one of the three inequalities_

\[\sup_{z\in K}\big{|}(\psi\circ\varrho^{\times 4}\circ\phi)(z)-z \overline{z}\big{|}<\varepsilon,\] \[\sup_{z\in K}\big{|}(\psi\circ\varrho^{\times 4}\circ\phi)(z)-z^{2} \big{|}<\varepsilon,\] \[\sup_{z\in K}\big{|}(\psi\circ\varrho^{\times 4}\circ\phi)(z)- \overline{z}^{2}\big{|}<\varepsilon\]

_holds true._

Proof.: Since \(\varrho\) is not \(\mathbb{R}\)-affine, there exists \(z_{0}\in\mathbb{C}\) such that \(\partial^{2}_{\mathrm{wirt}}\varrho(z_{0})\neq 0\), \(\partial_{\mathrm{wirt}}\overline{\partial}_{\mathrm{wirt}}\varrho(z_{0})\neq 0\), or \(\overline{\partial}^{2}_{\mathrm{wirt}}\varrho(z_{0})\neq 0\), see, e.g., Proposition B.2. Using the second-order Taylor expansion stated in Lemma 2.1, there is a function \(\Theta:\mathbb{C}\to\mathbb{C}\) satisfying \(\lim\limits_{z\to 0}\frac{\Theta(z)}{z^{2}}=0\) and

\[\varrho(z+z_{0})=\varrho(z_{0})+\partial_{\mathrm{wirt}}\varrho(z_{0})z+ \overline{\partial}_{\mathrm{wirt}}\varrho(z_{0})\overline{z}+\frac{1}{2} \partial^{2}_{\mathrm{wirt}}\varrho(z_{0})z^{2}+\partial_{\mathrm{wirt}} \overline{\partial}_{\mathrm{wirt}}\varrho(z_{0})z\overline{z}+\frac{1}{2} \overline{\partial}^{2}_{\mathrm{wirt}}\varrho(z_{0})\overline{z}^{2}+\Theta(z)\]

for every \(z\in\mathbb{C}\). Applying this identity to \(z=w\) and \(z=-w\) and adding up, we infer for any \(w\in\mathbb{C}\) that

\[\varrho(z_{0}+w)+\varrho(z_{0}-w)=2\varrho(z_{0})+\partial^{2}_{\mathrm{wirt }}\varrho(z_{0})w^{2}+2\partial_{\mathrm{wirt}}\overline{\partial}_{\mathrm{ wirt}}\varrho(z_{0})w\overline{w}+\overline{\partial}^{2}_{\mathrm{wirt}} \varrho(z_{0})\overline{w}^{2}+\Theta(w)+\Theta(-w).\]

Let \(h>0\) and \(z\in K\). If \(\partial_{\mathrm{wirt}}\overline{\partial}_{\mathrm{wirt}}\varrho(z_{0})\neq 0\), we see with \(w=hz\) and \(w=\mathrm{i}hz\) that

\[\frac{\varrho(z_{0}+hz)+\varrho(z_{0}-hz)+\varrho(z_{0}+\mathrm{ i}hz)+\varrho(z_{0}-\mathrm{i}hz)-4\varrho(z_{0})}{4h^{2}\partial_{\mathrm{wirt}} \overline{\partial}_{\mathrm{wirt}}\varrho(z_{0})}\] \[=z\overline{z}+\frac{\Theta(hz)+\Theta(-hz)+\Theta(\mathrm{i}hz)+ \Theta(-\mathrm{i}hz)}{4h^{2}\partial_{\mathrm{wirt}}\overline{\partial}_{ \mathrm{wirt}}\varrho(z_{0})}. \tag{3.5}\]

If \(\partial_{\mathrm{wirt}}\overline{\partial}_{\mathrm{wirt}}\varrho(z_{0})=0\) and \(\partial^{2}_{\mathrm{wirt}}\varrho(z_{0})\neq 0\), consider \(w=hz\) and \(w=\sqrt{\mathrm{i}}hz\), where \(\sqrt{\mathrm{i}}\) is a fixed square root of \(\mathrm{i}\):

\[\frac{\varrho(z_{0}+hz)+\varrho(z_{0}-hz)-\mathrm{i}\varrho(z_{0 }+\sqrt{\mathrm{i}}hz)-\mathrm{i}\varrho(z_{0}-\sqrt{\mathrm{i}}hz)+2(-1+ \mathrm{i})\varrho(z_{0})}{2h^{2}\partial^{2}_{\mathrm{wirt}}\varrho(z_{0})}\] \[=z^{2}+\frac{\Theta(hz)+\Theta(-hz)-\mathrm{i}\Theta(\sqrt{ \mathrm{i}}hz)-\mathrm{i}\Theta(-\sqrt{\mathrm{i}}hz)}{2h^{2}\partial^{2}_{ \mathrm{wirt}}\varrho(z_{0})}. \tag{3.6}\]

Last, if \(\partial^{2}_{\mathrm{wirt}}\varrho(z_{0})=\partial_{\mathrm{wirt}}\overline{ \partial}_{\mathrm{wirt}}\varrho(z_{0})=0\) consider \(w=hz\):

\[\frac{\varrho(z_{0}+hz)+\varrho(z_{0}-hz)-2\varrho(z_{0})}{h^{2}\overline{ \partial}^{2}_{\mathrm{wirt}}\varrho(z_{0})}=\overline{z}^{2}+\frac{\Theta(hz) +\Theta(-hz)}{h^{2}\overline{\partial}^{2}_{\mathrm{wirt}}\varrho(z_{0})}. \tag{3.7}\]

Since \(K\) is bounded, there is \(L>0\) with \(|z|\leq L\) for every \(z\in K\). For given \(\varepsilon^{\prime}>0\) there exists \(\delta>0\) such that

\[\left|\frac{\Theta(w)}{w^{2}}\right|<\frac{\varepsilon^{\prime}}{L^{2}}\]

for every \(w\in\mathbb{C}\setminus\{0\}\) with \(|w|<\delta\). Hence, we see for every \(h\in(0,\delta/L)\) and all \(z\in K\setminus\{0\}\) that

\[\left|\frac{\Theta(hz)}{h^{2}}\right|\leq L^{2}\cdot\left|\frac{\Theta(hz)}{(hz )^{2}}\right|<\varepsilon^{\prime}\]

where we used that \(|hz|<\delta\). Therefore, we conclude

\[\lim_{h\downarrow 0}\sup_{z\in K}\frac{\Theta(hz)}{h^{2}}=0,\]

using \(\Theta(0)=0\). The claim then follows from Equations (3.5) to (3.7).
In order to approximate arbitrary polynomials in the variables \(z_{1},\ldots,z_{n}\) and \(\overline{z_{1}},\ldots,\overline{z_{n}}\), we will compute iterative products of two complex numbers in Theorem 5.3. The following result enables the approximation of such products. An illustration of the CVNN blocks appearing in the proof are given in Figure 4.

**Proposition 3.3**.: _Let \(\varrho\in C^{2}(\mathbb{C};\mathbb{C})\) be not \(\mathbb{R}\)-affine, \(K\subseteq\mathbb{C}^{2}\) compact, and \(\varepsilon>0\). Then there exist \(\phi\in\mathrm{Aff}(\mathbb{C}^{2};\mathbb{C}^{12})\) and \(\psi\in\mathrm{Aff}(\mathbb{C}^{12};\mathbb{C})\) such that at least one of the three inequalities_

\[\sup_{(z,w)\in K}\left|(\psi\circ\varrho^{\times 12}\circ\phi)(z,w)- zw\right|<\varepsilon,\] \[\sup_{(z,w)\in K}\left|(\psi\circ\varrho^{\times 12}\circ\phi)(z,w)- z\overline{w}\right|<\varepsilon,\] \[\sup_{(z,w)\in K}\left|(\psi\circ\varrho^{\times 12}\circ\phi)(z,w)- \overline{zw}\right|<\varepsilon\]

_holds true._

Proof.: The main steps of the proof are to use a variant of the polarization identity to reconstruct the three multiplication operators from their values on the diagonal, and then to apply Proposition 3.2 to approximate the latter.

Precisely, the construction is as follows: If \(\zeta\mapsto\zeta\overline{\zeta}=\left|\zeta\right|^{2}\) can be approximated according to the first case of Proposition 3.2, use the identity

\[\left(\frac{1}{4}+\frac{\mathrm{i}}{4}\right)\left|z+w\right|^{2}+\left(- \frac{1}{4}+\frac{\mathrm{i}}{4}\right)\left|z-w\right|^{2}-\frac{\mathrm{i}}{ 2}\left|z-\mathrm{i}w\right|^{2}=z\overline{w}.\]

In order to approximate \((z,w)\mapsto z\overline{w}\), one needs \(4\) hidden neurons to approximate \(\zeta\mapsto\left|\zeta\right|^{2}\) for each of the \(3\) linear combinations of \(z\) and \(w\), resulting in a total amount of \(12\) hidden neurons.

If we have the second case of Proposition 3.2, we can approximate \(\zeta\mapsto\zeta^{2}\) using \(4\) hidden neurons. In this case, consider

\[\frac{1}{4}\left[(z+w)^{2}-(z-w)^{2}\right]=zw,\]

so that in total one needs \(8\) neurons in order to approximate \((z,w)\mapsto zw\). In the last case of Proposition 3.2 we can approximate \(\zeta\mapsto\overline{\zeta}^{2}\) using \(4\) hidden neurons. Considering

\[\frac{1}{4}\left[\overline{(z+w)}^{2}-\overline{(z-w)}^{2}\right]=\overline{ zw},\]

we infer that \((z,w)\mapsto\overline{zw}\) can be approximated using \(8\) hidden neurons.

Figure 3. Illustration of the neural network building block from Proposition 3.2. Neurons in the input and output layers are depicted in filled dots at the top and bottom, respectively. Applications of the activation function \(\varrho\) are shown as circles.
At the end of this section we want to introduce the fundamental concept of the _register model_. This construction has been heavily used in [15] to prove the real-valued counterpart of the theorem established in the present paper. The key idea is to transform a shallow neural network into a deep narrow network by "flipping" the shallow network and only performing one computation per layer. This is formalized in Proposition 3.4 and illustrated in Figure 5.

**Proposition 3.4**.: _Let \(n,m,\in\mathbb{N}\) and \(\varrho:\mathbb{C}\to\mathbb{C}\). Denote by \(\mathcal{I}_{n,m,n+m+1}^{\varrho}\) the set of register models_

\[T_{L}\circ\tilde{\varrho}\circ\ldots\circ\tilde{\varrho}\circ T_{1},\]

_where \(T_{1}\in\operatorname{Aff}(\mathbb{C}^{n};\mathbb{C}^{n+m+1})\), \(T_{L}\in\operatorname{Aff}(\mathbb{C}^{n+m+1};\mathbb{C}^{m})\), \(T_{\ell}\in\operatorname{Aff}(\mathbb{C}^{n+m+1};\mathbb{C}^{n+m+1})\) for \(\ell\in\{2,\ldots,L-1\}\), and the map \(\tilde{\varrho}\) acts on \(\mathbb{C}^{n+m+1}\) via_

\[\tilde{\varrho}:\mathbb{C}^{n+m+1}\to\mathbb{C}^{n+m+1},\;\;\;(\tilde{\varrho} (z_{1},\ldots,z_{n+m+1}))_{j}=\begin{cases}\varrho(z_{n+1}),&j=n+1,\\ z_{j},&j\neq n+1.\end{cases}\]

_Then \(\mathcal{S}\mathcal{N}_{n,m}^{\varrho}\subseteq\mathcal{I}_{n,m,n+m+1}^{\varrho}\)._

Proof.: Let \(f=V_{2}\circ\varrho^{\times(n+m+1)}\circ V_{1}\) with \(V_{1}\in\operatorname{Aff}(\mathbb{C}^{n};\mathbb{C}^{n+m+1})\) and \(V_{2}\in\operatorname{Aff}(\mathbb{C}^{n+m+1};\mathbb{C}^{m})\). For \(j\in\{1,\ldots,m\}\), the \(j\)th component function \(f_{j}\) of \(f\) can be written as

\[f_{j}(z)= \left(\sum_{k=1}^{n+m+1}c_{k,j}\varrho(a_{k}^{\top}z+b_{k})\right) +d_{j}\]

with suitably chosen \(a_{k}\in\mathbb{C}^{n}\) and \(c_{k,j},b_{k},d_{j}\in\mathbb{C}\). We define

\[T_{1}:\mathbb{C}^{n}\to\mathbb{C}^{n}\times\mathbb{C}\times\mathbb{C}^{m},\;\; \;\;T_{1}(z)=(z,a_{1}^{\top}z+b_{1},0).\]

For \(\ell\in\{2,\ldots,n+m+1\}\), we define

\[T_{\ell}:\mathbb{C}^{n}\times\mathbb{C}\times\mathbb{C}^{m}\to\mathbb{C}^{n} \times\mathbb{C}\times\mathbb{C}^{m},\;\;\;T_{\ell}(z,u,(w_{1},\ldots,w_{m}))= \begin{pmatrix}z\\ a_{\ell}^{\top}z+b_{\ell}\\ w_{1}+c_{\ell-1,1}u\\ w_{2}+c_{\ell-1,2}u\\ \vdots\\ w_{m}+c_{\ell-1,m}u\end{pmatrix}.\]

Figure 4. Illustration of the neural network building block from Proposition 3.3. Neurons in the input and output layers are depicted in filled dots at the top and bottom, respectively. Applications of the activation function \(\varrho\) are shown as circles. From the input values \(z\) and \(w\), three or two linear combinations are computed. Then the building block from Figure 3 is inserted to approximate \(z\mapsto z^{2}\), \(z\mapsto\overline{z}^{2}\), or \(z\mapsto z\overline{z}\). The results are again combined linearly.
Last, we set

\[T_{n+m+2}:\mathbb{C}^{n}\times\mathbb{C}\times\mathbb{C}^{m}\to\mathbb{C}^{m}, \quad T_{n+m+2}(z,u,(w_{1},\ldots,w_{m}))=\begin{pmatrix}w_{1}+c_{W,1}u+d_{1}\\ w_{2}+c_{W,2}u+d_{2}\\ \vdots\\ w_{m}+c_{W,m}u+d_{3}\end{pmatrix}.\]

Then we have

\[f=T_{n+m+2}\circ\tilde{\varrho}\circ\ldots\circ\tilde{\varrho}\circ T_{1},\]

which clearly yields \(f\in\mathcal{I}^{\varrho}_{n,m,n+m+1}\). 

## 4. The non-polyharmonic case

When the activation function \(\varrho\) is not polyharmonic, the universal approximation theorem for shallow CVNNs from [28, Theorem 1.3] is applicable. For convenience, we state the following special case relevant for our investigations.

**Theorem 4.1**.: _Let \(n\in\mathbb{N}\) and \(\varrho\in C(\mathbb{C};\mathbb{C})\). Then \(\mathcal{SN}^{\varrho}_{n,1}\) is universal if and only if \(\varrho\) is not polyharmonic._

In conjunction with suitable register models, it is possible to construct deep narrow networks to approximate a given continuous function. To do so, it is necessary to approximate the identity function. By assumption, the activation function \(\varrho\) is real differentiable at some point \(z_{0}\in\mathbb{C}\) with non-zero derivative. We consider three different cases. Firstly, if \(\varrho\) is even _complex_ differentiable at \(z_{0}\), Proposition 3.1 yields an approximation of \(\mathrm{id}_{\mathbb{C}}\) using shallow CVNNs with activation function \(\varrho\) and a width of \(1\). In that case, the register model can be used where the identity connections have to be replaced by sufficiently good approximations, resulting in a sufficient width of \(n+m+1\). Secondly, if \(\overline{\varrho}\) is

Figure 5. Illustration of the register model from Proposition 3.4. Neurons where the complex identity is used as activation function are visualized as squares, whereas neurons using \(\varrho\) as activation function are visualized using circles. The in-register neurons (squares on the left) store the input values and pass them to the computation neurons (middle circles). The result of the computations are added up and stored in the out-register neurons (squares on the right). The dashed box highlights one of the blocks that are later replaced using approximations of the complex identity.
complex differentiable at \(z_{0}\), we combine the fact that \(\mathcal{N}\overline{\mathcal{N}}_{n,m,n+m+1}^{\varrho}\) is universal and that \(\overline{\operatorname{id}_{\mathbb{C}}}\) can be approximated according to Proposition 3.1 to deduce the universality of \(\mathcal{N}\mathcal{N}_{n,m,n+m+1}^{\varrho}\). Lastly, if neither of the two former cases happens, we show that a width of \(2n+2m+1\) is sufficient for CVNNs with activation function \(\varrho\) in order to be universal, by combining the register model with the third building block described in Proposition 3.1.

**Theorem 4.2**.: _Let \(n,m\in\mathbb{N}\). Assume that \(\varrho\in C(\mathbb{C};\mathbb{C})\) is not polyharmonic and that there exists \(z_{0}\in\mathbb{C}\) such that \(\varrho\) is real differentiable at \(z_{0}\) with non-zero derivative._

_(i) If_ \(\partial_{\operatorname{wirt}}\varrho(z_{0})\neq 0\) _and_ \(\overline{\partial}_{\operatorname{wirt}}\varrho(z_{0})=0\)_, then the set_ \(\mathcal{N}\mathcal{N}_{n,m,n+m+1}^{\varrho}\) _is universal._

_(ii) If_ \(\partial_{\operatorname{wirt}}\varrho(z_{0})=0\) _and_ \(\overline{\partial}_{\operatorname{wirt}}\varrho(z_{0})\neq 0\)_, then the set_ \(\mathcal{N}\mathcal{N}_{n,m,n+m+1}^{\varrho}\) _is universal._

Proof.: Let \(f\in C(\mathbb{C}^{n};\mathbb{C}^{m})\), \(K\subseteq\mathbb{C}\) a compact subset, and \(\varepsilon>0\).

Using Theorem 4.1, Proposition 3.4, and the notation introduced therein, we infer that there is a function \(g\in\mathcal{I}_{n,m,n+m+1}^{\varrho}\) with

\[\sup_{z\in K}\left\|f(z)-g(z)\right\|_{\mathbb{C}^{m}}<\frac{\varepsilon}{2}.\]

This means that there exist \(T_{1}\in\operatorname{Aff}(\mathbb{C}^{n};\mathbb{C}^{n+m+1})\), \(T_{L}\in\operatorname{Aff}(\mathbb{C}^{n+m+1};\mathbb{C}^{m})\), and \(T_{\ell}\in\operatorname{Aff}(\mathbb{C}^{n+m+1};\mathbb{C}^{n+m+1})\) for \(\ell\in\{2,\ldots,L-1\}\) such that

\[g=T_{L}\circ\tilde{\varrho}\circ T_{L-1}\circ\ldots\circ\tilde{\varrho}\circ T _{1},\]

where \(\tilde{\varrho}:\mathbb{C}^{n+m+1}\to\mathbb{C}^{n+m+1}\) is given by

\[\left(\tilde{\varrho}(z_{1},\ldots,z_{n+m+1})\right)_{j}=\begin{cases}z_{j},& j\in\{1,\ldots,n+m\}\,,\\ \varrho(z_{n+m+1}),&j=n+m+1.\end{cases}\]

We first prove (i). Combining Proposition 3.1(i), Proposition A.2, and Proposition A.5, we deduce the existence of sequences \((\Psi_{j})_{j\in\mathbb{N}}\) and \((\Phi_{j})_{j\in\mathbb{N}}\) with \(\Psi_{j},\Phi_{j}\in\operatorname{Aff}(\mathbb{C}^{n+m+1};\mathbb{C}^{n+m+1})\) for \(j\in\mathbb{N}\), satisfying

\[\Psi_{j}\circ\varrho^{\times(n+m+1)}\circ\Phi_{j}\xrightarrow{j\to\infty} \tilde{\varrho}\]

locally uniformly. Proposition A.5 now implies that the sequence \((g_{j})_{j\in\mathbb{N}}\) given by

\[g_{j}=T_{L}\circ\Bigl{(}\Psi_{j}\circ\varrho^{\times(n+m+1)}\circ\Phi_{j} \Bigr{)}\circ T_{L-1}\circ\ldots\circ\Bigl{(}\Psi_{j}\circ\varrho^{\times(n+ m+1)}\circ\Phi_{j}\Bigr{)}\circ T_{1}\]

converges locally uniformly to \(g\) as \(j\to\infty\). Moreover, since the composition of affine maps is affine, we have \(g_{j}\in\mathcal{N}\mathcal{N}_{n,m,n+m+1}^{\varrho}\) for all \(j\in\mathbb{N}\). Claim (i) now follows from another application of Proposition A.2.

We now deal with (ii). By the fundamental properties of Wirtinger derivatives (see for instance [14, E. 1a]) we compute

\[\partial_{\operatorname{wirt}}\overline{\varrho}(z_{0})=\overline{\overline{ \partial}_{\operatorname{wirt}}\varrho(z_{0})}\neq 0\quad\text{and}\quad\overline{ \partial}_{\operatorname{wirt}}\overline{\varrho}(z_{0})=\overline{\partial _{\operatorname{wirt}}\varrho(z_{0})}=0.\]

It is immediate to see that \(\varrho\in C^{\infty}(\mathbb{C};\mathbb{C})\) if and only if \(\overline{\varrho}\in C^{\infty}(\mathbb{C};\mathbb{C})\). In that case, it holds \(\Delta^{m}\overline{\varrho}=\overline{\Delta^{m}\varrho}\) for any \(m\in\mathbb{N}_{0}\). These two properties imply that \(\overline{\varrho}\) is non-polyharmonic. From (i) we deduce that the set \(\mathcal{N}\mathcal{N}_{n,m,n+m+1}^{\varrho}\) is universal. We claim that the set \(\mathcal{N}\mathcal{N}_{n,m,n+m+1}^{\varrho}\) is universal, too. Indeed, let \(f\in\mathcal{N}\mathcal{N}_{n,m,n+m+1}^{\varrho}\) be arbitrary and consider the decomposition

\[f=V_{L}\circ\overline{\varrho}^{\times(n+m+1)}\circ V_{L-1}\circ\ldots\circ \overline{\varrho}^{\times(n+m+1)}\circ V_{1}\]

with \(V_{1}\in\operatorname{Aff}(\mathbb{C}^{n};\mathbb{C}^{n+m+1})\), \(V_{\ell}\in\operatorname{Aff}(\mathbb{C}^{n+m+1};\mathbb{C}^{n+m+1})\), and \(V_{L}\in\operatorname{Aff}(\mathbb{C}^{n+m+1};\mathbb{C}^{m})\) for every number \(\ell\in\{2,\ldots,L-1\}\). From Proposition 3.1(ii) we infer the existence of sequences \((\Phi_{j})_{j\in\mathbb{N}}\) and \((\Psi_{j})_{j\in\mathbb{N}}\) with \(\Phi_{j},\Psi_{j}\in\operatorname{Aff}(\mathbb{C}^{n+m+1};\mathbb{C}^{n+m+1})\) for \(j\in\mathbb{N}\) such that

\[\Psi_{j}\circ\varrho^{\times(n+m+1)}\circ\Phi_{j}\xrightarrow{j\to\infty} \overline{\operatorname{id}_{\mathbb{C}}}^{\times(n+m+1)}\]

locally uniformly. But then we see

\[V_{L}\circ\Bigl{(}\Psi_{j}\circ\varrho^{\times(n+m+1)}\circ\Phi_{j}\Bigr{)}\circ \varrho^{\times(n+m+1)}\circ V_{L-1}\circ\ldots\circ\Bigl{(}\Psi_{j}\circ \varrho^{\times(n+m+1)}\circ\Phi_{j}\Bigr{)}\circ\varrho^{\times(n+m+1)}\circ V _{1}\xrightarrow{j\to\infty}f\]
locally uniformly, where the left-hand side is an element of \(\mathcal{NN}^{\varrho}_{n,m,n+m+1}\) for every \(j\in\mathbb{N}\). Here we applied Proposition A.5. This shows that \(\mathcal{NN}^{\overline{\sigma}}_{n,m,n+m+1}\) is dense in \(\mathcal{NN}^{\varrho}_{n,m,n+m+1}\) with respect to local uniform convergence which in turn yields the universality of \(\mathcal{NN}^{\varrho}_{n,m,n+m+1}\), cf. Proposition A.2.

The proof of (iii) is analogous. Gluing together one copy of the complex identity \(\mathrm{id}_{\mathbb{C}}\) and \(n+m\) copies of the function \(\phi\) or (the projection onto the first component of) \(\psi\) of Proposition 3.1(iii), respectively, we construct sequences \((\Psi_{j})_{j\in\mathbb{N}},(\Phi_{j})_{j\in\mathbb{N}}\) satisfying \(\Psi_{j}\in\mathrm{Aff}(\mathbb{C}^{2n+2m+1};\mathbb{C}^{n+m+1})\) and \(\Phi_{j}\in\mathrm{Aff}(\mathbb{C}^{n+m+1};\mathbb{C}^{2n+2m+1})\) for \(j\in\mathbb{N}\) such that

\[\Psi_{j}\circ\varrho^{\times(2n+2m+1)}\circ\Phi_{j}\xrightarrow{j\to\infty} \tilde{\varrho}\]

locally uniformly. Because the building blocks consist of \(2\) neurons in this case, the resulting approximating neural network has width \(2n+2m+1\) instead of \(n+m+1\). An illustration of the replacement of the in-register and out-register neurons of the register model from Proposition 3.4 using a building block from Proposition 3.1 is given in Figure 6. 

Next we provide two examples of activation functions which are used in practice and to which Theorem 4.2 applies.

**Example 4.3**.: The modReLU function has for example been proposed in [2] as a generalization of the classical ReLU to the complex plane. For a parameter \(b<0\) it is defined as

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

An application of Theorem 4.2(iii) shows that for \(n,m\in\mathbb{N}\), and \(b<0\), the set \(\mathcal{NN}^{\mathrm{modReLU}_{b}}_{n,m,2n+2m+1}\) is universal.

To this end, let us verify the assumptions of Theorem 4.2 in detail. Since the continuity of \(\mathrm{modReLU}_{b}\) is immediate for \(z\in\mathbb{C}\) with \(|z|\neq-b\), it remains to check the case \(|z|=-b\). Take any sequence \((z_{j})_{j\in\mathbb{N}}\) with \(z_{j}\to z\) as \(j\to\infty\), where we assume without loss of generality \(|z_{j}|\geq-b\) for every \(j\in\mathbb{N}\). Then \(|\mathrm{modReLU}_{b}(z_{j})|=|z_{j}|+b\to|z|+b=0\) as \(j\to\infty\). This shows the continuity of \(\mathrm{modReLU}_{b}\).

In [10, Corollary 5.4] it is shown that for all \(z\in\mathbb{C}\) with \(|z|>-b\) and all \(k,\ell\in\mathbb{N}_{0}\) one has

\[\partial_{\mathrm{wirt}}^{k}\overline{\partial}_{\mathrm{wirt}}^{\ell}\, \mathrm{modReLU}_{b}(z)\neq 0.\]

This implies that \(\mathrm{modReLU}_{b}\) is non-polyharmonic and \(\partial_{\mathrm{wirt}}\,\mathrm{modReLU}_{b}(z)\neq 0\neq\overline{\partial}_{ \mathrm{wirt}}\,\mathrm{modReLU}_{b}(z)\) for all \(z\in\mathbb{C}\) with \(|z|>-b\).

Further note that the result from Theorem 4.2 cannot be used to reduce the sufficient width to \(n+m+1\) since it holds \(\partial_{\mathrm{wirt}}\,\mathrm{modReLU}_{b}(z)\neq 0\neq\overline{\partial}_{ \mathrm{wirt}}\,\mathrm{modReLU}_{b}(z)\) for all \(z\in\mathbb{C}\) with \(|z|>-b\), \(\partial_{\mathrm{wirt}}\,\mathrm{modReLU}_{b}(z)=\overline{\partial}_{ \mathrm{wirt}}\,\mathrm{modReLU}_{b}(z)=0\) for every \(z\in\mathbb{C}\) with \(|z|<-b\) and \(\mathrm{modReLU}_{b}\) is not differentiable in any \(z\in\mathbb{C}\) with \(|z|=-b\).

Figure 6. Illustration for the proof of Theorem 4.2: Replacement rules for the approximation of register models by CVNNs using building blocks.
**Example 4.4**.: The complex cardioid function has been used in [27] in the context of MRI fingerprinting, where complex-valued neural networks significantly outperformed their real-valued counterparts. It is defined as

\[\operatorname{card}:\mathbb{C}\to\mathbb{C},\quad\operatorname{card}(z):=\begin{cases} \frac{1}{2}\Big{(}1+\frac{\operatorname{Re}(z)}{|z|}\Big{)}\,z,&z\in\mathbb{C} \setminus\{0\}\,,\\ 0,&z=0.\end{cases}\]

An application of Theorem 4.2(i) shows that for \(n,m\in\mathbb{N}\) the set \(\mathcal{N}\mathcal{N}^{\operatorname{card}}_{n,m,n+m+1}\) is universal.

To this end, let us verify the assumptions of Theorem 4.2 in detail. The continuity of card on \(\mathbb{C}\setminus\{0\}\) is immediate. Further note that

\[|\operatorname{card}(z)|=\left|\frac{1}{2}\bigg{(}1+\frac{\operatorname{Re}(z )}{|z|}\bigg{)}\,z\right|\leq|z|\to 0\]

as \(z\to 0\), showing the continuity of card on the entire complex plane \(\mathbb{C}\). Now [10, Corollaries 5.6 and 5.7] show that card is non-polyharmonic and

\[\partial_{\operatorname{wirt}}\operatorname{card}(z)=\frac{1}{2}+\frac{1}{8} \cdot\frac{\overline{z}}{|z|}+\frac{3}{8}\cdot\frac{z}{|z|},\quad\overline{ \partial}_{\operatorname{wirt}}\operatorname{card}(z)=-\frac{1}{8}\cdot \frac{z^{3}}{\frac{1}{|z|}^{3}}+\frac{1}{8}\cdot\frac{z}{|z|}\]

for every \(z\in\mathbb{C}\setminus\{0\}\). Hence, we see \(\partial_{\operatorname{wirt}}\operatorname{card}(1)=1\) and \(\overline{\partial}_{\operatorname{wirt}}\operatorname{card}(1)=0\).

## 5. The polyharmonic case

In this section, we deal with activation functions \(\varrho:\mathbb{C}\to\mathbb{C}\) that are polyharmonic. However, it turns out that this property can be relaxed to only requiring that \(\varrho\in C^{2}(\mathbb{C};\mathbb{C})\) in order for the proofs to work. The main assumptions of this section can therefore be stated as follows.

**Assumption 5.1**.: _Let \(\varrho:\mathbb{C}\to\mathbb{C}\) be a function satisfying the following conditions:_

_(i)_ \(\varrho\in C^{2}(\mathbb{C};\mathbb{C})\)_,_

_(ii)_ \(\varrho\) _is not holomorphic,_

_(iii)_ \(\varrho\) _is not antiholomorphic,_

_(iv)_ \(\varrho\) _is not_ \(\mathbb{R}\)_-affine._

The Stone-Weierstrass theorem states that any continuous function can be arbitrarily well approximated by complex polynomials in \(z_{1},\dots,z_{n}\) and \(\overline{z_{1}},\dots,\overline{z_{n}}\) in the uniform norm on compact subsets of \(\mathbb{C}^{n}\). In order to show universality of CVNNs, it suffices therefore to show that such polynomials can be approximated by deep narrow CVNNs to arbitrary precision in the uniform norm on compact sets. We follow an approach similar to that of the register model by preserving the inputs and outputs from layer to layer while gradually performing multiplications to approximate the individual monomials.

Using Proposition 3.1 we derive the following Proposition 5.2. It states that the function \(z\mapsto(z,\overline{z})\) can be uniformly approximated on compact sets by a shallow network of width \(2\).

**Proposition 5.2**.: _Let \(\varrho\) satisfy Assumption 5.1, let \(K\subseteq\mathbb{C}\) be compact and \(\varepsilon>0\). Then there are maps \(\phi\in\operatorname{Aff}(\mathbb{C};\mathbb{C}^{2})\) and \(\psi\in\operatorname{Aff}(\mathbb{C}^{2};\mathbb{C}^{2})\) such that_

\[\sup_{z\in K}\left\|(\psi\circ\varrho^{\times 2}\circ\phi)(z)-(z,\overline{z}) \right\|_{\mathbb{C}^{2}}<\varepsilon.\]

Proof.: If there is a point \(z_{0}\in\mathbb{C}\) with \(\partial_{\operatorname{wirt}}\varrho(z_{0})\neq 0\neq\overline{\partial}_{ \operatorname{wirt}}\varrho(z_{0})\), we can directly apply Proposition 3.1(iii). If there is no such point \(z_{0}\), we can still find \(z_{1}\in\mathbb{C}\) with \(\partial_{\operatorname{wirt}}\varrho(z_{1})\neq 0\) and \(z_{2}\in\mathbb{C}\) with \(\overline{\partial}_{\operatorname{wirt}}\varrho(z_{2})\neq 0\), since \(\varrho\) is neither holomorphic nor antiholomorphic. By assumption of the nonexistence of \(z_{0}\in\mathbb{C}\) with \(\partial_{\operatorname{wirt}}\varrho(z_{0})\neq 0\neq\overline{\partial}_{ \operatorname{wirt}}\varrho(z_{0})\), it follows that \(\overline{\partial}_{\operatorname{wirt}}\varrho(z_{1})=0\) and \(\partial_{\operatorname{wirt}}\varrho(z_{2})=0\), and we can thus apply Proposition 3.1(i) and (ii). 

In fact, it is possible to show that under the conditions formulated in Assumption 5.1, one can always find a point \(z_{0}\in\mathbb{C}\) with \(\partial_{\operatorname{wirt}}\varrho(z_{0})\neq 0\neq\overline{\partial}_{ \operatorname{wirt}}\varrho(z_{0})\).

Proposition 5.2 is a central finding, since it implies that we can build from the activation function \(\varrho\) constructions that are similar to the register model construction from Proposition 3.4. Together with Proposition 3.3 from Section 3, this leads to the following proof of the sufficiency part of Theorem 1.1.
**Theorem 5.3**.: _Let \(n,m\in\mathbb{N}\). Assume that \(\varrho\in C(\mathbb{C};\mathbb{C})\) satisfies Assumption 5.1. Then \(\mathcal{N}\mathcal{N}^{\varrho}_{n,m,2n+2m+5}\) is universal._

_Moreover, if there is a point \(z_{0}\in\mathbb{C}\) with either_

\[\partial_{\mathrm{wirt}}\varrho(z_{0})\neq 0=\overline{\partial}_{\mathrm{wirt}} \varrho(z_{0})\quad\text{or}\quad\partial_{\mathrm{wirt}}\varrho(z_{0})=0 \neq\overline{\partial}_{\mathrm{wirt}}\varrho(z_{0}), \tag{5.1}\]

_then \(\mathcal{N}\mathcal{N}^{\varrho}_{n,m,n+m+4}\) is universal._

Proof.: We first provide a detailed proof of the universality of \(\mathcal{N}\mathcal{N}^{\varrho}_{n,m,2n+2m+5}\). Afterwards, we comment on how to adapt it to show that a width of \(n+m+4\) is sufficient if (5.1) is fulfilled.

From the Stone-Weierstrass theorem [9, Theorem 4.51], we know that the set of complex polynomials in \(z_{1},\ldots,z_{n}\) and \(\overline{z_{1}},\ldots,\overline{z_{n}}\) is dense in \(C(K;\mathbb{C})\) with respect to the uniform norm on the compact set \(K\subset\mathbb{C}^{n}\). Hence, it suffices to show that each function \(p:K\to\mathbb{C}^{m}\), whose components are polynomials in \(z_{1},\ldots,z_{n}\) and \(\overline{z_{1}},\ldots,\overline{z_{n}}\) can be uniformly approximated on \(K\) by CVNNs of width \(2n+2m+5\). Let \(p=(p_{1},\ldots,p_{m})\) be such a function and decompose \(p_{\ell}=\sum_{j=1}^{L_{\ell}}a_{j,\ell}p_{j,\ell}\) with each \(p_{j,\ell}\) being a monomial (i.e., a product of some of the variables \(z_{1},\ldots,z_{n},\overline{z_{1}},\ldots,\overline{z_{n}}\)), and \(a_{j,\ell}\in\mathbb{C}\) for \(\ell\in\{1,\ldots,m\}\) and \(j\in\{1,\ldots,L_{\ell}\}\). We now construct a CVNN that approximates \(p\) sufficiently well on \(K\).

The idea is to construct a register-like model where each layer consists of \(2n+m+1\) neurons with \(2n\)_in-register neurons_ storing the input values and their complex conjugates, \(m\)_out-register neurons_, and one _computation neuron_, see Figure 7.

For \(k\in\{1,2,3\}\), let \(\mathrm{mul}_{k}:\mathbb{C}^{2}\to\mathbb{C}\) be maps defined as \(\mathrm{mul}_{1}(z,w):=zw\), \(\mathrm{mul}_{2}(z,w):=z\overline{w}\), and \(\mathrm{mul}_{3}(z,w):=\overline{zw}\) for all \(z,w\in\mathbb{C}\). From Proposition 3.3 we know that there is \(\mathrm{mul}\in\{\mathrm{mul}_{1},\mathrm{mul}_{2},\mathrm{mul}_{3}\}\) that can be uniformly approximated by shallow CVNNs with activation function \(\varrho\) and a width of \(12\). The available multiplication \(\mathrm{mul}\) is fixed from now on. Each of the computation neurons takes two inputs and performs the multiplication \(\mathrm{mul}\).

We first sketch how to compute and save a single monomial of degree \(l\in\mathbb{N}\). Let \(q=z_{k_{\ell}}\cdot z_{k_{\ell-1}}\cdot\ldots\cdot z_{k_{1}}\) be such a monomial, where we assume for now that it includes no conjugated variables. We further assume that in every computation neuron we could perform a multiplication. The out-register neurons are initialized with \(0\) and the computation neuron is initialized with \(1\). Then the desired monomial can be computed iteratively as \(z_{k_{\ell}}\cdot(z_{k_{\ell-1}}\cdot\ldots(z_{k_{1}}\cdot 1)\ldots)\): in each layer, we pass to the computation neuron the currently needed variable \(z_{k_{j}}\) and the product obtained from the previous computational neuron. However, in general it might occur that the computational neuron only performs the operation \((z,w)\mapsto z\bar{w}\) or \((z,w)\mapsto\bar{z}\bar{w}\), and that the desired monomial also includes conjugated variables. Hence, the obtained monomial

\[\zeta_{k_{\ell}}\cdot\ldots\cdot\zeta_{k_{1}}:=\mathrm{mul}(z_{k_{\ell}}, \mathrm{mul}(z_{k_{\ell-1}},\ldots\mathrm{mul}(z_{k_{1}},1)\ldots))\]

might differ from \(q\).

However, by the basic properties of complex conjugation, for every \(j\in\{1,\ldots,\ell\}\), \(\zeta_{k_{j}}\) is either \(z_{k_{j}}\) or \(\bar{z}_{k_{j}}\). Since the input register saves both the variables and their conjugates, it suffices to pass, if necessary, \(\bar{z}_{k_{j}}\) instead of \(z_{k_{j}}\) from the input register to the computation neuron at the \(j\)th step. This ensures that it is always possible to compute the desired monomial.

Let us now describe how the entire polynomial \(p\) is computed: In each layer the current value of the computation neuron or its complex conjugate is multiplied with either the value of one of the in-register neurons or its complex conjugate, according to Proposition 3.3. This is performed repeatedly until the first monomial \(p_{1,1}\) is completed. Once the first monomial \(p_{1,1}\) is realized, it is then multiplied with its coefficient and stored into the first out-register neuron. This is then repeated for the other monomials \(p_{j,1}\) and the result is gradually added up in the first out-register neuron which then finally stores the desired polynomial for the first component \(p_{1}\). This procedure is then repeated for the other polynomials \(p_{2},\ldots,p_{m}\).

The approximation of this type of register model by CVNNs is done using the building blocks from Section 3. An illustration of the replacement rules is shown in Figure 8.

In the following we want to make this idea more precise. To this end, we set up a register model in Step 1, and show in Step 2 how this register model can be uniformly approximated by elements of \(\mathcal{N}\mathcal{N}^{\varrho}_{n,m,2n+2m+12}\) on compact sets. This yields universality of \(\mathcal{N}\mathcal{N}^{\varrho}_{n,m,2n+2m+12}\). In Step 2\({}^{\prime}\) we
provide a more detailed analysis of the arguments from Step 2 which lead to the claimed universality of \(\mathcal{NN}_{n,m,2n+2m+5}^{\varrho}\).

_Step 1_.: Fix \(\operatorname{mul}\in\{\operatorname{mul}_{1},\operatorname{mul}_{2}, \operatorname{mul}_{3}\}\). We can deduce that we can write \(p\) as a composition

\[p=T_{\operatorname{end}}\circ T_{N}\circ T_{N-1}\circ\ldots\circ T_{1}\circ T_ {\operatorname{init}} \tag{5.2}\]

for some number \(N\in\mathbb{N}\) with

\[T_{\operatorname{init}} :\mathbb{C}^{n}\to\mathbb{C}^{n}\times\mathbb{C}^{n}\times \mathbb{C}\times\mathbb{C}^{m}, z\mapsto(z,\overline{z},1,0),\] \[T_{\operatorname{end}} :\mathbb{C}^{n}\times\mathbb{C}^{n}\times\mathbb{C}\times \mathbb{C}^{m}\to\mathbb{C}^{m}, (z,u,w,v)\mapsto v\]

Figure 7. The register-like model from Theorem 5.3: Squares and diamonds depict the complex identity map and complex conjugation, respectively. Each circle stands for one of the multiplication operations \(\operatorname{mul}_{k}\) from the proof of Theorem 5.3. The choice of the connections from the in-register neurons (left) to the computation neuron (middle) depends on the monomial that is approximated and the operation \(\operatorname{mul}_{k}\) that is available (\(T_{\ell}\) of the first form). One of them is shown in the figure. The connections from the computation neuron (middle) to the out-register neuron (right) are drawn each time the computation of a monomial of one of the \(m\) output polynomials is completed (\(T_{\ell}\) of the second form). The dashed portions of the register model are to be replaced according to Figure 8.
and for every \(\ell\in\{1,\ldots,N\}\), the map \(T_{\ell}:\mathbb{C}^{2n+m+1}\to\mathbb{C}^{2n+m+1}\) has one of the two following forms: The first possible form is

\[T_{\ell}:\mathbb{C}^{n}\times\mathbb{C}^{n}\times\mathbb{C}\times\mathbb{C}^{m }\to\mathbb{C}^{n}\times\mathbb{C}^{n}\times\mathbb{C}\times\mathbb{C}^{m}, \quad(z,u,w,v)\mapsto(z,\overline{z},\operatorname{mul}((z,u)_{k},w),v)\]

where \((z,u)_{k}\) denotes the \(k\)th component of \((z,u)\in\mathbb{C}^{n}\times\mathbb{C}^{n}\) for some \(k=k(\ell)\in\{1,\ldots,2n\}\). This function realizes the multiplication of the current value of the computation neuron with one of the values of the in-registers, choosing either a variable or its complex conjugation, depending on \(k\). The second possible form is

\[T_{\ell}:\mathbb{C}^{n}\times\mathbb{C}^{n}\times\mathbb{C}\times\mathbb{C}^{m }\to\mathbb{C}^{n}\times\mathbb{C}^{n}\times\mathbb{C}\times\mathbb{C}^{m}, \quad(z,u,w,v)\mapsto(z,u,1,v+a_{j,k}we_{k})\]

for some \(k\in\{1,\ldots,m\}\) and \(j\in\{1,\ldots,L_{k}\}\). Here, \(e_{k}\) denotes the \(k\)th standard basis vector of \(\mathbb{C}^{m}\). This is the function that adds a finished monomial to the corresponding out-register neuron.

_Step \(2\)._ To conclude, we see that every function \(T_{\ell}\) occurring in the composition (5.2) is either an affine map (if it has the second form) or has the property that there are sequences \((\phi_{j}^{\ell})_{j\in\mathbb{N}}\) and \((\psi_{j}^{\ell})_{j\in\mathbb{N}}\) with \(\phi_{j}^{\ell}\in\operatorname{Aff}(\mathbb{C}^{2n+m+1};\mathbb{C}^{2n+2m+12})\) and \(\psi_{j}^{\ell}\in\operatorname{Aff}(\mathbb{C}^{2n+2m+12};\mathbb{C}^{2n+m+1})\) such that

\[\psi_{j}^{\ell}\circ\varrho^{\times(2n+2m+12)}\circ\phi_{j}^{\ell}\xrightarrow{ j\to\infty}T_{\ell}\quad\text{locally uniformly},\]

where we applied Propositions 3.3 and 5.2. Similarly, one can find sequences \((\phi_{j}^{\text{init}})_{j\in\mathbb{N}}\) and \((\psi_{j}^{\text{init}})_{j\in\mathbb{N}}\) with \(\phi_{j}^{\text{init}}\in\operatorname{Aff}(\mathbb{C}^{n};\mathbb{C}^{2n})\) and \(\psi_{j}^{\text{init}}\in\operatorname{Aff}(\mathbb{C}^{2n};\mathbb{C}^{2n+m+1})\) satisfying

\[\psi_{j}^{\text{init}}\circ\varrho^{\times 2n}\circ\phi_{j}^{\text{init}} \xrightarrow{j\to\infty}T_{\text{init}}\quad\text{locally uniformly}.\]

Using Proposition A.5, we obtain the universality of \(\mathcal{N}\mathcal{N}_{2n+2m+12}^{g}\).

_Step \(2^{\prime}\)._ In the previous step, we approximated in the sense of Proposition A.5 the complex identity neurons and the complex conjugation neurons by the elementary building blocks given in Proposition 3.1, and the multiplication operation mul by the building block given by Proposition 3.3. The latter is a shallow CVNN of width \(12\). In order to further reduce the width, we will first approximate this shallow CVNN by an appropriate register model according to Proposition 3.4, and only then replace all complex identities and conjugations by the elementary building blocks.

Each of the functions \(T_{1},\ldots,T_{N}\) occurring in the composition (5.2) is either an affine map (if it has the second form) or has the following property: For each \(\ell\in\{1,\ldots,N\}\), there are sequences

Figure 8. Replacement rules for the highlighted portions in Figure 4(b).
and \((\psi_{j}^{\ell})_{j\in\mathbb{N}}\) with \(\phi_{j}^{\ell}\in\operatorname{Aff}(\mathbb{C}^{2};\mathbb{C}^{12})\) and \(\psi_{j}^{\ell}\in\operatorname{Aff}(\mathbb{C}^{12};\mathbb{C})\) such that \(F_{j}^{\ell}\xrightarrow{j\to\infty}T_{\ell}\) locally uniformly, where we define

\[F_{j}^{\ell}:\mathbb{C}^{n}\times\mathbb{C}^{n}\times\mathbb{C}\times\mathbb{C} ^{m}\to\mathbb{C}^{n}\times\mathbb{C}^{n}\times\mathbb{C}\times\mathbb{C}^{m}, \quad(z,u,w,v)\mapsto(z,\overline{z},\widetilde{F}_{j}^{\ell}((z,u)_{k},w),v)\]

and \(\widetilde{F}_{j}^{\ell}=\psi_{j}^{\ell}\circ\varrho^{\times 12}\circ\phi_{j}^{ \ell}:\mathbb{C}^{2}\to\mathbb{C}\) with \(k=k(\ell)\) from the definition of \(T_{\ell}\).

Note that \(\widetilde{F}_{j}^{\ell}\in\mathcal{SN}_{2,1}^{\varrho}\subseteq\mathcal{I}_ {2,1,4}^{\varrho}\) for all \(j,\ell\in\mathbb{N}\) thanks to Proposition 3.4. Hence, every function \(F_{j}^{\ell}\) can be represented by a ("large") register model with input and output dimension both equal to \(2n+m+1\) and with \(2n+m+4\) neurons per hidden layer. These \(2n+m+4\) neurons can be grouped as follows: \(n\) in-register neurons use the complex identity as activation function and store the \(n\) input values, \(n\) in-register neurons use the complex conjugation as activation function and store the complex conjugates of those input values, \(m\) out-register neurons use the complex identity as activation function and carry the output values of the original polynomial, and \(4\) neurons for the ("small") register model that represents \(\widetilde{F}_{j}^{\ell}\), of which two in-register neurons and one out-register neurons use the complex identity, and one computation neuron uses \(\varrho\) as activation function. Note that one of the two in-register neurons of the small register model stores one of the input variables of the large register model and can therefore be dropped. This results in total in a register model with \(n+m+2\) identity neurons, \(n\) conjugation neurons, and \(1\) neuron using \(\varrho\) as its activation function per hidden layer.

Making this more precise, we write

\[F_{j}^{\ell}=V_{K}\circ\tilde{\varrho}\circ V_{K-1}\circ\ldots\circ V_{2} \circ\tilde{\varrho}\circ V_{1},\]

where \(\tilde{\varrho}\) is as in Proposition 3.4, i.e., it applies \(\varrho\) to the \((2n+2)\)th input variable, and the identity to all others, and for \(r\in\{2,\ldots,K-1\}\), we write

\[V_{1}:\mathbb{C}^{n}\times\mathbb{C}^{n}\times\mathbb{C}\times \mathbb{C}^{m}\to\mathbb{C}^{n}\times\mathbb{C}^{n}\times\mathbb{C}^{3}\times \mathbb{C}^{m}, (z,u,w,v)\mapsto(z,\overline{z},\eta_{1}(z,u,w),v),\] \[V_{r}:\mathbb{C}^{n}\times\mathbb{C}^{n}\times\mathbb{C}^{3} \times\mathbb{C}^{m}\to\mathbb{C}^{n}\times\mathbb{C}^{n}\times\mathbb{C} \times\mathbb{C}^{m}, (z,u,w,v)\mapsto(z,\overline{z},\eta_{r}(z,u,w),v)\] \[V_{K}:\mathbb{C}^{n}\times\mathbb{C}^{n}\times\mathbb{C}^{3} \times\mathbb{C}^{m}\to\mathbb{C}^{n}\times\mathbb{C}^{n}\times\mathbb{C} \times\mathbb{C}^{m}, (z,u,w,v)\mapsto(z,\overline{z},\eta_{K}(z,u,w),v),\]

where \(\eta_{1}\in\operatorname{Aff}(\mathbb{C}^{2n+1};\mathbb{C}^{3})\), \(\eta_{r}\in\operatorname{Aff}(\mathbb{C}^{2n+3};\mathbb{C}^{3})\), and \(\eta_{K}\in\operatorname{Aff}(\mathbb{C}^{2n+3};\mathbb{C})\).

Now we approximate the functions \(\tilde{\varrho}\circ V_{r}\) for \(\ell\in\{1,\ldots,r\}\) and \(V_{K}\) by shallow CVNNs in the sense of Proposition A.5. Using Proposition 5.2 in the large register model, we approximate each of the \(n\) in-register pairs of input values and their complex conjugates, and each of the out-register neurons by a shallow CVNN with \(2\) hidden neurons. Similarly, the remaining in-register neuron and the out-register neuron of the small register model are approximated in the same way. This results in shallow CVNNs with \(2n+2m+(2\cdot 2+1)=2n+2m+5\) hidden neurons. Formally, we approximate

* \(\tilde{\varrho}\circ V_{1}\) by a sequence of CVNNs \(\psi\circ\varrho^{\times(2n+2m+5)}\circ\phi\) with \(\phi\in\operatorname{Aff}(\mathbb{C}^{2n+m+1};\mathbb{C}^{2n+2m+5})\) and \(\psi\in\operatorname{Aff}(\mathbb{C}^{2n+2m+5};\mathbb{C}^{2n+m+3})\),
* \(\tilde{\varrho}\circ V_{r}\) by a sequence of CVNNs \(\psi\circ\varrho^{\times(2n+2m+5)}\circ\phi\) with \(\phi\in\operatorname{Aff}(\mathbb{C}^{2n+m+3};\mathbb{C}^{2n+2m+5})\) and \(\psi\in\operatorname{Aff}(\mathbb{C}^{2n+2m+5};\mathbb{C}^{2n+m+3})\) for \(r\in\{2,\ldots,K-1\}\),
* \(V_{K}\) by a sequence of CVNNs \(\psi\circ\varrho^{\times(2n+2m+2)}\circ\phi\) with \(\phi\in\operatorname{Aff}(\mathbb{C}^{2n+m+3};\mathbb{C}^{2n+2m+2})\) and \(\psi\in\operatorname{Aff}(\mathbb{C}^{2n+2m+2};\mathbb{C}^{2n+m+1})\).

Overall, we deduce that each function \(F_{j}^{\ell}\) with \(j\in\mathbb{N}\) and \(\ell\in\{1,\ldots,N\}\) can be uniformly approximated using CVNNs from \(\mathcal{N}\mathcal{N}_{2n+m+1,2n+m+1,2n+2m+5}^{\varrho}\). Another application of Proposition A.5 shows that each function \(T_{1},\ldots,T_{N}\) appearing in (5.2) can be uniformly approximated using CVNNs from \(\mathcal{N}\mathcal{N}_{2n+m+1,2n+m+1,2n+2m+5}^{\varrho}\). Similar arguments apply to \(T_{\text{init}}\) and \(T_{\text{end}}\) in (5.2), and this shows that the polynomial \(p\) can be uniformly approximated by CVNNs from \(\mathcal{N}\mathcal{N}_{n,m,2n+2m+5}^{\varrho}\). Since \(p\) was arbitrary, this together with the Stone-Weierstrass theorem shows the universality of \(\mathcal{N}\mathcal{N}_{n,m,2n+2m+5}^{\varrho}\).

We now discuss in what way the preceding proof has to be adapted in order to show the universality of \(\mathcal{N}\mathcal{N}_{n,m,n+m+4}^{\varrho}\) if (5.1) is fulfilled.

First assume \(\partial_{\text{wirt}}\varrho(z_{0})\neq 0=\overline{\partial}_{\text{wirt}} \varrho(z_{0})\). Proposition 3.1(i) shows that we can approximate \(\operatorname{id}_{\mathbb{C}}\) arbitrarily well using shallow CVNNs of width \(1\) using \(\varrho\) as activation function. We then construct a register-model in the spirit of the first part of Step 1 that has in each layer \(n\) in-register neurons,
out-register neurons, one computation neuron realizing the available multiplication \(\operatorname{mul}_{k}\) and a _single_ conjugation neuron that stores the conjugated version of one of the in-registers. The idea is that we do not need to store the conjugates of all the inputs in every layer. In fact, it suffices to include one conjugation neuron in which the conjugate of one of the inputs is stored if that input is needed in the current computation step. Approximating the multiplication mul first by shallow networks according to Proposition 3.3 and then replacing these shallow networks by deep register models according to Proposition 3.4 yields that we approximate a given polynomial by large variants of CVNNs with a width of \(n+m+1+4\) where \(n+m+3\) neurons use the identity activation function, \(1\) neuron the conjugation and \(1\) neuron \(\varrho\) as activation function. Note also here that we may omit one of the identity neurons since its value is already contained in either the \(n\) in-registers or the conjugation neuron. This leads in total to a network of width \(n+m+4\) with \(n+m+2\) identity neurons, one conjugation neuron and one neuron that uses \(\varrho\) as activation function. Now we approximate \(n+m+1\) of these identity neurons in the spirit of Proposition 3.1(i) using shallow CVNNs with activation \(\varrho\) of width \(1\) and the remaining identity neuron together with the conjugation neuron in the spirit of Proposition 5.2 using a shallow CVNN with \(\varrho\) activation and a width of \(2\). This yields the universality of \(\mathcal{NN}_{n,m,n+m+4}^{\varrho}\).

If on the other hand \(\partial_{\operatorname{wirt}}\varrho(z_{0})=0\neq\overline{\partial}_{ \operatorname{wirt}}\varrho(z_{0})\) we argue as in the proof of Theorem 4.2(ii): By combining the previous part which yields the universality of \(\mathcal{NN}_{n,m,n+m+4}^{\overline{\varrho}}\) and Proposition 3.1(ii) we deduce the universality of \(\mathcal{NN}_{n,m,n+m+4}^{\varrho}\). Notice that \(\varrho\) satisfies Assumption 5.1 iff \(\overline{\varrho}\) does. 

## 6. Necessity of our assumptions

The proof of Theorem 1.1 is not yet complete. So far, we have proven that activation functions \(\varrho\in C(\mathbb{C};\mathbb{C})\) which are neither holomorphic, nor antiholomorphic, nor \(\mathbb{R}\)-affine yield universality of CVNNs of width \(2n+2m+5\), with indicated improvements under additional assumptions. The necessity part is done in this section: If \(\varrho\) is holomorphic, antiholomorphic, or \(\mathbb{R}\)-affine, then even the set of CVNNs using activation function \(\varrho\) with _arbitrary widths and depths_ is not universal, cf. Theorem 6.3. Furthermore, Theorem 1.1 states that under the mentioned constraints on the activation function, a width of \(2n+2m+5\) is sufficient for universality of CVNNs with input dimension \(n\) and output dimension \(m\). But could we have done better? In Theorem 6.4 below, we show that for a family of real-valued activation functions, the set \(\mathcal{NN}_{n,m,W}^{\varrho}\) is not universal when \(W<\max\left\{2n,2m\right\}\). In our final result Theorem 6.5, we show that real differentiability of the activation function at one point with non-vanishing derivative is not necessary for the universal approximation property of deep narrow CVNNs.

We prepare the proof of Theorem 6.3 by two lemmas, the first of which is about uniform convergence of \(\mathbb{R}\)-affine functions.

**Lemma 6.1**.: _Let \(n,m\in\mathbb{N}\), \((f_{k})_{k\in\mathbb{N}}\) be a sequence of \(\mathbb{R}\)-affine functions from \(\mathbb{R}^{n}\) to \(\mathbb{R}^{m}\) and \(f:\mathbb{R}^{n}\to\mathbb{R}^{m}\). Let \((f_{k})_{k\in\mathbb{N}}\) converge locally uniformly to \(f\). Then \(f\) is \(\mathbb{R}\)-affine too._

Proof.: Let \(A_{k}\in\mathbb{R}^{m\times n}\) and \(b_{k}\in\mathbb{R}^{m}\) such that \(f_{k}(x)=A_{k}x+b_{k}\). Let \(b:=f(0)\). Then we have \(b_{k}=f_{k}(0)\to f(0)=b\). Furthermore we see for every \(j\in\left\{1,\ldots,n\right\}\) that

\[\left\|A_{k}e_{j}-A_{\ell}e_{j}\right\|_{\mathbb{R}^{m}}\leq\left\|A_{k}e_{j} +b_{k}-A_{\ell}e_{j}-b_{\ell}\right\|_{\mathbb{R}^{m}}+\left\|b_{k}-b_{\ell} \right\|_{\mathbb{R}^{m}}\to 0\]

as \(k,\ell\to\infty\), uniformly over \(j\), meaning \(\max_{j\in\left\{1,\ldots,n\right\}}\left\|A_{k}e_{j}-A_{\ell}e_{j}\right\|_{ \mathbb{R}^{m}}\to 0\) as \(k,\ell\to\infty\). Consequently, \((A_{k})_{k\in\mathbb{N}}\) is a Cauchy sequence and thus converges to some \(A\in\mathbb{R}^{m\times n}\). We claim \(f(x)=Ax+b\) for every \(x\in\mathbb{R}^{n}\). Indeed, this follows from

\[\left\|A_{k}x+b_{k}-Ax+b\right\|_{\mathbb{R}^{m}}\leq\left\|A_{k}x-Ax\right\|_ {\mathbb{R}^{m}}+\left\|b_{k}-b\right\|_{\mathbb{R}^{m}}\leq\left\|A_{k}-A \right\|_{\mathbb{R}^{m\times n}}\left\|x\right\|_{\mathbb{R}^{m}}+\left\|b_{k}- b\right\|_{\mathbb{R}^{m}}\to 0.\]

as \(k\to\infty\). 

Our second lemma in preparation of the proof of Theorem 6.3 concerns locally uniform limits of sequences of functions that are either holomorphic or antiholomorphic.

**Lemma 6.2**.: _Let \(\mathcal{F}:=\left\{F:\mathbb{C}\to\mathbb{C}\;:\;F\;\text{holomorphic or antiholomorphic}\right\}\) and \((f_{k})_{k\in\mathbb{N}}\) be a sequence of functions with \(f_{k}\in\mathcal{F}\) for every \(k\in\mathbb{N}\). Let \(f:\mathbb{C}\to\mathbb{C}\) be such that \(f_{k}\to f\) locally uniformly. Then it holds \(f\in\mathcal{F}\)._
Proof.: We distinguish two cases:

1. If there is a subsequence of \((f_{k})_{k\in\mathbb{N}}\) consisting of holomorphic functions, the limit \(f\) of this subsequence also has to be holomorphic (see for instance [26, Theorem 10.28]).
2. If there is a subsequence of \((f_{k})_{k\in\mathbb{N}}\) consisting of antiholomorphic functions, the limit \(f\) of this subsequence also has to be antiholomorphic, where we again apply [26, Theorem 10.28] to the complex conjugates of the functions in this subsequence.

The necessity part of Theorem 1.1 is covered by the following theorem.

**Theorem 6.3**.: _Let \(n,m\in\mathbb{N}\) and \(\varrho:\mathbb{C}\to\mathbb{C}\) be holomorphic or antiholomorphic or \(\mathbb{R}\)-affine. Then \(\mathcal{N}\mathcal{N}^{\varrho}_{n,m}:=\bigcup\limits_{W\in\mathbb{N}} \mathcal{N}\mathcal{N}^{\varrho}_{n,m,W}\) is not universal._

Proof.: It suffices to show the claim for \(m=1\), since local uniform approximation in \(\mathbb{C}^{m}\) means componentwise local uniform approximation.

We start with the case \(n=1\). If \(\varrho\) is holomorphic or antiholomorphic it follows that all the elements of \(\mathcal{N}\mathcal{N}^{\varrho}_{1,1}\) are holomorphic or antiholomorphic (see, e.g., [28, Proof of Eq. (4.15), p. 28]. But then it follows from Lemma 6.2 and Proposition A.2 that \(\mathcal{N}\mathcal{N}^{\varrho}_{1,1}\) is not universal. If \(\varrho\) is \(\mathbb{R}\)-affine, each element of \(\mathcal{N}\mathcal{N}^{\varrho}_{1,1}\) is \(\mathbb{R}\)-affine (as the composition of \(\mathbb{R}\)-affine functions). By Lemma 6.1 and Proposition A.2 it follows that \(\mathcal{N}\mathcal{N}^{\varrho}_{1,1}\) is not universal.

The case \(n>1\) can be reduced to the case \(n=1\) in the following way: Assume that \(\mathcal{N}\mathcal{N}^{\varrho}_{n,1}\) is universal and pick any arbitrary function \(f\in C(\mathbb{C};\mathbb{C})\). Let \(\pi:\mathbb{C}^{n}\to\mathbb{C}\), \(\pi(z_{1},\ldots,z_{n})=z_{1}\) and \(\widetilde{\pi}:\mathbb{C}\to\mathbb{C}^{n}\), \(\widetilde{\pi}(z)=(z,0,\ldots,0)\). Note that it holds \(\pi\circ\widetilde{\pi}=\mathrm{id}_{\mathbb{C}}\). By assumption, there is a sequence \((g_{k})_{k\in\mathbb{N}}\) with \(g_{k}\in\mathcal{N}\mathcal{N}^{\varrho}_{n,1}\) satisfying \(g_{k}\in\mathcal{N}\mathcal{N}^{\varrho}_{n,1}\) for \(k\in\mathbb{N}\) and \(g_{k}\to f\circ\pi\) locally uniformly. From Proposition A.5 it follows \(g_{k}\circ\widetilde{\pi}\to f\) locally uniformly. Since \(g_{k}\circ\widetilde{\pi}\in\mathcal{N}\mathcal{N}^{\varrho}_{1,1}\) for every \(k\in\mathbb{N}\) it follows that \(\mathcal{N}\mathcal{N}^{\varrho}_{1,1}\) is universal, in contradiction to what has just been shown. 

In the previous sections, we showed that for a large class of activation functions a width of \(2n+2m+5\) is sufficient for universality of CVNNs with input dimension \(n\) and output dimension \(m\). Following the lines of [5, Lemma 1], we show next that for some activation functions, a width of at least \(\max\left\{2n,2m\right\}\) is necessary to guarantee universality.

**Theorem 6.4**.: _Let \(n,m\in\mathbb{N}\)._

1. _Let_ \(\phi\in C(\mathbb{R};\mathbb{C})\)_, and_ \(\varrho:\mathbb{C}\to\mathbb{C}\) _be given by_ \(\varrho(z):=\phi(\mathrm{Re}(z))\)_. Then_ \(\mathcal{N}\mathcal{N}^{\varrho}_{n,m,2n-1}\) _is not universal._
2. _Let_ \(\varrho:\mathbb{C}\to\mathbb{R}\)_. Then_ \(\mathcal{N}\mathcal{N}^{\varrho}_{n,m,2m-1}\) _is not universal._

Proof.: We start with (i). Let \(K:=[-2,2]^{n}+\mathrm{i}[-2,2]^{n}\subseteq\mathbb{C}^{n}\) and \(f(z):=\left(\left\|z\right\|_{\mathbb{C}^{n}},0,\ldots,0\right)\) for \(z\in\mathbb{C}^{n}\). Let \(g\in\mathcal{N}\mathcal{N}^{\varrho}_{n,m,2n-1}\) be arbitrary. From the definition of \(\varrho\), it follows that we may write \(g\) as

\[g(z)=\psi(\mathrm{Re}(Vz)+b),\]

where \(\psi:\mathbb{C}^{2n-1}\to\mathbb{C}^{m}\) is some function, \(V\in\mathbb{C}^{(2n-1)\times n}\), \(b\in\mathbb{R}^{2n-1}\), and the real part \(\mathrm{Re}\) is taken componentwise. Interpreting \(\mathrm{Re}\circ V\) as an \(\mathbb{R}\)-linear function from \(\mathbb{R}^{2n}\) to \(\mathbb{R}^{2n-1}\) we conclude from \(2n-1<2n\) that there exists \(v\in\mathbb{C}^{n}\) with \(\left\|v\right\|_{\mathbb{C}^{n}}=1\) satisfying \(\mathrm{Re}(Vv)=0\) and hence

\[g(z+v)=g(z)\text{ for any }z\in\mathbb{C}^{n}. \tag{6.1}\]

We set \(A:=B_{0.1}(0),B:=B_{0.1}(v)\subseteq\mathbb{C}^{n}\) and compute

\[\int_{K}\left\|f(z)-g(z)\right\|_{\mathbb{C}^{m}}\mathrm{d}z \geq\int_{A}\left\|f(z)-g(z)\right\|_{\mathbb{C}^{m}}\mathrm{d}z+ \int_{B}\left\|f(z)-g(z)\right\|_{\mathbb{C}^{m}}\mathrm{d}z\] \[\stackrel{{ B=\underline{A}+v}}{{\longrightarrow}}\int_{A} \left(\left\|f(z)-g(z)\right\|_{\mathbb{C}^{m}}+\left\|f(z+v)-g(z+v)\right\|_{ \mathbb{C}^{m}}\right)\mathrm{d}z\] \[\stackrel{{\eqref{eq:C}}}{{\geq}}\int_{A}\left\|f(z )-f(z+v)\right\|_{\mathbb{C}^{m}}\mathrm{d}z\geq 0.8\cdot\lambda^{2n}(A)\]
with \(\lambda^{2n}\) denoting the \(2n\)-dimensional Lebesgue measure. In the last inequality we used

\[\left|\left\|z\right\|_{\mathbb{C}^{n}}-\left\|z+v\right\|_{\mathbb{C}^{n}}\right| \geq\left\|z+v\right\|_{\mathbb{C}^{n}}-\left\|z\right\|_{\mathbb{C}^{n}}\geq \left\|v\right\|_{\mathbb{C}^{n}}-2\left\|z\right\|_{\mathbb{C}^{n}}\geq 0.8.\]

Hence it follows that \(\mathcal{NN}^{\varrho}_{n,m,2n-1}\) is not dense in \(C(K;\mathbb{C}^{m})\) with respect to the \(L^{1}\)-norm and thus, using Holder's inequality, it follows that \(\mathcal{NN}^{\varrho}_{n,m,2n-1}\) is not dense in \(C(K;\mathbb{C}^{m})\) with respect to the \(L^{p}\)-norm for any \(p\in[1,\infty]\), so in particular for \(p=\infty\) which shows that \(\mathcal{NN}^{\varrho}_{n,m,2n-1}\) is not universal.

Next, we prove (ii). To this end, we construct a function \(f\in C(\mathbb{C}^{n};\mathbb{C}^{m})\), a compact set \(K\subset\mathbb{C}^{n}\), and a number \(\varepsilon>0\) such that

\[\sup_{z\in K}\left\|f(z)-g(z)\right\|_{\mathbb{C}^{m}}\geq\varepsilon\]

for all \(g\in\mathcal{NN}^{\varrho}_{n,m,2m-1}\). For a moment, fix \(g\in\mathcal{NN}^{\varrho}_{n,m,2m-1}\). Since the activation function \(\varrho\) is real-valued, the output of the last but one layer of \(g\) is a function \(\psi:\mathbb{C}^{n}\to\mathbb{R}^{2m-1}\). Also, there are a matrix \(V\in\mathbb{C}^{m\times(2m-1)}\) and a vector \(b\in\mathbb{C}^{m}\) such that \(g(z)=V\psi(z)+b\). With \(\mathbb{C}^{m}\cong\mathbb{R}^{2m}\), we may view \(V\) as a linear map \(\mathbb{R}^{2m-1}\to\mathbb{R}^{2m}\), and the range \(\{g(z)\,:\,z\in\mathbb{C}\}\) of \(g\) is thus contained in a \((2m-1)\)-dimensional affine subspace \(U=U(g)\) of \(\mathbb{R}^{2m}\). As

\[\sup_{z\in K}\left\|f(z)-g(z)\right\|_{\mathbb{R}^{2m}}\geq\sup_{z\in K}\inf_{ u\in U(g)}\left\|f(z)-u\right\|_{\mathbb{R}^{2m}},\]

it is sufficient for our purposes to find a function \(f\in C(\mathbb{C}^{n};\mathbb{C}^{m})\), a compact set \(K\subset\mathbb{C}^{n}\), and a number \(\varepsilon>0\) such that

\[\inf_{U}\sup_{z\in K}\inf_{u\in U}\left\|f(z)-u\right\|_{\mathbb{R}^{2m}}\geq\varepsilon\]

where the outermost infimum traverses the \((2m-1)\)-dimensional affine subspaces \(U\) of \(\mathbb{R}^{2m}\). This is achieved by a function \(f\) whose range \(\{f(z)\,:\,z\in\mathbb{C}\}\) is not contained in any \((2m-1)\)-dimensional affine subspace \(U\) of \(\mathbb{R}^{2m}\). A semi-explicit construction is like this:

Let \(K:=\{(\lambda,0,\ldots,0)\,:\,\lambda\in\mathbb{R},0\leq\lambda\leq 1\}\subseteq \mathbb{C}^{n}\) and

\[f_{1}:\mathbb{C}^{n}\to[0,1],\quad f_{1}(z_{1},\ldots,z_{n})=\max\{0,\min\{1, \operatorname{Re}(z_{1})\}\}.\]

Further let \(f_{2}:[0,1]\to\mathbb{C}^{m}\) be a parameterization of a curve that along the edges of the cube \(Q:=[0,1]^{m}+\mathrm{i}[0,1]^{m}\subseteq\mathbb{C}^{m}\cong\mathbb{R}^{2m}\) passes through all of its vertices \(\{0,1\}^{m}+\mathrm{i}\,\{0,1\}^{m}\subseteq\mathbb{C}^{m}\cong\mathbb{R}^{ 2m}\), and \(f=f_{2}\circ f_{1}\). From [3, Corollary 2.5], we deduce

\[\inf_{U}\sup_{z\in K}\inf_{u\in U}\left\|f(z)-u\right\|_{\mathbb{R}^{2m}}\geq \frac{1}{2}\]

and this finishes the proof. 

Note that there are non-polyharmonic functions like \(\varrho(z)=\mathrm{e}^{\mathrm{Re}(z)}\) but also polyharmonic functions that are neither holomorphic, nor anti-holomorphic, nor \(\mathbb{R}\)-affine like \(\varrho(z)=\mathrm{Re}(z)^{2}\) that meet the assumptions made in Theorem 6.4.

Following the ideas from [15, Proposition 4.15] we also want to add a short note on the necessity of the differentiability of the activation function. It turns out that the differentiability of the activation function is _not_ a necessary condition for the fact that narrow networks with width \(n+m+1\) using this activation function have the universal approximation property. The proof is in fact identical to the proof presented in [15]. However, we include a detailed proof to clarify that the reasoning also works in the case of activation functions \(\mathbb{C}\to\mathbb{C}\).

**Theorem 6.5**.: _Take any function \(w\in C(\mathbb{C};\mathbb{C})\) which is bounded and nowhere real differentiable. Then \(\varrho(z):=\sin(z)+w(z)\exp(-z)\) is also nowhere differentiable and \(\mathcal{NN}^{\varrho}_{n,m,n+m+1}\) is universal._

Proof.: Since \(\varrho\) is non-polyharmonic (since it is nowhere differentiable) it suffices to show that the identity function can be uniformly approximated on compact sets using compositions of the form \(\psi\circ\varrho\circ\phi\) with \(\phi,\psi\in\mathrm{Aff}(\mathbb{C};\mathbb{C})\). Then the statement can be derived similarly to the proof of Theorem 4.2(i).

Therefore, take any compact set \(K\subseteq\mathbb{C}\) and \(\varepsilon>0\). Choose \(M_{1}>0\) with \(\left|z\right|\leq M_{1}\) for every \(z\in K\). Take \(h>0\) arbitrary and consider

\[\sup_{z\in K\setminus\{0\}}\left|\frac{\sin(hz)-hz}{h}\right|\leq\sup_{z\in K \setminus\{0\}}M_{1}\left|\frac{\sin(hz)}{hz}-1\right|\to 0\]
as \(h\to 0\). Therefore, we may take \(h>0\) with

\[\left|\frac{\sin(hz)-hz}{h}\right|<\frac{\varepsilon}{2}\]

for every \(z\in K\). Furthermore, choose \(M_{2}>0\) with \(|w(z)|\leq M_{2}\) for every \(z\in K\) and pick \(k\in\mathbb{N}\) large enough such that

\[\frac{\left|\exp(-hz)\right|\left|\exp(-2\pi k)\right|}{h}<\frac{\varepsilon} {2M_{2}}\]

for all \(z\in K\). Hence we derive

\[\left|\frac{\sin(hz+2\pi k)+w(hz+2\pi k)\exp(-hz-2\pi k)}{h}-z\right|\] \[= \left|\frac{\sin(hz)+w(hz+2\pi k)\exp(-hz-2\pi k)}{h}-z\right|\] \[\leq \left|\frac{\sin(hz)-hz}{h}\right|+M_{2}\frac{\left|\exp(-hz) \right|\left|\exp(-2\pi k)\right|}{h}<\varepsilon.\]

Thus, we get the claim by defining \(\phi(z):=hz+2\pi k\) and \(\psi(z):=\frac{1}{h}z\). 

## Appendix A Topological notes on locally uniform convergence

In this appendix, we discuss the relationship between locally uniform convergence, the compact open topology, and the universal approximation property introduced in Definition 2.4. Note that [21, Appendix B] is another account on the same topic.

Although locally uniform convergence can be studied more generally for functions defined on a topological space and taking values in a metric space, we restrict ourselves to functions \(\mathbb{C}^{n}\to\mathbb{C}^{m}\).

**Definition A.1**.: Let \((f_{k})_{k\in\mathbb{N}}\) be a sequence of functions \(f_{k}:\mathbb{C}^{n}\to\mathbb{C}^{m}\) and \(f:\mathbb{C}^{n}\to\mathbb{C}^{m}\). The sequence \((f_{k})_{k\in\mathbb{N}}\) converges locally uniformly to \(f\), if for every compact set \(K\subseteq\mathbb{C}^{n}\) we have

\[\sup_{z\in K}\left\|f_{k}(z)-f(z)\right\|_{\mathbb{C}^{m}}\xrightarrow{k\to \infty}0.\]

There is a certain equivalence between locally uniform convergence and the universal approximation property introduced in Definition 2.4.

**Proposition A.2**.: _Let \(\mathcal{F}\subseteq C(\mathbb{C}^{n};\mathbb{C}^{m})\) and \(f\in C(\mathbb{C}^{n};\mathbb{C}^{m})\). Then the following are equivalent:_

_(i)_ _For every compact set_ \(K\subseteq\mathbb{C}^{n}\) _and_ \(\varepsilon>0\)_, there is a function_ \(g\in\mathcal{F}\) _satisfying_

\[\left\|f-g\right\|_{C(K;\mathbb{C}^{m})}<\varepsilon.\]

_(ii)_ _There is a sequence_ \((f_{k})_{k\in\mathbb{N}}\) _with_ \(f_{k}\in\mathcal{F}\) _for_ \(k\in\mathbb{N}\) _such that_ \((f_{k})_{k\in\mathbb{N}}\) _converges locally uniformly to_ \(f\)_._

Proof.: We start with the implication (i)\(\Rightarrow\)(ii). Let \(f\in C(\mathbb{C}^{n};\mathbb{C}^{m})\). For every \(k\in\mathbb{N}\), choose \(f_{k}\in\mathcal{F}\) with

\[\left\|f_{k}-f\right\|_{C(\overline{B}_{k}(0);\mathbb{C}^{m})}\leq\frac{1}{k}.\]

Then \((f_{k})_{k\in\mathbb{N}}\) converges locally uniformly to \(f\), since every compact set \(K\subseteq\mathbb{C}^{n}\) is contained in \(\overline{B}_{k}(0)\) for all \(k\geq J\) for some \(J\in\mathbb{N}\).

Now we show the implication (ii)\(\Rightarrow\)(i). For any compact set \(K\subseteq\mathbb{C}^{n}\) and \(\varepsilon>0\) we know by definition of locally uniform convergence that there is \(k\in\mathbb{N}\) satisfying

\[\left\|f_{k}-f\right\|_{C(K;\mathbb{C}^{m})}<\varepsilon.\]

Since \(f_{k}\in\mathcal{F}\), this shows (i). 

In particular, Proposition A.2 yields the following equivalence:

(i') The set \(\mathcal{F}\) has the universal approximation property.
* For every \(f\in C(\mathbb{C}^{n};\mathbb{C}^{m})\), there is a sequence \((f_{k})_{k\in\mathbb{N}}\) of elements \(f_{k}\in\mathcal{F}\) such that \((f_{k})_{k\in\mathbb{N}}\) converges locally uniformly to \(f\).

Next, we show that locally uniform convergence of sequences \((f_{k})_{k\in\mathbb{N}}\) of elements \(f_{k}\in C(\mathbb{C}^{n};\mathbb{C}^{m})\) coincides with convergence with respect to the _compact-open topology_, cf. [8, Definition XII.1.1]. Hence, the compact-open topology is the topology in charge when we speak about universality of a set of continuous functions.

**Definition A.3**.: For each pair of sets \(A\subseteq\mathbb{C}^{n},B\subseteq\mathbb{C}^{m}\), we denote

\[(A,B):=\left\{f\in C(\mathbb{C}^{n};\mathbb{C}^{m})\::\:f(A)\subseteq B\right\}.\]

The _compact-open topology_ on \(C(\mathbb{C}^{n};\mathbb{C}^{m})\) is then the smallest topology containing the sets \((K,V)\), where \(K\subseteq\mathbb{C}^{n}\) is compact and \(V\subseteq\mathbb{C}^{m}\) is open.

That this topology indeed induces locally uniform convergence is a direct consequence of [8, Theorem XII.7.2] and \(\mathbb{C}^{m}\) being a metric space.

**Proposition A.4**.: _Let \((f_{k})_{k\in\mathbb{N}}\) be a sequence of functions with \(f_{k}\in C(\mathbb{C}^{n};\mathbb{C}^{m})\) and \(f\in C(\mathbb{C}^{n};\mathbb{C}^{m})\). Then the following statements are equivalent._

* _The sequence_ \((f_{k})_{k\in\mathbb{N}}\) _converges to_ \(f\) _in the compact-open topology._
* _The sequence_ \((f_{k})_{k\in\mathbb{N}}\) _converges to_ \(f\) _locally uniformly._

In the present paper, it is of particular importance that the composition of functions is compatible with locally uniform convergence.

**Proposition A.5**.: _Let \((f_{k})_{k\in\mathbb{N}}\) and \((g_{k})_{k\in\mathbb{N}}\) be two sequences of functions with \(f_{k}\in C(\mathbb{C}^{n_{1}};\mathbb{C}^{n_{2}})\) and \(g_{k}\in C(\mathbb{C}^{n_{2}};\mathbb{C}^{n_{3}})\) for \(k\in\mathbb{N}\). Let \(f\in C(\mathbb{C}^{n_{1}};\mathbb{C}^{n_{2}})\) and \(g\in C(\mathbb{C}^{n_{2}};\mathbb{C}^{n_{3}})\) such that \(f_{k}\to f\) and \(g_{k}\to g\) locally uniformly. Then we have_

\[g_{k}\circ f_{k}\xrightarrow{k\to\infty}g\circ f\]

_locally uniformly._

Proof.: Using [8, Theorem XII.2.2], we know that the map

\[C(\mathbb{C}^{n_{1}};\mathbb{C}^{n_{2}})\times C(\mathbb{C}^{n_{2}};\mathbb{ C}^{n_{3}})\to C(\mathbb{C}^{n_{1}};\mathbb{C}^{n_{3}}),\quad(h_{1},h_{2}) \mapsto h_{2}\circ h_{1}\]

is continuous, where each space \(C(\mathbb{C}^{n_{j}};\mathbb{C}^{n_{k}})\) is equipped with the compact-open topology and Cartesian products of spaces are equipped with the product topology. Note here that we use the fact that \(\mathbb{C}^{n_{1}}\) and \(\mathbb{C}^{n_{3}}\) are Hausdorff spaces and \(\mathbb{C}^{n_{2}}\) is locally compact. Then the claim follows from Proposition A.4. 

Note that the statement of Proposition A.5 can inductively be extended to the composition of \(L\) functions, where \(L\) is any natural number.

## Appendix B Taylor expansion using Wirtinger derivatives

In this appendix we give some details about the Taylor expansion introduced in Lemma 2.1. Furthermore we show that an activation function which is not \(\mathbb{R}\)-affine necessarily admits a point where one of the second-order Wirtinger derivatives does not vanish. We begin by restating and proving Lemma 2.1.

**Lemma B.1**.: _Let \(\varrho\in C(\mathbb{C};\mathbb{C})\) and \(z,z_{0}\in\mathbb{C}\). If \(\varrho\) is real differentiable in \(z_{0}\), then_

\[\varrho(z+z_{0})=\varrho(z_{0})+\partial_{\rm{wirt}}\varrho(z_{0})z+\overline {\partial}_{\rm{wirt}}\varrho(z_{0})\overline{z}+\Theta_{1}(z)\] (B.1)

_for a function \(\Theta_{1}:\mathbb{C}\to\mathbb{C}\) with \(\lim_{\mathbb{C}\setminus\{0\}\ni z\to 0}\frac{\Theta_{1}(z)}{z}=0\). If \(\varrho\in C^{2}(\mathbb{C};\mathbb{C})\), then_

\[\varrho(z+z_{0})=\varrho(z_{0})+\partial_{\rm{wirt}}\varrho(z_{0})z+\overline {\partial}_{\rm{wirt}}\varrho(z_{0})\overline{z}+\frac{1}{2}\partial_{\rm{wirt }}^{2}\varrho(z_{0})z^{2}+\partial_{\rm{wirt}}\overline{\partial}_{\rm{wirt}} \varrho(z_{0})z\overline{z}+\frac{1}{2}\overline{\partial}_{\rm{wirt}}^{2} \varrho(z_{0})\overline{z}^{2}+\Theta_{2}(z)\] (B.2)

_for a function \(\Theta_{2}:\mathbb{C}\to\mathbb{C}\) with \(\lim_{\mathbb{C}\setminus\{0\}\ni z\to 0}\frac{\Theta_{2}(z)}{z^{2}}=0\)._
Proof.: Equation (B.1) follows from the definition of real differentiability (2.1) by using

\[\frac{\partial\varrho}{\partial x}(z_{0})\operatorname{Re}(z)+\frac{\partial \varrho}{\partial y}(z_{0})\operatorname{Im}(z)=\frac{\partial\varrho}{ \partial x}(z_{0})\cdot\frac{1}{2}(z+\overline{z})+\frac{\partial\varrho}{ \partial y}\cdot\frac{1}{2\mathrm{i}}(z-\overline{z})=\partial_{\mathrm{wirt} }\varrho(z_{0})z+\overline{\partial}_{\mathrm{wirt}}\varrho(z_{0})\overline{z}.\]

In order to prove (B.2) we use the second-order Taylor expansion of \(\varrho\) around \(z_{0}\) which can be found for instance in [1, Theorem VII.5.11] and obtain

\[\varrho(z+z_{0})=\varrho(z_{0})+\frac{\partial\varrho}{\partial x}(z_{0})x+ \frac{\partial\varrho}{\partial y}(z_{0})y+\frac{1}{2}\frac{\partial^{2} \varrho}{\partial x^{2}}(z_{0})x^{2}+\frac{\partial^{2}\varrho}{\partial x \partial y}(z_{0})xy+\frac{1}{2}\frac{\partial^{2}\varrho}{\partial y^{2}}(z_ {0})y^{2}+\Theta_{2}(z)\]

where \(\Theta_{2}:\mathbb{C}\to\mathbb{C}\) satisfies \(\lim_{\mathbb{C}\setminus\{0\}\ni z\to 0}\frac{\Theta_{2}(z)}{z^{2}}=0\). Furthermore, we use the notation \(x=\operatorname{Re}(z)\) and \(y=\operatorname{Im}(z)\). Letting \(x=\frac{1}{2}(z+\overline{z})\), \(y=\frac{1}{2\mathrm{i}}(z-\overline{z})\) and using

\[\frac{1}{4}\cdot\begin{pmatrix}1&-2\mathrm{i}&-1\\ 1&0&1\\ 1&2\mathrm{i}&-1\end{pmatrix}\begin{pmatrix}\frac{\partial^{2}}{\partial x^{2} }\\ \frac{\partial^{2}}{\partial x\partial y}\\ \frac{\partial^{2}}{\partial y^{2}}\end{pmatrix}=\begin{pmatrix}\partial_{ \mathrm{wirt}}^{2}\\ \partial_{\mathrm{wirt}}^{2}\overline{\partial}_{\mathrm{wirt}}\\ \overline{\partial}_{\mathrm{wirt}}^{2}\end{pmatrix}\] (B.3)

yields the claim. 

The following Proposition is required in the proof of Proposition 3.2.

**Proposition B.2**.: _Let \(\varrho\in C^{2}(\mathbb{C};\mathbb{C})\) be not \(\mathbb{R}\)-affine. Then there is a point \(z_{0}\in\mathbb{C}\) such that either \(\partial_{\mathrm{wirt}}^{2}\varrho(z_{0})\neq 0\), \(\partial_{\mathrm{wirt}}\overline{\partial}_{\mathrm{wirt}}\varrho(z_{0})\neq 0\) or \(\overline{\partial}_{\mathrm{wirt}}^{2}\varrho(z_{0})\neq 0\)._

Proof.: Assume \(\partial_{\mathrm{wirt}}^{2}\varrho\equiv\partial_{\mathrm{wirt}}\overline{ \partial}_{\mathrm{wirt}}\varrho\equiv\overline{\partial}_{\mathrm{wirt}}^{2} \varrho\equiv 0\). From the fact that the matrix on the left-hand side in (B.3) is invertible it follows \(\frac{\partial^{2}\varrho}{\partial x^{2}}\equiv\frac{\partial^{2}\varrho}{ \partial x\partial y}\equiv\frac{\partial^{2}\varrho}{\partial y^{2}}\equiv 0\). Since \(\varrho\) is \(\mathbb{R}\)-affine if and only \(\operatorname{Re}(\varrho)\) and \(\operatorname{Im}(\varrho)\) are both \(\mathbb{R}\)-affine, we may assume that \(\varrho\) is real-valued. It is a well-known fact that a \(C^{1}\)-function with vanishing gradient is necessarily constant. Applying this fact to \(\frac{\partial\varrho}{\partial x}\) and \(\frac{\partial\varrho}{\partial y}\) separately shows

\[\nabla\varrho\equiv a\]

for a constant \(a\in\mathbb{R}^{2}\). Let \(f(z):=z^{\top}a\) where \(z\in\mathbb{C}\) is treated as an element of \(\mathbb{R}^{2}\). Then the gradient of \(\varrho-f\) vanishes identically and hence it holds \(\varrho-f\equiv b\) for a constant \(b\in\mathbb{R}\). This yields

\[\varrho(z)=z^{\top}a+b\quad\text{for all $z\in\mathbb{C}$}.\]

But then \(\varrho\) is \(\mathbb{R}\)-affine. 

**Acknowledgements.** The authors thank Felix Voigtlaender for helpful comments. PG acknowledges support by the German Science Foundation (DFG) in the context of the Emmy Noether junior research group VO 2594/1-1.
