About optimization of methods for mixed derivatives of bivariate functions

††footnotetext: Key words. Numerical differentiation, Legendre polynomials,
truncation method, minimal radius of Galerkin information.

Y.V. Semenova $\\!\\!{}^{{\dagger}}$, S.G. Solodky
$\\!\\!{}^{{\dagger},{\ddagger}}$

$\\!\\!{}^{{\dagger}}\\!\\!$ Institute of Mathematics, National Academy of
Sciences of Ukraine, Kyiv

$\\!\\!{}^{{\ddagger}}\\!\\!$ University of Giessen, Department of
Mathematics, Giessen, Germany

> Abstract.

> The problem of optimal recovering high-order mixed derivatives of bivariate
> functions with finite smoothness is studied. On the basis of the truncation
> method, an algorithm for numerical differentiation is constructed, which is
> order-optimal both in the sense of accuracy and in terms of the amount of
> involved Galerkin information.

##  1\. Description of the problem

Currently, many research activities on the problem of stable numerical
differentiation have been taking place due to the importance of this tool in
such areas of science and technology as finance, mathematical physics, image
processing, analytical chemistry, viscous elastic mechanics, reliability
analysis, pattern recognition, and many others. Among these investigations we
highlight [4], which is the first publication on numerical differentiation in
terms of the theory of ill-posed problems. Further research [4] has been
continued in numerous publications on numerical differentiation (see, for
example, [21], [30], [7], [8], [1], [9], [12], [31], [32], [10], [24]),
covering different classes of the functions and the types of proposed methods.
Despite the abundance of works on this topic, the problem for recovery of high
order derivatives was considered only in a few publications, among which we
note [5], [22], [11], [19], [12] and [24]. In particular, the results of [24]
have opened perspective for further investigation of numerical methods for
recovery of high order derivatives. Namely as the main criteria of method’s
efficiency have been taken its ability to achieve the optimal order of
accuracy by using the minimal amount of discrete information. Note that
particular these aspects of numerical differentiation remains still
insufficiently studied. The present paper continues the research of [24], [23]
and proposes a numerical method for recovering the high order mixed
derivatives of smooth bivariate functions. The method is not only stable to
small perturbations of the input data, but is also optimal in terms of
accuracy and quantity of involved Fourier-Legendre coefficients, and also has
a simple numerical implementation.

Let $\\{\varphi_{k}(t)\\}_{k=0}^{\infty}$ be the system of Legendre
polynomials orthonormal on $[-1,1]$ as

$\varphi_{k}(t)=\sqrt{k+1/2}(2^{k}k!)^{-1}\frac{d^{k}}{dt^{k}}[(t^{2}-1)^{k}],\quad
k=0,1,2,\ldots.$

By $L_{2}=L_{2}(Q)$ we mean space of square-summable on $Q=[-1,1]^{2}$
functions $f(t,\tau)$ with inner product

$\langle f,g\rangle=\int_{-1}^{1}\int_{-1}^{1}f(t,\tau)g(t,\tau)d\tau dt$

and standard norm

$\|f\|_{L_{2}}^{2}=\sum_{k,j=0}^{\infty}|\langle
f,\varphi_{k,j}\rangle|^{2}<\infty,$

where

$\langle
f,\varphi_{k,j}\rangle=\int_{-1}^{1}\int_{-1}^{1}f(t,\tau)\varphi_{k}(t)\varphi_{j}(\tau)d\tau
dt,\quad k,j=0,1,2,\ldots,$

are Fourier-Legendre coefficients of $f$. Let $\ell_{p}$, $1\leq p\leq\infty$,
be the space of numerical sequences
$\overline{x}=\\{x_{k,j}\\}_{k,j\in\mathbb{N}_{0}}$,
$\mathbb{N}_{0}=\\{0\\}\bigcup\mathbb{N}$, such that the corresponding
relation

$\|\overline{x}\|_{\ell_{p}}:=\left\\{\begin{array}[]{cl}\bigg{(}\sum\limits_{k,j\in\mathbb{N}_{0}}|x_{k,j}|^{p}\bigg{)}^{\frac{1}{p}}<\infty,&1\leq
p<\infty,\\\ \\\
\sup\limits_{k,j\in\mathbb{N}_{0}}|x_{k,j}|<\infty,&p=\infty,\end{array}\right.$

is fulfilled.

We introduce the space of functions

$L_{s,2}^{\mu}(Q)=\\{f\in
L_{2}(Q):\quad\|f\|_{s,\mu}^{s}:=\sum_{k,j=0}^{\infty}({\underline{k\cdot
j}})^{s\mu}|\langle f,\varphi_{k,j}\rangle|^{s}<\infty\\},$

where $\mu>0$, $1\leq s<\infty$, $\underline{k}=\max\\{1,k\\}$,
$k=0,1,2,\dots$. Note that in the future we will use the same notations both
for space and for a unite ball from this space:
$L_{s,2}^{\mu}=L_{s,2}^{\mu}(Q)=\\{f\in L_{s,2}^{\mu}\\!:\|f\|_{s,\mu}\leq
1\\}$, what we call a class of functions. What exactly is meant by
$L_{s,2}^{\mu}$, space or class, will be clear depending on the context in
each case. It should be noted that $L^{\mu}_{s,2}$ is generalization of the
class of bivariate functions with dominating mixed partial derivatives.
Moreover, let $C=C(Q)$ be the space of continuous on $Q$ bivariate functions.

We represent a function $f(t,\tau)$ from $L_{s,2}^{\mu}$ as

$f(t,\tau)=\sum_{k,j=0}^{\infty}\langle
f,\varphi_{k,j}\rangle\varphi_{k}(t)\varphi_{j}(\tau),$

and by its mixed derivative we mean the following series

$f^{(r_{1},r_{2})}(t,\tau)=\sum_{k,j=r}^{\infty}\langle
f,\varphi_{k,j}\rangle\varphi^{(r_{1})}_{k}(t)\varphi^{(r_{2})}_{j}(\tau),\quad
r_{1},r_{2}=1,2,\ldots.$ (1.1)

Assume that instead of the exact values of the Fourier-Legendre coefficients
$\langle f,\varphi_{k,j}\rangle$ only some their perturbations are known with
the error level $\delta$ in the metrics of $\ell_{p}$, $1\leq p\leq\infty$.
More accurately, we assume that there is a sequence of numbers
$\overline{f^{\delta}}=\\{\langle
f^{\delta},\varphi_{k,j}\rangle\\}_{k,j\in\mathbb{N}_{0}}$ such that for
$\overline{\xi}=\\{\xi_{k,j}\\}_{k,j\in\mathbb{N}_{0}}$, where
$\xi_{k,j}=\langle f-f_{\delta},\varphi_{k,j}\rangle$, and for some $1\leq
p\leq\infty$ the relation

$\|\overline{\xi}\|_{\ell_{p}}\leq\delta,\quad 0<\delta<1,$ (1.2)

is true.

The research of this work is devoted to the optimization of methods for
recovering the derivative (1.1) of functions from classes $L^{\mu}_{s,2}$.
Further, we give a strict statement of the problem to be studied. In the
coordinate space $[r_{1},\infty)\times[r_{2},\infty)$ we take an arbitrary
bounded domain $\Omega$. By $\text{card}\vskip 6.0pt plus 2.0pt minus
2.0pt(\Omega)$ we mean the number of points that make up $\Omega$ and by the
information vector $G(\Omega,\overline{f}^{\delta})\in\mathbb{R}^{N}$,
$\text{card}\vskip 6.0pt plus 2.0pt minus 2.0pt(\Omega)=N$, we take the set of
perturbed values of Fourier-Legendre coefficients $\left\\{\langle
f^{\delta},\varphi_{k,j}\rangle\right\\}_{(k,j)\in\Omega}$.

Let $X=L_{2}(Q)$ or $X=C(Q)$. By numerical differentiation algorithm we mean
any mapping $\psi^{(r_{1},r_{2})}=\psi^{(r_{1},r_{2})}(\Omega)$ that
corresponds to the information vector $G(\Omega,\overline{f}^{\delta})$ an
element $\psi^{(r_{1},r_{2})}(G(\Omega,\overline{f}^{\delta}))\in X$, which is
taken as an approximation to the derivative (1.1) of function $f$ from the
class $L_{s,2}^{\mu}$. We denote by $\Psi(\Omega)$ the set of all algorithms
$\psi^{(r_{1},r_{2})}(\Omega):\,\mathbb{R}^{N}\rightarrow X$, that use the
same information vector $G(\Omega,\overline{f}^{\delta})$.

We do not require from algorithms which belong to $\Psi(\Omega)$, in generally
speaking, either linearity or even stability. The only condition for
algorithms from $\Psi(\Omega)$ is to use input information in the form of
perturbed values of the coefficients Fourier-Legendre with indices from the
domain $\Omega$ of the coordinate plane. Such the general understanding of the
algorithm is explained by the desire to compare the widest possible range of
possible methods of numerical differentiation.

The error of the algorithm $\psi^{(r_{1},r_{2})}$ on the class $L^{\mu}_{s,2}$
is determined by the quantity

$\varepsilon_{\delta}(L^{\mu}_{s,2},\psi^{(r_{1},r_{2})}(\Omega),X,\ell_{p})=\sup\limits_{f\in
L^{\mu}_{s,2},\,\|f\|_{s,\mu}\leq 1}\
\,\sup\limits_{\overline{f}^{\delta}:\,(\ref{perturbation})}\|f^{(r_{1},r_{2})}-\psi^{(r_{1},r_{2})}(G(\Omega,\overline{f}^{\delta}))\|_{X}.$

The minimal radius of the Galerkin information for the problem of numerical
differentiation on the class $L^{\mu}_{s,2}$ is given by

$R^{(r_{1},r_{2})}_{N,\delta}(L^{\mu}_{s,2},X,\ell_{p})=\inf\limits_{\Omega:\,\text{card}\vskip
6.0pt plus 2.0pt minus 2.0pt(\Omega)\leq N}\
\,\inf\limits_{\psi^{(r_{1},r_{2})}\in\Psi(\Omega)}\varepsilon_{\delta}(L^{\mu}_{s,2},\psi^{(r_{1},r_{2})}(\Omega),X,\ell_{p}).$

This value $R^{(r_{1},r_{2})}_{N,\delta}(L^{\mu}_{s,2},X,\ell_{p})$ describes
the minimal possible accuracy in the metric of $X$, which can be achieved by
numerical differentiation of arbitrary function $f\in L_{s,2}^{\mu}$ , while
using not more than $N$ values of its Fourier-Legendre coefficients that are
$\delta$-perturbed in the $\ell_{p}$ metric. Note that the minimal radius of
Galerkin information in the problem of recovering the first partial derivative
was studied in [27], and for other types of ill-posed problems, similar
studies were previously carried out in [18], [13]. It should be added that the
minimal radius characterizes the information complexity of the considered
problem and is traditionally studied within the framework of the IBC-theory
(Information- Based Complexity Theory), the foundations of which are laid in
monographs [29] and [28].

The aim of our research is to find sharp estimates (in the power scale) for
the quantities $R^{(r_{1},r_{2})}_{N,\delta}(L^{\mu}_{s,2},C,\ell_{p})$ and
$R^{(r_{1},r_{2})}_{N,\delta}(L^{\mu}_{s,2},L_{2},\ell_{p})$.

##  2\. Truncation method. Error estimate in $L_{2}$– metric

It should be noted that at the moment a number of approaches were developed
for numerical differentiation (see, for example, [3], [2], [20], [22], and
also see [25] and its bibliography). All these methods are accepted to divide
into three groups (see [22]): difference methods, interpolation methods and
regularization methods. As is known, the first two types of methods have their
advantage in the simplicity of implementation, but they guarantee satisfactory
accuracy only in the case of exactly given input data about the differentiable
function. In the same time regularization methods give stable approximations
to the desired derivatives in the case of perturbed input data but most of
them (for example, the Tikhonov method and its various variations) are quite
complicated for numerical realization in view of their integral form and
require hard-to-implement rules for determination of regularization parameters
(see [22]). Recently in [24] a concise numerical method, called the truncation
method, has been proposed as stable and simple approach to numerical
differentiation of multivariable functions. The essence of this method is to
replace the Fourier series (1.1) with a finite Fourier sum using perturbed
data $\langle f_{\delta},\varphi_{k,j}\rangle$. In the truncation method to
ensure the stability of the approximation and achieve the required order
accuracy, it is necessary to choose properly the discretization parameter,
which here serves as a regularization parameter. So, the process of
regularization in method under consideration consists in matching the
discretization parameter with the perturbation level $\delta$ of the input
data. Simplicity of implementation is the main advantage of this method.

In the case of an arbitrary bounded domain $\Omega$ of the coordinate plane
$[r_{1},\infty)\times[r_{2},\infty)$, the truncation method for
differentiating functions of two variables has the form

$\mathcal{D}_{\Omega}^{(r_{1},r_{2})}f_{\delta}(t,\tau)=\sum_{(k,j)\in\Omega}\langle
f_{\delta},\varphi_{k,j}\rangle\varphi_{k}^{(r_{1})}(t)\varphi_{j}^{(r_{2})}(\tau).$

In order to increase the efficiency of the approach under study, we take a
hyperbolic cross as the domain $\Omega$ of the following form

$\Omega=\Gamma_{n,\gamma}:=\\{(k,j):\ k\cdot j^{\gamma}\leq n,\quad
k=r_{1},\ldots,n/r_{2}^{\gamma},\quad
j=r_{2},\ldots,(n/r_{1})^{1/\gamma}\\},\qquad\gamma\geq 1.$

Then the proposed version of the truncation method can be written as

$\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f_{\delta}(t,\tau)=\sum_{k=r_{1}}^{n/r_{2}^{\gamma}}\,\sum_{j=r_{2}}^{(n/k)^{1/\gamma}}\,\langle
f_{\delta},\varphi_{k,j}\rangle\varphi^{(r_{1})}_{k}(t)\varphi^{(r_{2})}_{j}(\tau).$
(2.1)

We note that earlier the idea of a hyperbolic cross in the problem of
numerical differentiation was used in the papers [26], [27], [24] (for more
details about usage of hyperbolic cross in solving the other types of ill-
posed problems see [18], [13], [15]).

The approximation properties of the method (2.1) will be investigated in
Sections 2 and 3 while in Section 4 it will be established that the method
(2.1) is order-optimal in the sense of the minimal radius of the Galerkin
information.

The parameters $n$ and $\gamma$ in (2.1) should be chosen depending on
$\delta$, $p$, $s$, $r_{1},r_{2}$ and $\mu$ so as to minimize the error of the
method $\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}$:

$f^{(r_{1},r_{2})}(t,\tau)-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f_{\delta}(t,\tau)$
$=\left(f^{(r_{1},r_{2})}(t,\tau)-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f(t,\tau)\right)+\left(\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f(t,\tau)-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f_{\delta}(t,\tau)\right).$
(2.2)

For the first difference on the right-hand side of (2.2), the representation

$f^{(r_{1},r_{2})}(t,\tau)-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f(t,\tau)=\triangle_{1}(t,\tau)+\triangle_{2}(t,\tau)$
(2.3)

holds true, where

$\triangle_{1}(t,\tau)=\sum_{k=n/r_{2}^{\gamma}+1}^{\infty}\quad\sum_{j=r_{2}}^{\infty}\langle
f,\varphi_{kj}\rangle\varphi^{(r_{1})}_{k}(t)\,\varphi^{(r_{2})}_{j}(\tau),$
(2.4)
$\triangle_{2}(t,\tau)=\sum_{k=r_{1}}^{n/r_{2}^{\gamma}}\quad\sum_{j=(n/k)^{1/\gamma}+1}^{\infty}\langle
f,\varphi_{kj}\rangle\varphi^{(r_{1})}_{k}(t)\,\varphi^{(r_{2})}_{j}(\tau).$
(2.5)

For our calculations, we need the following formula (see Lemma 18 [16])

$\varphi_{k}^{\prime}(t)\,=2\,\sqrt{k+1/2}\mathop{{\sum}^{*}}\limits_{l=0}^{k-1}\sqrt{l+1/2}\,\varphi_{l}(t),\
\ \ k\in\mathbb{N},$ (2.6)

where in aggregate
$\mathop{{\sum}^{*}}\limits_{l=0}^{k-1}\sqrt{l+1/2}\,\varphi_{l}(t)\,$ the
summation is extended over only those terms for which $k+l$ is odd.

Let us estimate the error of the method (2.1) in the metric of $L_{2}$. Upper
bounds for difference (2.3) are contained in the following statement.

###### Lemma 2.1

Let $f\in L^{\mu}_{s,2}$, $1\leq s<\infty$, $r_{1}\geq r_{2}$,
$\mu>2r_{1}-1/s+1/2$.

I) If $r_{1}=r_{2}$, then for $\gamma=1$ it holds true

$\|f^{(r_{1},r_{1})}-\mathcal{D}_{n,1}^{(r_{1},r_{1})}f\|_{L_{2}}\leq
c\|f\|_{s,\mu}n^{-\mu+2r_{1}-1/s+1/2}\ln^{3/2-1/s}n.$

II) If $r_{1}>r_{2}$, then
a) for any
$\gamma\in\left[1,\frac{\mu-2r_{2}+1/s-1/2}{\mu-2r_{1}+1/s+1/2}\right)\bigcup\left(\frac{\mu-2r_{2}+1/s-1/2}{\mu-2r_{1}+1/s+1/2},\frac{\mu-2r_{2}+1/s+1/2}{\mu-2r_{1}+1/s+1/2}\right)\bigcup\left(\frac{\mu-2r_{2}+1/s+1/2}{\mu-2r_{1}+1/s+1/2},\frac{\mu-2r_{2}+1/s-1/2}{\mu-2r_{1}+1/s-1/2}\right)$
it holds true

$\|f^{(r_{1},r_{2})}-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f\|_{L_{2}}\leq
c\|f\|_{s,\mu}n^{-\mu+2r_{1}-1/s+1/2},$

b) for $\gamma=\frac{\mu-2r_{2}+1/s-1/2}{\mu-2r_{1}+1/s+1/2}$ and
$\gamma=\frac{\mu-2r_{2}+1/s-1/2}{\mu-2r_{1}+1/s-1/2}$ it holds true

$\|f^{(r_{1},r_{2})}-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f\|_{L_{2}}\leq
c\|f\|_{s,\mu}n^{-\mu+2r_{1}-1/s+1/2}\ln^{1/2}n,$

c) for $\gamma=\frac{\mu-2r_{2}+1/s+1/2}{\mu-2r_{1}+1/s+1/2}$ it holds true

$\|f^{(r_{1},r_{2})}-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f\|_{L_{2}}\leq
c\|f\|_{s,\mu}n^{-\mu+2r_{1}-1/s+1/2}\ln^{1-1/s}n.$

Proof. Using the formula (2.6), from (2.4) we have

$\triangle_{1}(t,\tau)=2^{r_{1}+r_{2}}\sum_{k=n/r_{2}^{\gamma}+1}^{\infty}\quad\sum_{j=r_{2}}^{\infty}\sqrt{k+1/2}\sqrt{j+1/2}\>\langle
f,\varphi_{kj}\>\rangle$
$\times\mathop{{\sum}^{*}}\limits_{l_{1}=r_{1}-1}^{k-1}(l_{1}+1/2)\>\mathop{{\sum}^{*}}\limits_{l_{2}=r_{1}-2}^{l_{1}-1}(l_{2}+1/2)\ldots\mathop{{\sum}^{*}}\limits_{l_{r_{1}-1}=1}^{l_{r_{1}-2}-1}(l_{r_{1}-1}+1/2)\mathop{{\sum}^{*}}\limits_{l_{r_{1}}=0}^{l_{r_{1}-1}-1}\sqrt{l_{r_{1}}+1/2}\,\varphi_{l_{r_{1}}}(t)$
$\times\mathop{{\sum}^{*}}\limits_{m_{1}=r_{2}-1}^{j-1}(m_{1}+1/2)\mathop{{\sum}^{*}}\limits_{m_{2}=r_{2}-2}^{m_{1}-1}(m_{2}+1/2)\ldots\mathop{{\sum}^{*}}\limits_{m_{r_{2}-1}=1}^{m_{r_{2}-2}-1}(m_{r_{2}-1}+1/2)\mathop{{\sum}^{*}}\limits_{m_{r_{2}}=0}^{m_{r_{2}-1}-1}\sqrt{m_{r_{2}}+1/2}\>\varphi_{m_{r_{2}}}(\tau).$

Further, we change the order of summation and get

$\triangle_{1}(t,\tau)=\triangle_{11}(t,\tau)+\triangle_{12}(t,\tau),$

where

$\triangle_{11}(t,\tau)=2^{r_{1}+r_{2}}\mathop{{\sum}^{*}}\limits_{l_{r_{1}}=0}^{n/r_{2}^{\gamma}-r_{1}+1}\sqrt{l_{r_{1}}+1/2}\>\varphi_{l_{r_{1}}}(t)\mathop{{\sum}^{*}}\limits_{m_{r_{2}}=0}^{\infty}\sqrt{m_{r_{2}}+1/2}\>\varphi_{m_{r_{2}}}(\tau)$
$\times\sum_{k=n/r_{2}^{\gamma}+1}^{\infty}\quad\sum_{j=m_{r_{2}}+r_{2}}^{\infty}\sqrt{k+1/2}\sqrt{j+1/2}\>\langle
f,\varphi_{kj}\>\rangle B^{r_{1},r_{2}}_{k,j},$ (2.7)
$\triangle_{12}(t,\tau)=2^{r_{1}+r_{2}}\mathop{{\sum}^{*}}\limits_{l_{r_{1}}=n/r_{2}^{\gamma}-r_{1}+2}^{\infty}\sqrt{l_{r_{1}}+1/2}\>\varphi_{l_{r_{1}}}(t)\mathop{{\sum}^{*}}\limits_{m_{r_{2}}=0}^{\infty}\sqrt{m_{r_{2}}+1/2}\>\varphi_{m_{r_{2}}}(\tau)$
$\times\sum_{k=l_{r_{1}}+r_{1}}^{\infty}\quad\sum_{j=m_{r_{2}}+r_{2}}^{\infty}\sqrt{k+1/2}\sqrt{j+1/2}\>\langle
f,\varphi_{kj}\>\rangle B^{r_{1},r_{2}}_{k,j}$ (2.8)

and

$B^{r_{1},r_{2}}_{k,j}:=\mathop{{\sum}^{*}}\limits_{l_{1}=l_{r_{1}}+r_{1}-1}^{k-1}(l_{1}+1/2)\mathop{{\sum}^{*}}\limits_{l_{2}=l_{r_{1}}+r_{1}-2}^{l_{1}-1}(l_{2}+1/2)\ldots\mathop{{\sum}^{*}}\limits_{l_{{r_{1}}-1}=l_{r_{1}}+1}^{l_{{r_{1}}-2}-1}(l_{{r_{1}}-1}+1/2)$
$\times\mathop{{\sum}^{*}}\limits_{m_{1}=m_{r_{2}}+r_{2}-1}^{j-1}(m_{1}+1/2)\mathop{{\sum}^{*}}\limits_{m_{2}=m_{r_{2}}+r_{2}-2}^{m_{1}-1}(m_{2}+1/2)\ldots\mathop{{\sum}^{*}}\limits_{m_{{r_{2}}-1}=m_{r_{2}}+1}^{m_{{r_{2}}-2}-1}(m_{{r_{2}}-1}+1/2)$
$\leq ck^{2(r_{1}-1)}j^{2(r_{2}-1)}.$ (2.9)

Let’s $1<s<\infty.$ At first we bound $\triangle_{1}(t,\tau)$. Using Hölder
inequality and (2.9), for $\mu>2r_{1}+\frac{s-1}{s}-1/2$ we have

$\|\triangle_{11}\|_{L_{2}}^{2}\leq
4^{r_{1}+r_{2}}\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}^{\gamma}-r_{1}+1}(l_{r_{1}}+1/2)\mathop{{\sum}}\limits_{m_{r_{2}}=0}^{\infty}(m_{r_{2}}+1/2)$
$\times\left(\sum_{k=n/r_{2}^{\gamma}+1}^{\infty}\quad\sum_{j=m_{r_{2}}+r_{2}}^{\infty}(kj)^{\mu}\>|\langle
f,\varphi_{kj}\>\rangle|\frac{B_{k,j}^{r_{1},r_{2}}}{(kj)^{\mu-1/2}}\right)^{2}\leq
c\|f\|_{s,\mu}^{2}\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}^{\gamma}-r_{1}+1}(l_{r_{1}}+1/2)\mathop{{\sum}}\limits_{m_{r_{2}}=0}^{\infty}(m_{r_{2}}+1/2)$
$\times\left(\sum_{k=n/r_{2}^{\gamma}+1}^{\infty}k^{-(\mu-2r_{1}+3/2)s/(s-1)}\sum_{j=m_{r_{2}}+r_{2}}^{\infty}j^{-(\mu-2r_{2}+3/2)s/(s-1)}\right)^{2(s-1)/s}$
$\leq
c\|f\|_{s,\mu}^{2}n^{-2(\mu-2r_{1}+3/2)+\frac{2(s-1)}{s}}\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}^{\gamma}-r_{1}+1}(l_{r_{1}}+1/2)\mathop{{\sum}}\limits_{m_{r_{2}}=0}^{\infty}(m_{r_{2}}+1/2)^{-2(\mu-2r_{2}+3/2)+\frac{2(s-1)}{s}+1}$
$\leq c\|f\|_{s,\mu}^{2}n^{-2(\mu-2r_{1}+1/s-1/2)},$
$\|\triangle_{12}\|_{L_{2}}^{2}\leq
c\|f\|_{s,\mu}^{2}\mathop{{\sum}}\limits_{l_{r_{1}}=n/r_{2}^{\gamma}-r_{1}+2}^{\infty}(l_{r_{1}}+1/2)\mathop{{\sum}}\limits_{m_{r_{2}}=0}^{\infty}(m_{r_{2}}+1/2)$
$\times\left(\sum_{k=l_{r_{1}}+r_{1}}^{\infty}k^{-(\mu-2r_{1}+3/2)s/(s-1)}\sum_{j=m_{r_{2}}+r_{2}}^{\infty}j^{-(\mu-2r_{2}+3/2)s/(s-1)}\right)^{2(s-1)/s}$
$\leq
c\|f\|_{s,\mu}^{2}\mathop{{\sum}}\limits_{l_{r_{1}}=n/r_{2}^{\gamma}-r_{1}+2}^{\infty}(l_{r_{1}}+1/2)^{-2(\mu-2r_{1}+3/2)+\frac{2(s-1)}{s}+1}\mathop{{\sum}}\limits_{m_{r_{2}}=0}^{\infty}(m_{r_{2}}+1/2)^{-2(\mu-2r_{2}+3/2)+\frac{2(s-1)}{s}+1}$
$\leq c\|f\|_{s,\mu}^{2}n^{-2(\mu-2r_{1}+1/s-1/2)}.$

Combining the estimates for $\triangle_{11}(t,\tau)$ and
$\triangle_{12}(t,\tau)$ we obtain

$\|\triangle_{1}\|_{L_{2}}\leq c\|f\|_{s,\mu}n^{-\mu+2r_{1}-1/s+1/2}.$

Next, using the formula (2.6), from (2.5) we have

$\triangle_{2}(t,\tau)=2^{r_{1}+r_{2}}\sum_{k=r_{1}}^{n/r_{2}^{\gamma}}\quad\sum_{j=(n/k)^{1/\gamma}+1}^{\infty}\sqrt{k+1/2}\sqrt{j+1/2}\>\langle
f,\varphi_{kj}\rangle$
$\times\mathop{{\sum}^{*}}\limits_{l_{1}=r_{1}-1}^{k-1}(l_{1}+1/2)\mathop{{\sum}^{*}}\limits_{l_{2}=r_{1}-2}^{l_{1}-1}(l_{2}+1/2)\ldots\mathop{{\sum}^{*}}\limits_{l_{{r_{1}}-1}=1}^{l_{{r_{1}}-2}-1}(l_{{r_{1}}-1}+1/2)\mathop{{\sum}^{*}}\limits_{l_{r_{1}}=0}^{l_{{r_{1}}-1}-1}\sqrt{l_{r_{1}}+1/2}\>\varphi_{l_{r_{1}}}(t)$
$\times\mathop{{\sum}^{*}}\limits_{m_{1}={r_{2}}-1}^{j-1}(m_{1}+1/2)\mathop{{\sum}^{*}}\limits_{m_{2}={r_{2}}-2}^{m_{1}-1}(m_{2}+1/2)\ldots\mathop{{\sum}^{*}}\limits_{m_{{r_{2}}-1}=1}^{m_{{r_{2}}-2}-1}(m_{{r_{2}}-1}+1/2)\mathop{{\sum}^{*}}\limits_{m_{r_{2}}=0}^{m_{{r_{2}}-1}-1}\sqrt{m_{r_{2}}+1/2}\>\varphi_{m_{r_{2}}}(\tau).$

Further, we change the order of summation and get

$\triangle_{2}(t,\tau)=\triangle_{21}(t,\tau)+\triangle_{22}(t,\tau)+\triangle_{23}(t,\tau),$

where

$\triangle_{21}(t,\tau)=2^{r_{1}+r_{2}}\mathop{{\sum}^{*}}\limits_{l_{r_{1}}=0}^{n/r_{2}^{\gamma}-r_{1}}\sqrt{l_{r_{1}}+1/2}\,\varphi_{l_{r_{1}}}(t)\mathop{{\sum}^{*}}\limits_{m_{r_{2}}=0}^{(\frac{n}{l_{r_{1}}+r_{1}})^{1/\gamma}-r_{2}+1}\sqrt{m_{r_{2}}+1/2}\>\varphi_{m_{r_{2}}}(\tau)$
$\times\sum_{k=l_{r_{1}}+r_{1}}^{\frac{n}{(\underline{m}_{r_{2}}+r_{2}-1)^{\gamma}}}\quad\sum_{j=(n/k)^{1/\gamma}+1}^{\infty}\sqrt{k+1/2}\sqrt{j+1/2}\>\langle
f,\varphi_{kj}\rangle\>B^{r_{1},r_{2}}_{k,j},$
$\triangle_{22}(t,\tau)=2^{r_{1}+r_{2}}\mathop{{\sum}^{*}}\limits_{l_{r_{1}}=0}^{n/r_{2}^{\gamma}-r_{1}}\sqrt{l_{r_{1}}+1/2}\,\varphi_{l_{r_{1}}}(t)\mathop{{\sum}^{*}}\limits_{m_{r_{2}}=2}^{(\frac{n}{l_{r_{1}}+r_{1}})^{1/\gamma}-r_{2}+2}\sqrt{m_{r_{2}}+1/2}\>\varphi_{m_{r_{2}}}(\tau)$
$\times\sum_{k=\frac{n}{(m_{r_{2}}+r_{2}-2)^{\gamma}}}^{n/r_{2}^{\gamma}}\quad\sum_{j=m_{r_{2}}+r_{2}}^{\infty}\sqrt{k+1/2}\sqrt{j+1/2}\>\langle
f,\varphi_{kj}\>\rangle B^{r_{1},r_{2}}_{k,j},$
$\triangle_{23}(t,\tau)=2^{r_{1}+r_{2}}\mathop{{\sum}^{*}}\limits_{l_{r_{1}}=0}^{n/r_{2}^{\gamma}-r_{1}}\sqrt{l_{r_{1}}+1/2}\,\varphi_{l_{r_{1}}}(t)\mathop{{\sum}^{*}}\limits_{m_{r_{2}}=(\frac{n}{l_{r_{1}}+r_{1}})^{1/\gamma}-r_{2}+3}^{\infty}\sqrt{m_{r_{2}}+1/2}\>\varphi_{m_{r_{2}}}(\tau)$
$\times\sum_{k=l_{r_{1}}+r_{1}}^{n/r_{2}^{\gamma}}\quad\sum_{j=m_{r_{2}}+r_{2}}^{\infty}\sqrt{k+1/2}\sqrt{j+1/2}\>\langle
f,\varphi_{kj}\>\rangle B^{r_{1},r_{2}}_{k,j}.$

Let as before $1<s<\infty$. Now we estimate the norm of $\triangle_{2}$ in the
case of $r_{1}>r_{2}$ and $\gamma=1$:

$\|\triangle_{21}\|_{L_{2}}^{2}\leq
c\|f\|_{s,\mu}^{2}n^{-2(\mu-2r_{1}+3/2)+\frac{2(s-1)}{s}}\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}-r_{1}}(l_{r_{1}}+1/2)\mathop{{\sum}}\limits_{m_{r_{2}}=0}^{\frac{n}{l_{r_{1}}+r_{1}}-r_{2}+1}(m_{r_{2}}+1/2)$
$\times\left(\sum_{k=l_{r_{1}}+r_{1}}^{\frac{n}{\underline{m}_{r_{2}}+r_{2}-1}}k^{2(r_{1}-r_{2})s/(s-1)-1}\right)^{2(s-1)/s}\leq
c\|f\|_{s,\mu}^{2}n^{-2(\mu-2r_{1}+3/2)+\frac{2(s-1)}{s}}$
$\times\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}-r_{1}}(l_{r_{1}}+1/2)\mathop{{\sum}}\limits_{m_{r_{2}}=0}^{\frac{n}{l_{r_{1}}+r_{1}}-r_{2}+1}(m_{r_{2}}+1/2)^{1-4(r_{1}-r_{2})}$
$\leq
c\|f\|_{s,\mu}^{2}n^{-2(\mu-2r_{1}+3/2)+\frac{2(s-1)}{s}}\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}-r_{1}}(l_{r_{1}}+1/2)=c\|f\|_{s,\mu}^{2}n^{-2(\mu-2r_{1}+1/s-1/2)},$
$\|\triangle_{22}\|_{L_{2}}^{2}\leq
c\|f\|_{s,\mu}^{2}\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}-r_{1}}(l_{r_{1}}+1/2)\mathop{{\sum}}\limits_{m_{r_{2}}=2}^{\frac{n}{l_{r_{1}}+r_{1}}-r_{2}+2}(m_{r_{2}}+1/2)$
$\times\left(\sum_{k=\frac{n}{m_{r_{2}}+r_{2}-2}}^{n/r_{2}}k^{-(\mu-2r_{1}+3/2)s/(s-1)}\sum_{j=m_{r_{2}}+r_{2}}^{\infty}j^{-(\mu-2r_{2}+3/2)s/(s-1)}\right)^{2(s-1)/s}$
$\leq
c\|f\|_{s,\mu}^{2}n^{-2(\mu-2r_{1}+3/2)+\frac{2(s-1)}{s}}\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}-r_{1}}(l_{r_{1}}+1/2)\mathop{{\sum}}\limits_{m_{r_{2}}=2}^{\frac{n}{l_{r_{1}}+r_{1}}-r_{2}+2}(m_{r_{2}}+1/2)^{1-4(r_{1}-r_{2})}$
$\leq c\|f\|_{s,\mu}^{2}n^{-2(\mu-2r_{1}+1/s-1/2)},$
$\|\triangle_{23}\|_{L_{2}}^{2}\leq
c\|f\|_{s,\mu}^{2}\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}-r_{1}}(l_{r_{1}}+1/2)\mathop{{\sum}}\limits_{m_{r_{2}}=\frac{n}{l_{r_{1}}+r_{1}}-r_{2}+3}^{\infty}(m_{r_{2}}+1/2)$
$\times\left(\sum_{k=l_{r_{1}}+r_{1}}^{n/r_{2}}k^{-(\mu-2r_{1}+3/2)s/(s-1)}\sum_{j=m_{r_{2}}+r_{2}}^{\infty}j^{-(\mu-2r_{2}+3/2)s/(s-1)}\right)^{2(s-1)/s}$
$\leq
c\|f\|_{s,\mu}^{2}\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}-r_{1}}(l_{r_{1}}+1/2)^{1-2(\mu-2r_{1}+3/2)+\frac{2(s-1)}{s}}\mathop{{\sum}}\limits_{m_{r_{2}}=\frac{n}{l_{r_{1}}+r_{1}}-r_{2}+3}^{\infty}(m_{r_{2}}+1/2)^{1-2(\mu-2r_{2}+3/2)+\frac{2(s-1)}{s}}$
$\leq
c\|f\|_{s,\mu}^{2}n^{-2(\mu-2r_{2}+3/2)+\frac{2(s-1)}{s}+2}\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}-r_{1}}(l_{r_{1}}+1/2)^{4(r_{1}-r_{2})-1}$
$\leq c\|f\|_{s,\mu}^{2}n^{-2(\mu-2r_{1}+1/s-1/2)}.$

Combining the estimates for $\triangle_{21}(t,\tau)$, $\triangle_{22}(t,\tau)$
and $\triangle_{23}(t,\tau)$ we obtain

$\|\triangle_{2}\|_{L_{2}}\leq c\,\|f\|_{s,\mu}n^{-\mu+2r_{1}-1/s+1/2}.$

To estimate $\|\triangle_{2}\|_{L_{2}}$ for $r_{1}=r_{2}$ or $\gamma>1$, it is
enough to repeat an analogous line of reasoning. The combination of (2.3) and
bounds for the norms of $\triangle_{1}$, $\triangle_{2}$ makes it possible to
establish the desired inequality in the case $1<s<\infty$.

For $s=1$ the assertion of Lemma is proved similarly.

$\Box$

The following statement contains estimates for the second difference from the
right-hand side of (2.2) in the metric of $L_{2}$.

###### Lemma 2.2

Let $f\in L_{2}(Q)$ and let the condition (1.2) be satisfied.

I) If $r_{1}=r_{2}$, then
a) for $\gamma=1$ it holds true

$\|\mathcal{D}^{(r_{1},r_{1})}_{n,1}f-\mathcal{D}^{(r_{1},r_{1})}_{n,1}f_{\delta}\|_{L_{2}}\leq
c\delta n^{2r_{1}-1/p+1/2}\ln^{3/2-1/p}n,$

b) for any $1<\gamma<\frac{2r_{1}-1/2+(p-1)/p}{2r_{1}-3/2+(p-1)/p}$ it holds
true

$\|\mathcal{D}^{(r_{1},r_{1})}_{n,\gamma}f-\mathcal{D}^{(r_{1},r_{1})}_{n,\gamma}f_{\delta}\|_{L_{2}}\leq
c\delta n^{2r_{1}-1/p+1/2},$

c) for $\gamma=\frac{2r_{1}-1/2+(p-1)/p}{2r_{1}-3/2+(p-1)/p}$ it holds true

$\|\mathcal{D}^{(r_{1},r_{1})}_{n,\gamma}f-\mathcal{D}^{(r_{1},r_{1})}_{n,\gamma}f_{\delta}\|_{L_{2}}\leq
c\delta n^{2r_{1}-1/p+1/2}\ln^{1/2}n.$

II) If $r_{1}>r_{2}$, then for any $\gamma\geq 1$ it holds true

$\|\mathcal{D}^{(r_{1},r_{2})}_{n,\gamma}f-\mathcal{D}^{(r_{1},r_{2})}_{n,\gamma}f_{\delta}\|_{L_{2}}\leq
c\delta n^{2r_{1}-1/p+1/2}.$

Proof. Let’s write down the representation

$\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f(t,\tau)-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f_{\delta}(t,\tau)=\sum_{k=r_{1}}^{n/r_{2}^{\gamma}}\,\sum_{j=r_{2}}^{(n/k)^{1/\gamma}}\,\langle
f-f_{\delta},\varphi_{k,j}\rangle\varphi^{(r_{1})}_{k}(t)\varphi^{(r_{2})}_{j}(\tau).$

Using the formula (2.6), we get

$\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f(t,\tau)-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f_{\delta}(t,\tau)=2^{r_{1}+r_{2}}\sum_{k=r_{1}}^{n/r_{2}^{\gamma}}\,\sum_{j=r_{2}}^{(n/k)^{1/\gamma}}\sqrt{k+1/2}\sqrt{j+1/2}\>\langle
f-f_{\delta},\varphi_{kj}\>\rangle$
$\times\mathop{{\sum}^{*}}\limits_{l_{1}=r_{1}-1}^{k-1}(l_{1}+1/2)\>\mathop{{\sum}^{*}}\limits_{l_{2}=r_{1}-2}^{l_{1}-1}(l_{2}+1/2)\>\ldots\mathop{{\sum}^{*}}\limits_{l_{{r_{1}}-1}=1}^{l_{{r_{1}}-2}-1}(l_{{r_{1}}-1}+1/2)\>\mathop{{\sum}^{*}}\limits_{l_{r_{1}}=0}^{l_{{r_{1}}-1}-1}\sqrt{l_{r_{1}}+1/2}\>\varphi_{l_{r_{1}}}(t)$
$\times\mathop{{\sum}^{*}}\limits_{m_{1}={r_{2}}-1}^{j-1}(m_{1}+1/2)\>\mathop{{\sum}^{*}}\limits_{m_{2}={r_{2}}-2}^{m_{1}-1}(m_{2}+1/2)\>\ldots\mathop{{\sum}^{*}}\limits_{m_{{r_{2}}-1}=1}^{m_{{r_{2}}-2}-1}(m_{{r_{2}}-1}+1/2)\>\mathop{{\sum}^{*}}\limits_{m_{r_{2}}=0}^{m_{{r_{2}}-1}-1}\sqrt{m_{r_{2}}+1/2}\>\varphi_{m_{r_{2}}}(\tau).$

Further, we change the order of summation and get

$\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f(t,\tau)-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f_{\delta}(t,\tau)=2^{r_{1}+r_{2}}\,\mathop{{\sum}^{*}}\limits_{l_{r_{1}}=0}^{n/r_{2}^{\gamma}-r_{1}}\sqrt{l_{r_{1}}+1/2}\,\varphi_{l_{r_{1}}}(t)\mathop{{\sum}^{*}}\limits_{m_{r_{2}}=0}^{(\frac{n}{l_{r_{1}}+r_{1}})^{1/\gamma}-r_{2}}\sqrt{m_{r_{2}}+1/2}\>\varphi_{m_{r_{2}}}(\tau)$
$\times\sum_{k=l_{r_{1}}+r_{1}}^{\frac{n}{(m_{r_{2}}+r_{2})^{\gamma}}}\quad\sum_{j=m_{r_{2}}+r_{2}}^{(\frac{n}{k})^{1/\gamma}}\sqrt{k+1/2}\sqrt{j+1/2}\,\langle
f-f_{\delta},\varphi_{kj}\>\rangle B^{r_{1},r_{2}}_{k,j}.$

Let $1<p<\infty$ first. Then, using the Hölder inequality and the estimate
(2.9), we find

$\|\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f_{\delta}\|_{L_{2}}^{2}\leq
4^{r_{1}+r_{2}}\,\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}^{\gamma}-r_{1}}(l_{r_{1}}+1/2)\,\mathop{{\sum}}\limits_{m_{r_{2}}=0}^{(\frac{n}{l_{r_{1}}+r_{1}})^{1/\gamma}-r_{2}}(m_{r_{2}}+1/2)$
$\times\left(\sum_{k=l_{r_{1}}+r_{1}}^{\frac{n}{(m_{r_{2}}+r_{2})^{\gamma}}}\quad\sum_{j=m_{r_{2}}+r_{2}}^{(\frac{n}{k})^{1/\gamma}}\sqrt{k+1/2}\sqrt{j+1/2}\,|\langle
f-f_{\delta},\varphi_{kj}\>\rangle|B^{r_{1},r_{2}}_{k,j}\right)^{2}$ $\leq
c\delta^{2}\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}^{\gamma}-r_{1}}(l_{r_{1}}+1/2)\,\mathop{{\sum}}\limits_{m_{r_{2}}=0}^{(\frac{n}{l_{r_{1}}+r_{1}})^{1/\gamma}-r_{2}}(m_{r_{2}}+1/2)$
$\times\left(\sum_{k=l_{r_{1}}+r_{1}}^{\frac{n}{(m_{r_{2}}+r_{2})^{\gamma}}}k^{(2r_{1}-3/2)p/(p-1)}\sum_{j=m_{r_{2}}+r_{2}}^{(\frac{n}{k})^{1/\gamma}}j^{(2r_{2}-3/2)p/(p-1)}\right)^{2(p-1)/p}$
$\leq
c\delta^{2}\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}^{\gamma}-r_{1}}(l_{r_{1}}+1/2)\,\mathop{{\sum}}\limits_{m_{r_{2}}=0}^{(\frac{n}{l_{r_{1}}+r_{1}})^{1/\gamma}-r_{2}}(m_{r_{2}}+1/2)$
$\times\left(n^{\frac{(2r_{2}-3/2)p}{\gamma(p-1)}+1/\gamma}\sum_{k=l_{r_{1}}+r_{1}}^{\frac{n}{(m_{r_{2}}+r_{2})^{\gamma}}}k^{\frac{(2r_{1}-3/2)p}{p-1}-\frac{(2r_{2}-3/2)p}{\gamma(p-1)}-1/\gamma}\right)^{2(p-1)/p}.$
(2.10)

Whence it follows for $r_{1}>r_{2}$ and any $\gamma\geq 1$ that

$\|\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f_{\delta}\|_{L_{2}}^{2}\leq
c\delta^{2}n^{4r_{1}-3+\frac{2(p-1)}{p}}\,\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}^{\gamma}-r_{1}}(l_{r_{1}}+1/2)=c\delta^{2}n^{4r_{1}-2/p+1}.$

which was required to prove.

Let’s now $r_{1}=r_{2}$ and any $\gamma=1$. Then from (2.10) we obtain

$\|\mathcal{D}_{n,1}^{(r_{1},r_{1})}f-\mathcal{D}_{n,1}^{(r_{1},r_{1})}f_{\delta}\|_{L_{2}}^{2}\leq
c\delta^{2}n^{4r_{1}-3+\frac{2(p-1)}{p}}\,\ln^{\frac{2(p-1)}{p}}n\,\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{1}-r_{1}}(l_{r_{1}}+1/2)\mathop{{\sum}}\limits_{m_{r_{1}}=0}^{\frac{n}{l_{r_{1}}+r_{1}}-r_{1}}(m_{r_{1}}+1/2)$
$\leq
c\delta^{2}n^{4r_{1}-1+\frac{2(p-1)}{p}}\,\ln^{\frac{2(p-1)}{p}}n\,\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}-r_{1}}\frac{1}{l_{r_{1}}+1/2}=c\delta^{2}n^{4r_{1}-2/p+1}\,\ln^{3-2/p}n.$

To estimate
$\|\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f_{\delta}\|_{L_{2}}$
for $\gamma>1$ and also for $p=1$, $p=\infty$, it is enough to repeat an
analogous line of reasoning.

$\Box$

The combination of Lemmas 2.1 and 2.2 gives

###### Theorem 2.3

Let $f\in L^{\mu}_{s,2}$, $1\leq s<\infty$, $r_{1}\geq r_{2}$,
$\mu>2r_{1}+1/2-1/s$, and let the condition (1.2) be satisfied.

I) If $r_{1}=r_{2}$, then for
$n\asymp\left(\delta/\ln^{1/p-1/s}\frac{1}{\delta}\right)^{-\frac{1}{\mu-1/p+1/s}}$
and $\gamma=1$ it holds true

$\|f^{(r_{1},r_{1})}-\mathcal{D}^{(r_{1},r_{1})}_{n,1}f_{\delta}\|_{L_{2}}\leq
c\left(\delta\ln^{1/s-1/p}\frac{1}{\delta}\right)^{\frac{\mu-2r_{1}+1/s-1/2}{\mu-1/p+1/s}}\ln^{3/2-1/s}\frac{1}{\delta}.$

II) If $r_{1}>r_{2}$, then
a) for $n\asymp\delta^{-\frac{1}{\mu-1/p+1/s}}$ and any
$\gamma\in\left[1,\frac{\mu-2r_{2}+1/s-1/2}{\mu-2r_{1}+1/s+1/2}\right)\bigcup\left(\frac{\mu-2r_{2}+1/s-1/2}{\mu-2r_{1}+1/s+1/2},\frac{\mu-2r_{2}+1/s+1/2}{\mu-2r_{1}+1/s+1/2}\right)$
$\bigcup\left(\frac{\mu-2r_{2}+1/s+1/2}{\mu-2r_{1}+1/s+1/2},\frac{\mu-2r_{2}+1/s-1/2}{\mu-2r_{1}+1/s-1/2}\right)$
it holds true

$\|f^{(r_{1},r_{2})}-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f_{\delta}\|_{L_{2}}\leq
c\delta^{\frac{\mu-2r_{1}+1/s-1/2}{\mu-1/p+1/s}},$

b) for
$n\asymp\delta^{-\frac{1}{\mu-1/p+1/s}}\ln^{\frac{1}{2(\mu-1/p+1/s)}}\frac{1}{\delta}$
and
$\,\gamma=\frac{\mu-2r_{2}+1/s-1/2}{\mu-2r_{1}+1/s+1/2},\quad\gamma=\frac{\mu-2r_{2}+1/s-1/2}{\mu-2r_{1}+1/s-1/2}\,$
it holds true

$\|f^{(r_{1},r_{2})}-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f_{\delta}\|_{L_{2}}\leq
c\left(\delta/\ln^{1/2}\frac{1}{\delta}\right)^{\frac{\mu-2r_{1}+1/s-1/2}{\mu-1/p+1/s}}\ln^{1/2}\frac{1}{\delta},$

c) for
$n\asymp\left(\delta/\ln^{1-1/s}\frac{1}{\delta}\right)^{-\frac{1}{\mu-1/p+1/s}}$
and $\,\gamma=\frac{\mu-2r_{2}+1/s+1/2}{\mu-2r_{1}+1/s+1/2}\,$ it holds true

$\|f^{(r_{1},r_{2})}-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f_{\delta}\|_{L_{2}}\leq
c\left(\delta/\ln^{1-1/s}\frac{1}{\delta}\right)^{\frac{\mu-2r_{1}+1/s-1/2}{\mu-1/p+1/s}}\ln^{1-1/s}\frac{1}{\delta}.$

##  3\. Truncation method. Error estimate in the metric of $C$

Now we have to bound the error of (2.1) in the metric of $C$. Upper estimates
for the norm of difference (2.3) are contained in the following statement.

###### Lemma 3.1

Let $f\in L^{\mu}_{s,2}$, $1\leq s<\infty$, $r_{1}\geq r_{2}$,
$\mu>2r_{1}+3/2-1/s$.

I) If $r_{1}=r_{2}$, then for $\gamma=1$ it holds true

$\|f^{(r_{1},r_{1})}-\mathcal{D}_{n,1}^{(r_{1},r_{1})}f\|_{C}\leq
c\|f\|_{s,\mu}n^{-\mu+2r_{1}-1/s+3/2}\ln^{2-1/s}n.$

II) If $r_{1}=r_{2}+1$, then
a) for any
$\gamma\in\left(1,\frac{\mu-2r_{1}+1/s+5/2}{\mu-2r_{1}+1/s+1/2}\right)\bigcup\left(\frac{\mu-2r_{1}+1/s+5/2}{\mu-2r_{1}+1/s+1/2},\frac{\mu-2r_{1}+1/s+1/2}{\mu-2r_{1}+1/s-3/2}\right)$
it holds true

$\|f^{(r_{1},r_{1}-1)}-\mathcal{D}_{n,\gamma}^{(r_{1},r_{1}-1)}f\|_{C}\leq
c\|f\|_{s,\mu}n^{-\mu+2r_{1}-1/s+3/2},$

b) for $\gamma=1$ and $\gamma=\frac{\mu-2r_{1}+1/s+1/2}{\mu-2r_{1}+1/s-3/2}$
it holds true

$\|f^{(r_{1},r_{1}-1)}-\mathcal{D}_{n,\gamma}^{(r_{1},r_{1}-1)}f\|_{C}\leq
c\|f\|_{s,\mu}n^{-\mu+2r_{1}-1/s+3/2}\ln n,$

c) for $\gamma=\frac{\mu-2r_{1}+1/s+5/2}{\mu-2r_{1}+1/s+1/2}$ it holds true

$\|f^{(r_{1},r_{1}-1)}-\mathcal{D}_{n,\gamma}^{(r_{1},r_{1}-1)}f\|_{C}\leq
c\|f\|_{s,\mu}n^{-\mu+2r_{1}-1/s+3/2}\ln^{1-1/s}n.$

III) If $r_{1}>r_{2}+1$, then
a) for any
$\gamma\in\left[1,\frac{\mu-2r_{2}+1/s-3/2}{\mu-2r_{1}+1/s+1/2}\right)\bigcup\left(\frac{\mu-2r_{2}+1/s-3/2}{\mu-2r_{1}+1/s+1/2},\frac{\mu-2r_{2}+1/s+1/2}{\mu-2r_{1}+1/s+1/2}\right)$
$\bigcup\left(\frac{\mu-2r_{2}+1/s+1/2}{\mu-2r_{1}+1/s+1/2},\frac{\mu-2r_{2}+1/s-3/2}{\mu-2r_{1}+1/s-3/2}\right)$
it holds true

$\|f^{(r_{1},r_{2})}-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f\|_{C}\leq
c\|f\|_{s,\mu}n^{-\mu+2r_{1}-1/s+3/2},$

b) for $\gamma=\frac{\mu-2r_{2}+1/s-3/2}{\mu-2r_{1}+1/s+1/2}$ and
$\gamma=\frac{\mu-2r_{2}+1/s-3/2}{\mu-2r_{1}+1/s-3/2}$ it holds true

$\|f^{(r_{1},r_{2})}-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f\|_{C}\leq
c\|f\|_{s,\mu}n^{-\mu+2r_{1}-1/s+3/2}\ln n,$

c) for $\gamma=\frac{\mu-2r_{2}+1/s+1/2}{\mu-2r_{1}+1/s+1/2}$ it holds true

$\|f^{(r_{1},r_{2})}-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f\|_{C}\leq
c\|f\|_{s,\mu}n^{-\mu+2r_{1}-1/s+3/2}\ln^{1-1/s}n.$

Proof. Let’s start with the case $1<s<\infty$. Using (2.9), from (2.7) and
(2.8) we get

$\|\triangle_{11}\|_{C}\leq
c\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}^{\gamma}-r_{1}+1}(l_{r_{1}}+1/2)\mathop{{\sum}}\limits_{m_{r_{2}}=0}^{\infty}(m_{r_{2}}+1/2)$
$\times\sum_{k=n/r_{2}^{\gamma}+1}^{\infty}\quad\sum_{j=m_{r_{2}}+r_{2}}^{\infty}(kj)^{\mu}\>|\langle
f,\varphi_{kj}\>\rangle|\frac{B_{k,j}^{r_{1},r_{2}}}{(kj)^{\mu-1/2}}\leq
c\|f\|_{s,\mu}\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}^{\gamma}-r_{1}+1}(l_{r_{1}}+1/2)\mathop{{\sum}}\limits_{m_{r_{2}}=0}^{\infty}(m_{r_{2}}+1/2)$
$\times\left(\sum_{k=n/r_{2}^{\gamma}+1}^{\infty}k^{-(\mu-2r_{1}+3/2)s/(s-1)}\sum_{j=m_{r_{2}}+r_{2}}^{\infty}j^{-(\mu-2r_{2}+3/2)s/(s-1)}\right)^{(s-1)/s}$
$\leq
c\|f\|_{s,\mu}n^{-(\mu-2r_{1}+3/2)+\frac{s-1}{s}}\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}^{\gamma}-r_{1}+1}(l_{r_{1}}+1/2)\mathop{{\sum}}\limits_{m_{r_{2}}=0}^{\infty}(m_{r_{2}}+1/2)^{1-(\mu-2r_{2}+3/2)+\frac{s-1}{s}}$
$\leq c\|f\|_{s,\mu}n^{-\mu+2r_{1}-1/s+3/2},$ $\|\triangle_{12}\|_{C}\leq
c\|f\|_{s,\mu}\mathop{{\sum}}\limits_{l_{r_{1}}=n/r_{2}^{\gamma}-r_{1}+2}^{\infty}(l_{r_{1}}+1/2)\mathop{{\sum}}\limits_{m_{r_{2}}=0}^{\infty}(m_{r_{2}}+1/2)$
$\times\left(\sum_{k=l_{r_{1}}+r_{1}}^{\infty}k^{-(\mu-2r_{1}+3/2)s/(s-1)}\sum_{j=m_{r_{2}}+r_{2}}^{\infty}j^{-(\mu-2r_{2}+3/2)s/(s-1)}\right)^{(s-1)/s}$
$\leq
c\|f\|_{s,\mu}\mathop{{\sum}}\limits_{l_{r_{1}}=n/r_{2}^{\gamma}-r_{1}+2}^{\infty}(l_{r_{1}}+1/2)^{1-(\mu-2r_{1}+3/2)+\frac{s-1}{s}}\mathop{{\sum}}\limits_{m_{r_{2}}=0}^{\infty}(m_{r_{2}}+1/2)^{1-(\mu-2r_{2}+3/2)+\frac{s-1}{s}}$
$\leq c\|f\|_{s,\mu}n^{-\mu+2r_{1}-1/s+3/2}.$

Thus, we get

$\|\triangle_{1}\|_{C}\leq c\|f\|_{s,\mu}n^{-\mu+2r_{1}-1/s+3/2}.$

Further, we estimate the norm of $\triangle_{2}$ in the case of $r_{1}>r_{2}$
and $\gamma=1$:

$\|\triangle_{21}\|_{C}\leq
c\|f\|_{s,\mu}n^{-\mu+2r_{2}-3/2+\frac{s-1}{s}}\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}-r_{1}}(l_{r_{1}}+1/2)\mathop{{\sum}}\limits_{m_{r_{2}}=0}^{\frac{n}{l_{r_{1}}+r_{1}}-r_{2}+1}(m_{r_{2}}+1/2)$
$\times\left(\sum_{k=l_{r_{1}}+r_{1}}^{\frac{n}{\underline{m}_{r_{2}}+r_{2}-1}}k^{2(r_{1}-r_{2})s/(s-1)-1}\right)^{(s-1)/s}\leq
c\|f\|_{s,\mu}n^{-\mu+2r_{1}-3/2+\frac{s-1}{s}}$
$\times\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}-r_{1}}(l_{r_{1}}+1/2)\mathop{{\sum}}\limits_{m_{r_{2}}=0}^{\frac{n}{l_{r_{1}}+r_{1}}-r_{2}+1}(m_{r_{2}}+1/2)^{1-2(r_{1}-r_{2})}$
$\leq c\|f\|_{s,\mu}\left\\{\begin{array}[]{cl}n^{-\mu+2r_{1}-1/s+3/2}\,\ln
n,&r_{1}=r_{2}+1,\\\
n^{-\mu+2r_{1}-1/s+3/2},&r_{1}>r_{2}+1,\end{array}\right.$
$\|\triangle_{22}\|_{C}\leq
c\|f\|_{s,\mu}\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}-r_{1}}(l_{r_{1}}+1/2)\mathop{{\sum}}\limits_{m_{r_{2}}=2}^{\frac{n}{l_{r_{1}}+r_{1}}-r_{2}+2}(m_{r_{2}}+1/2)$
$\times\left(\sum_{k=\frac{n}{m_{r_{2}}+r_{2}-2}}^{n/r_{2}}k^{-(\mu-2r_{1}+3/2)s/(s-1)}\sum_{j=m_{r_{2}}+r_{2}}^{\infty}j^{-(\mu-2r_{2}+3/2)s/(s-1)}\right)^{(s-1)/s}$
$\leq
c\|f\|_{s,\mu}n^{-\mu+2r_{1}-3/2+\frac{s-1}{s}}\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}-r_{1}}(l_{r_{1}}+1/2)\mathop{{\sum}}\limits_{m_{r_{2}}=2}^{\frac{n}{l_{r_{1}}+r_{1}}-r_{2}+2}(m_{r_{2}}+1/2)^{1-2(r_{1}-r_{2})}$
$\leq c\|f\|_{s,\mu}\left\\{\begin{array}[]{cl}n^{-\mu+2r_{1}-1/s+3/2}\,\ln
n,&r_{1}=r_{2}+1,\\\
n^{-\mu+2r_{1}-1/s+3/2},&r_{1}>r_{2}+1,\end{array}\right.$
$\|\triangle_{23}\|_{C}\leq
c\|f\|_{s,\mu}\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}-r_{1}}(l_{r_{1}}+1/2)\mathop{{\sum}}\limits_{m_{r_{2}}=\frac{n}{l_{r_{1}}+r_{1}}-r_{2}+3}^{\infty}(m_{r_{2}}+1/2)$
$\times\left(\sum_{k=l_{r_{1}}+r_{1}}^{n/r_{2}}k^{-(\mu-2r_{1}+3/2)s/(s-1)}\sum_{j=m_{r_{2}}+r_{2}}^{\infty}j^{-(\mu-2r_{2}+3/2)s/(s-1)}\right)^{(s-1)/s}$
$\leq
c\|f\|_{s,\mu}\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}-r_{1}}(l_{r_{1}}+1/2)^{1-(\mu-2r_{1}+3/2)+\frac{s-1}{s}}\mathop{{\sum}}\limits_{m_{r_{2}}=\frac{n}{l_{r_{1}}+r_{1}}-r_{2}+3}^{\infty}(m_{r_{2}}+1/2)^{1-(\mu-2r_{2}+3/2)+\frac{s-1}{s}}$
$\leq
c\|f\|_{s,\mu}n^{-(\mu-2r_{2}+3/2)+\frac{s-1}{s}+2}\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}-r_{1}}(l_{r_{1}}+1/2)^{2(r_{1}-r_{2})-1}$
$\leq c\|f\|_{s,\mu}n^{-\mu+2r_{1}-1/s+3/2}.$

Combining the estimates for $\triangle_{21}(t,\tau)$, $\triangle_{22}(t,\tau)$
and $\triangle_{23}(t,\tau)$ we obtain

$\|\triangle_{2}\|_{C}\leq
c\|f\|_{s,\mu}\left\\{\begin{array}[]{cl}n^{-\mu+2r_{1}-1/s+3/2}\,\ln
n,&r_{1}=r_{2}+1,\\\
n^{-\mu+2r_{1}-1/s+3/2},&r_{1}>r_{2}+1,\end{array}\right..$

To estimate $\|\triangle_{2}\|_{C}$ for $r_{1}=r_{2}$ or $\gamma>1$, it is
enough to repeat an analogous line of reasoning. The combination of (2.3) and
bounds for the norms of $\triangle_{1}$, $\triangle_{2}$ makes it possible to
establish the desired inequality in the case of $1<s<\infty$.

For $s=1$ the assertion of Lemma is proved similarly.

$\Box$

The following statement contains upper estimates for the second difference
from the right-hand side of (2.2) in the metric of $C$.

###### Lemma 3.2

Let $f\in L_{2}(Q)$ and let the condition (1.2) be satisfied.

I) If $r_{1}=r_{2}$, then
a) for $\gamma=1$ it holds true

$\|\mathcal{D}^{(r_{1},r_{1})}_{n,1}f-\mathcal{D}^{(r_{1},r_{1})}_{n,1}f_{\delta}\|_{C}\leq
c\delta n^{2r_{1}-1/p+3/2}\ln^{2-1/p}n,$

b) for any $\quad 1<\gamma<\frac{2r_{1}+1/2+(p-1)/p}{2r_{1}-3/2+(p-1)/p}$ it
holds true

$\|\mathcal{D}^{(r_{1},r_{1})}_{n,\gamma}f-\mathcal{D}^{(r_{1},r_{1})}_{n,\gamma}f_{\delta}\|_{C}\leq
c\delta n^{2r_{1}-1/p+3/2},$

c) for $\gamma=\frac{2r_{1}+1/2+(p-1)/p}{2r_{1}-3/2+(p-1)/p}$ it holds true

$\|\mathcal{D}^{(r_{1},r_{1})}_{n,\gamma}f-\mathcal{D}^{(r_{1},r_{1})}_{n,\gamma}f_{\delta}\|_{C}\leq
c\delta n^{2r_{1}-1/p+3/2}\ln n.$

II) If $r_{1}=r_{2}+1$, then
a) for $\gamma=1$ it holds true

$\|\mathcal{D}^{(r_{1},r_{1}-1)}_{n,\gamma}f-\mathcal{D}^{(r_{1},r_{1}-1)}_{n,\gamma}f_{\delta}\|_{C}\leq
c\delta n^{2r_{1}-1/p+3/2}\ln n,$

b) for any $\gamma>1$ it holds true

$\|\mathcal{D}^{(r_{1},r_{1}-1)}_{n,\gamma}f-\mathcal{D}^{(r_{1},r_{1}-1)}_{n,\gamma}f_{\delta}\|_{C}\leq
c\delta n^{2r_{1}-1/p+3/2}.$

III) If $r_{1}>r_{2}+1$, then for any $\gamma\geq 1$ it holds true

$\|\mathcal{D}^{(r_{1},r_{2})}_{n,\gamma}f-\mathcal{D}^{(r_{1},r_{2})}_{n,\gamma}f_{\delta}\|_{C}\leq
c\delta n^{2r_{1}-1/p+3/2}.$

Proof. Let $1<p<\infty$ first. Then, using the Hölder inequality and the
estimate (2.9), we find

$\|\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f_{\delta}\|_{C}\leq
2^{r_{1}+r_{2}}\,\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}^{\gamma}-r_{1}}(l_{r_{1}}+1/2)\,\mathop{{\sum}}\limits_{m_{r_{2}}=0}^{(\frac{n}{l_{r_{1}}+r_{1}})^{1/\gamma}-r_{2}}(m_{r_{2}}+1/2)$
$\times\sum_{k=l_{r_{1}}+r_{1}}^{\frac{n}{(m_{r_{2}}+r_{2})^{\gamma}}}\quad\sum_{j=m_{r_{2}}+r_{2}}^{(\frac{n}{k})^{1/\gamma}}\sqrt{k+1/2}\sqrt{j+1/2}\,|\langle
f-f_{\delta},\varphi_{kj}\>\rangle|B^{r_{1},r_{2}}_{k,j}$ $\leq
c\delta\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}^{\gamma}-r_{1}}(l_{r_{1}}+1/2)\,\mathop{{\sum}}\limits_{m_{r_{2}}=0}^{(\frac{n}{l_{r_{1}}+r_{1}})^{1/\gamma}-r_{2}}(m_{r_{2}}+1/2)$
$\times\left(\sum_{k=l_{r_{1}}+r_{1}}^{\frac{n}{(m_{r_{2}}+r_{2})^{\gamma}}}k^{(2r_{1}-3/2)p/(p-1)}\sum_{j=m_{r_{2}}+r_{2}}^{(\frac{n}{k})^{1/\gamma}}j^{(2r_{2}-3/2)p/(p-1)}\right)^{(p-1)/p}$
$\leq c\delta n^{(2r_{2}-3/2)/\gamma+\frac{p-1}{\gamma
p}}\,\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}^{\gamma}-r_{1}}(l_{r_{1}}+1/2)\,\mathop{{\sum}}\limits_{m_{r_{2}}=0}^{(\frac{n}{l_{r_{1}}+r_{1}})^{1/\gamma}-r_{2}}(m_{r_{2}}+1/2)$
$\times\left(\sum_{k=l_{r_{1}}+r_{1}}^{\frac{n}{(m_{r_{2}}+r_{2})^{\gamma}}}k^{\frac{(2r_{1}-3/2)p}{p-1}-\frac{(2r_{2}-3/2)p}{\gamma(p-1)}-1/\gamma}\right)^{(p-1)/p}$
$\leq c\delta
n^{2r_{1}-3/2+\frac{p-1}{p}}\,\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}^{\gamma}-r_{1}}(l_{r_{1}}+1/2)\,$
$\times\mathop{{\sum}}\limits_{m_{r_{2}}=0}^{(\frac{n}{l_{r_{1}}+r_{1}})^{1/\gamma}-r_{2}}(m_{r_{2}}+1/2)^{1-\gamma(2r_{1}-3/2)+2r_{2}-3/2+\frac{p-1}{p}-\frac{\gamma(p-1)}{p}}.$
(3.1)

Whence it follows for $r_{1}=r_{2}+1$ and $\gamma=1$ that

$\|\mathcal{D}_{n,1}^{(r_{1},r_{1}-1)}f-\mathcal{D}_{n,1}^{(r_{1},r_{1}-1)}f_{\delta}\|_{C}\leq
c\delta n^{2r_{1}-3/2+\frac{p-1}{p}}\,\ln
n\,\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{2}^{\gamma}-r_{1}}(l_{r_{1}}+1/2)=c\delta
n^{2r_{1}-1/p+3/2}\ln n,$

which was required to prove.

Let’s now $r_{1}=r_{2}+1$, $\gamma>1$ or $r_{1}>r_{2}+1$, $\gamma\geq 1$. Then
from (3.1) we obtain the desired estimate

$\|\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f_{\delta}\|_{C}\leq
c\delta
n^{2r_{1}-3/2+\frac{p-1}{p}}\,\mathop{{\sum}}\limits_{l_{r_{1}}=0}^{n/r_{1}-r_{1}}(l_{r_{1}}+1/2)\leq
c\delta n^{2r_{1}-1/p+3/2}.$

To estimate
$\|\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f_{\delta}\|_{C}$
for $r_{1}=r_{2}$, it is enough to repeat an analogous line of reasoning.

In the case of $p=1$ and $p=\infty$ the assertion of Lemma is proved
similarly.

$\Box$

The combination of Lemmas 3.1 and 3.2 gives

###### Theorem 3.3

Let $f\in L^{\mu}_{s,2}$, $1\leq s<\infty$, $r_{1}\geq r_{2}$,
$\mu>2r_{1}-1/s+3/2$, and let the condition (1.2) be satisfied.

I) If $r_{1}=r_{2}$, then for
$n\asymp\left(\delta/\ln^{1/p-1/s}\frac{1}{\delta}\right)^{-\frac{1}{\mu-1/p+1/s}}$
and $\gamma=1$ it holds true

$\|f^{(r_{1},r_{1})}-\mathcal{D}^{(r_{1},r_{1})}_{n,1}f_{\delta}\|_{C}\leq
c\left(\delta\ln^{1/s-1/p}\frac{1}{\delta}\right)^{\frac{\mu-2r_{1}+1/s-3/2}{\mu-1/p+1/s}}\ln^{2-1/s}\frac{1}{\delta}.$

II) If $r_{1}=r_{2}+1$, then
a) for $n\asymp\delta^{-\frac{1}{\mu-1/p+1/s}}$ and $\gamma=1$ it holds true

$\|f^{(r_{1},r_{1}-1)}-\mathcal{D}_{n,1}^{(r_{1},r_{1}-1)}f_{\delta}\|_{C}\leq
c\delta^{\frac{\mu-2r_{1}+1/s-3/2}{\mu-1/p+1/s}}\ln\frac{1}{\delta},$

b) for $n\asymp\delta^{-\frac{1}{\mu-1/p+1/s}}$ and any
$\gamma\in\left(1,\frac{\mu-2r_{1}+1/s+5/2}{\mu-2r_{1}+1/s+1/2}\right)\bigcup\left(\frac{\mu-2r_{1}+1/s+5/2}{\mu-2r_{1}+1/s+1/2},\frac{\mu-2r_{1}+1/s+1/2}{\mu-2r_{1}+1/s-3/2}\right)$
it holds true

$\|f^{(r_{1},r_{1}-1)}-\mathcal{D}_{n,\gamma}^{(r_{1},r_{1}-1)}f_{\delta}\|_{C}\leq
c\delta^{\frac{\mu-2r_{1}+1/s-3/2}{\mu-1/p+1/s}},$

c) for
$n\asymp\left(\delta/\ln\frac{1}{\delta}\right)^{-\frac{1}{\mu-1/p+1/s}}$ and
$\,\gamma=\frac{\mu-2r_{1}+1/s+1/2}{\mu-2r_{1}+1/s-3/2}\,$ it holds true

$\|f^{(r_{1},r_{1}-1)}-\mathcal{D}_{n,\gamma}^{(r_{1},r_{1}-1)}f_{\delta}\|_{C}\leq
c\left(\delta/\ln\frac{1}{\delta}\right)^{\frac{\mu-2r_{1}+1/s-3/2}{\mu-1/p+1/s}}\ln\frac{1}{\delta},$

d) for
$n\asymp\left(\delta/\ln^{1-1/s}\frac{1}{\delta}\right)^{-\frac{1}{\mu-1/p+1/s}}$
and $\,\gamma=\frac{\mu-2r_{1}+1/s+5/2}{\mu-2r_{1}+1/s+1/2}\,$ it holds true

$\|f^{(r_{1},r_{1}-1)}-\mathcal{D}_{n,\gamma}^{(r_{1},r_{1}-1)}f_{\delta}\|_{C}\leq
c\left(\delta/\ln^{1-1/s}\frac{1}{\delta}\right)^{\frac{\mu-2r_{1}+1/s-3/2}{\mu-1/p+1/s}}\ln^{1-1/s}\frac{1}{\delta}.$

III) If $r_{1}>r_{2}+1$, then
a) for $n\asymp\delta^{-\frac{1}{\mu-1/p+1/s}}$ and any
$\gamma\in\left[1,\frac{\mu-2r_{2}+1/s-3/2}{\mu-2r_{1}+1/s+1/2}\right)\bigcup\left(\frac{\mu-2r_{2}+1/s-3/2}{\mu-2r_{1}+1/s+1/2},\frac{\mu-2r_{2}+1/s+1/2}{\mu-2r_{1}+1/s+1/2}\right)$
$\bigcup\left(\frac{\mu-2r_{2}+1/s+1/2}{\mu-2r_{1}+1/s+1/2},\frac{\mu-2r_{2}+1/s-3/2}{\mu-2r_{1}+1/s-3/2}\right)$
it holds true

$\|f^{(r_{1},r_{2})}-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f_{\delta}\|_{C}\leq
c\delta^{\frac{\mu-2r_{1}+1/s-3/2}{\mu-1/p+1/s}},$

b) for
$n\asymp\left(\delta/\ln\frac{1}{\delta}\right)^{-\frac{1}{\mu-1/p+1/s}}$ and
$\,\gamma=\frac{\mu-2r_{2}+1/s-3/2}{\mu-2r_{1}+1/s+1/2},\quad\gamma=\frac{\mu-2r_{2}+1/s-3/2}{\mu-2r_{1}+1/s-3/2}\,$
it holds true

$\|f^{(r_{1},r_{2})}-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f_{\delta}\|_{C}\leq
c\left(\delta/\ln\frac{1}{\delta}\right)^{\frac{\mu-2r_{1}+1/s-3/2}{\mu-1/p+1/s}}\ln\frac{1}{\delta},$

c) for
$n\asymp\left(\delta/\ln^{1-1/s}\frac{1}{\delta}\right)^{-\frac{1}{\mu-1/p+1/s}}$
and $\,\gamma=\frac{\mu-2r_{2}+1/s+1/2}{\mu-2r_{1}+1/s+1/2}\,$ it holds true

$\|f^{(r_{1},r_{2})}-\mathcal{D}_{n,\gamma}^{(r_{1},r_{2})}f_{\delta}\|_{C}\leq
c\left(\delta/\ln^{1-1/s}\frac{1}{\delta}\right)^{\frac{\mu-2r_{1}+1/s-3/2}{\mu-1/p+1/s}}\ln^{1-1/s}\frac{1}{\delta}.$

###### Remark 3.4

The $\mathcal{D}^{(r,r)}_{n}$ (2.1) method was studied earlier (see [23]) for
the problem of numerical differentiation of functions from $L^{\mu}_{s,2}$ in
the case of $r=1$ and $p=s=2$. Thus, the results of theorems 2.3 and 3.3
generalize studies [23] for the case of arbitrary $r,p,s$.

##  4\. Minimal radius of Galerkin information

Let us turn to finding sharp estimates (in the power scale) for the minimal
radius. First, we establish a lower estimate for the quantity
$R_{N,\delta}^{(r_{1},r_{2})}(L^{\mu}_{s,2},C,\ell_{p})$. We fix an
arbitrarily chosen domain $\hat{\Omega}$, $\text{card}\vskip 6.0pt plus 2.0pt
minus 2.0pt(\hat{\Omega})\leq N$, of the coordinate plane
$[r_{1},\infty)\times[r_{2},\infty)$ and build an auxiliary function

$f_{1}(t,\tau)=\widetilde{c}\,\bigg{(}\varphi_{0}(t)\varphi_{0}(\tau)\,+\,N^{-\mu-1/s}\,r_{2}^{-\mu}\varphi_{r_{2}}(\tau)\mathop{{\sum}\,^{\prime}}\limits_{k=N+r_{1}}^{3N+r_{1}}\varphi_{k}(t)\bigg{)},$

where the sum $\mathop{{\sum}\,^{\prime}}\limits_{k=N+r_{1}}^{3N+r_{1}}$ is
taken over any $N$ pairwise distinct functions $\varphi_{k}(t)$ such that
$N+r_{1}\leq k\leq 3N+r_{1}$ and $(k,r_{2})\notin\hat{\Omega}$. Obviously,
there is at least one set of such functions.

Now we estimate the norm of $f_{1}$ in the space metric $L^{\mu}_{s,2}$:

$\|f_{1}\|^{s}_{s,\mu}={\widetilde{c}}^{\,s}\,\bigg{(}1+N^{-s\mu-1}\,\mathop{{\sum}\,^{\prime}}\limits_{k=N+r_{1}}^{3N+r_{1}}k^{s\mu}\bigg{)}\leq{\widetilde{c}}^{\,s}\,\bigg{(}1+4^{s\mu}\bigg{)}.$

Whence it follows that to satisfy the condition $\|f_{1}\|_{s,\mu}\leq 1$ it
suffices to take

${\widetilde{c}}=\bigg{(}1+4^{s\mu}\bigg{)}^{-1/s}.$

Next, we take another function from the class $L^{\mu}_{s,2}$:

$f_{2}(t,\tau)=\widetilde{c}\,\varphi_{0}(t)\varphi_{0}(\tau).$

Let us find a lower bound for the quantity
$\|f_{1}^{(r_{1},r_{2})}-f_{2}^{(r_{1},r_{2})}\|_{C}$. For this we need
formulas

$\varphi_{r}^{(r)}(t)=\frac{\sqrt{r+1/2}}{2^{r-1/2}}\ \frac{(2r)!}{r!}\
\varphi_{0}(t),\quad f_{2}^{(r_{1},r_{2})}(t,\tau)\equiv 0,$
$f_{1}^{(r_{1},r_{2})}(t,\tau)=\frac{\widetilde{c}}{r_{2}^{\mu}}\,{N^{-\mu-1/s}}\,\varphi_{r_{2}}^{(r_{2})}(\tau)\mathop{{\sum}\,^{\prime}}\limits_{k=N+r_{1}}^{3N+r_{1}}\varphi_{k}^{(r_{1})}(t)$
$=2^{r_{1}}\,\frac{\widetilde{c}}{r_{2}^{\mu}}\,{N^{-\mu-1/s}}\,\varphi_{r_{2}}^{(r_{2})}(\tau)\mathop{{\sum}\,^{\prime}}\limits_{k=N+r_{1}}^{3N+r_{1}}\sqrt{k+1/2}\mathop{{\sum}^{*}}\limits_{l_{1}=r_{1}-1}^{k-1}(l_{1}+1/2)\mathop{{\sum}^{*}}\limits_{l_{2}=r_{1}-2}^{l_{1}-1}(l_{2}+1/2)$
$\ldots\mathop{{\sum}^{*}}\limits_{l_{r_{1}-1}=1}^{l_{r_{1}-2}-1}(l_{r_{1}-1}+1/2)\mathop{{\sum}^{*}}\limits_{l_{r_{1}}=0}^{l_{r_{1}-1}-1}\sqrt{l_{r_{1}}+1/2}\,\varphi_{l_{r_{1}}}(t).$
(4.1)

It is easy to see that

$\|f_{1}^{(r_{1},r_{2})}-f_{2}^{(r_{1},r_{2})}\|_{C}\geq|f_{1}^{(r_{1},r_{2})}(1,1)|\geq\overline{c}\,N^{-\mu+2r_{1}-1/s+3/2},$

where

$\overline{c}=\frac{\widetilde{c}\sqrt{r_{2}+1/2}}{2^{r_{1}+r_{2}}r_{2}^{\mu}}\,\frac{(2r_{2})!}{r_{1}!r_{2}!}.$

Since for any $1\leq p\leq\infty$ it holds true

$\|\overline{f}_{1}-\overline{f}_{2}\|_{\ell_{p}}=\frac{\widetilde{c}}{r_{2}^{\mu}}\,N^{-\mu-1/s+1/p},$

then in the case of $N^{-\mu-1/s+1/p}\leq r_{2}^{\mu}\delta/{\widetilde{c}}$
under $\delta$-perturbations of the functions $f_{1}$ and $f_{2}$ can be
considered

$f^{\delta}_{1}(t,\tau)=f_{2}(t,\tau),\qquad
f^{\delta}_{2}(t,\tau)=f_{1}(t,\tau).$

Let us find the upper bound for
$\|f_{1}^{(r_{1},r_{2})}-f_{2}^{(r_{1},r_{2})}\|_{C}$. Taking into account the
relation
$G(\hat{\Omega},\overline{f}_{1}^{\delta})=G(\hat{\Omega},\overline{f}_{2}^{\delta})$,
for any $\psi^{(r_{1},r_{2})}(\hat{\Omega})\in\Psi(\hat{\Omega})$ we find

$\|f_{1}^{(r_{1},r_{2})}-f_{2}^{(r_{1},r_{2})}\|_{C}\leq\|f_{1}^{(r_{1},r_{2})}-\psi^{(r_{1},r_{2})}(G(\hat{\Omega},\overline{f}_{1}^{\delta}))\|_{C}+\|f_{2}^{(r_{1},r_{2})}-\psi^{(r_{1},r_{2})}(G(\hat{\Omega},\overline{f}_{2}^{\delta}))\|_{C}\leq$
$\leq 2\,\sup\limits_{f\in L^{\mu}_{s,2},\|f\|_{s,\mu}\leq 1}\
\sup\limits_{\overline{f^{\delta}}:\,(\ref{perturbation})}\|f^{(r_{1},r_{2})}-\psi^{(r_{1},r_{2})}(G(\hat{\Omega},\overline{f}^{\delta}))\|_{C}=:2\,\varepsilon_{\delta}(L^{\mu}_{s,2},\psi^{(r_{1},r_{2})}(\hat{\Omega}),C,\ell_{p}).$

That is

$\varepsilon_{\delta}(L^{\mu}_{s,2},\psi^{(r_{1},r_{2})}(\hat{\Omega}),C,\ell_{p})\geq\frac{\overline{c}}{2}\,N^{-\mu+2r_{1}-1/s+3/2}.$

From the fact that the domain $\hat{\Omega}$ and the algorithm
$\psi^{(r_{1},r_{2})}(\hat{\Omega})\in\Psi(\hat{\Omega})$ are arbitrary,
follows that

$R_{N,\delta}^{(r_{1},r_{2})}(L^{\mu}_{s,2},C,\ell_{p})\geq\frac{\overline{c}}{2}\,N^{-\mu+2r_{1}-1/s+3/2}.$

Thus, the following assertion is proved.

###### Theorem 4.1

Let $1\leq s<\infty$, $r_{1}\geq r_{2}$, $\mu>2r_{1}-1/s+3/2$, $1\leq
p\leq\infty$. Then for any
$N\geq\Big{(}r_{2}^{\mu}\delta/\widetilde{c}\Big{)}^{-1/(\mu+1/s-1/p)}$ it
holds true

$R_{N,\delta}^{(r_{1},r_{2})}(L^{\mu}_{s,2},C,\ell_{p})\geq\frac{\overline{c}}{2}\,N^{-\mu+2r_{1}-1/s+3/2}.$

The following assertion contains sharp estimates (in the power scale) for the
minimal radius in the uniform metric.

###### Theorem 4.2

Let $f\in L^{\mu}_{s,2}$, $1\leq s<\infty$, $r_{1}\geq r_{2}$,
$\mu>2r_{1}+1/2-3/s$.

I) If $r_{1}=r_{2}$, then for
$N\asymp\left(\delta/\ln^{\mu}\frac{1}{\delta}\right)^{-\frac{1}{\mu-1/p+1/s}}$
it holds true

$N^{-\mu+2r_{1}-1/s+3/2}\preceq
R_{N,\delta}^{(r_{1},r_{1})}(L^{\mu}_{s,2},C,\ell_{p})\preceq
N^{-\mu+2r_{1}-1/s+3/2}\ln^{\mu-2r_{1}+1/2}N,$
$\left(\delta/\ln^{\mu}\frac{1}{\delta}\right)^{\frac{\mu-2r_{1}+1/s-3/2}{\mu-1/p+1/s}}\preceq
R_{N,\delta}^{(r_{1},r_{1})}(L^{\mu}_{s,2},C,\ell_{p})\preceq\left(\delta\ln^{1/s-1/p}\frac{1}{\delta}\right)^{\frac{\mu-2r_{1}+1/s-3/2}{\mu-1/p+1/s}}\ln^{2-1/s}\frac{1}{\delta}.$

The upper bounds are implemented by the method
$\mathcal{D}^{(r_{1},r_{1})}_{n,\gamma}$ (2.1) with
$n\asymp\left(\delta/\ln^{1/p-1/s}\frac{1}{\delta}\right)^{-\frac{1}{\mu-1/p+1/s}}$
and $\gamma=1$.

II) If $r_{1}=r_{2}+1$, then for $N\asymp\delta^{-\frac{1}{\mu-1/p+1/s}}$ it
holds true

$R_{N,\delta}^{(r_{1},r_{1}-1)}(L^{\mu}_{s,2},C,\ell_{p})\asymp
N^{-\mu+2r_{1}-1/s+3/2}\asymp\delta^{\frac{\mu-2r_{1}+1/s-3/2}{\mu-1/p+1/s}}.$

The order-optimal bounds are implemented by the method
$\mathcal{D}^{(r_{1},r_{1}-1)}_{n,\gamma}$ (2.1) with
$n\asymp\delta^{-\frac{1}{\mu-1/p+1/s}}$ and any
$\gamma\in\left(1,\frac{\mu-2r_{1}+1/s+5/2}{\mu-2r_{1}+1/s+1/2}\right)\bigcup\left(\frac{\mu-2r_{1}+1/s+5/2}{\mu-2r_{1}+1/s+1/2},\frac{\mu-2r_{1}+1/s+1/2}{\mu-2r_{1}+1/s-3/2}\right)$
.

III) If $r_{1}>r_{2}+1$, then for $N\asymp\delta^{-\frac{1}{\mu-1/p+1/s}}$ it
holds true

$R_{N,\delta}^{(r_{1},r_{2})}(L^{\mu}_{s,2},C,\ell_{p})\asymp
N^{-\mu+2r_{1}-1/s+3/2}\asymp\delta^{\frac{\mu-2r_{1}+1/s-3/2}{\mu-1/p+1/s}}.$

The order-optimal bounds are implemented by the method
$\mathcal{D}^{(r_{1},r_{2})}_{n,\gamma}$ (2.1) for
$n\asymp\delta^{-\frac{1}{\mu-1/p+1/s}}$ and any
$\gamma\in\left(1,\frac{\mu-2r_{2}+1/s-3/2}{\mu-2r_{1}+1/s+1/2}\right)\bigcup\left(\frac{\mu-2r_{2}+1/s-3/2}{\mu-2r_{1}+1/s+1/2},\frac{\mu-2r_{2}+1/s+1/2}{\mu-2r_{1}+1/s+1/2}\right)\bigcup\left(\frac{\mu-2r_{2}+1/s+1/2}{\mu-2r_{1}+1/s+1/2},\frac{\mu-2r_{2}+1/s-3/2}{\mu-2r_{1}+1/s-3/2}\right)$
.

Proof. The upper bounds for $R_{N,\delta}^{(r,r)}(L^{\mu}_{s,2},C,\ell_{p})$
follow from Theorem 3.3. The lower bound is found in Theorem 4.1.

$\Box$

Let’s turn to estimating the minimal radius in the integral metric.

###### Theorem 4.3

Let $1\leq s<\infty$, $r_{1}\geq r_{2}$, $\mu>2r_{1}-1/s+1/2$, $1\leq
p\leq\infty$. Then for any
$N\geq\Big{(}r_{2}^{\mu}\delta/\widetilde{c}\Big{)}^{-1/(\mu+1/s-1/p)}$ it
holds true

$R_{N,\delta}^{(r_{1},r_{2})}(L^{\mu}_{s,2},L_{2},\ell_{p})\geq\overline{\overline{c}}\,N^{-\mu+2r_{1}-1/s+1/2},$

where
$\overline{\overline{c}}=\frac{\widetilde{c}\sqrt{r_{2}+1/2}}{2^{3r_{1}+r_{2}-3/2}\,r_{2}^{\mu}}\frac{(2r_{2})!}{(r_{1}-1)!r_{2}!}$
.

Proof The proof of Theorem 4.3 almost completely coincides with the proof of
Theorem 4.1, including the form of the auxiliary functions $f_{1}$,
$f_{1}^{\delta}$, $f_{2}$, $f_{2}^{\delta}$. The only difference is in the
lower estimate of the norm of $f_{1}^{(r_{1},r_{2})}-f_{2}^{(r_{1},r_{2})}$.
Changing the order of summation in (4.1) yields to the representation

$f_{1}^{(r_{1},r_{2})}(t,\tau)=2^{r_{1}}\,\frac{\widetilde{c}}{r_{2}^{\mu}}\,{N^{-\mu-1/s}}\,\varphi_{r_{2}}^{(r_{2})}(\tau)$
$\times\Big{(}\mathop{{\sum}^{*}}\limits_{l_{r_{1}}=0}^{N}\sqrt{l_{r_{1}}+1/2}\,\varphi_{l_{r_{1}}}(t)\mathop{{\sum}\,^{\prime}}\limits_{k=N+r_{1}}^{3N+r_{1}}\sqrt{k+1/2}+\mathop{{\sum}^{*}}\limits_{l_{r_{1}}=N+1}^{3N}\sqrt{l_{r_{1}}+1/2}\,\varphi_{l_{r_{1}}}(t)\mathop{{\sum}\,^{\prime}}\limits_{k=l_{r_{1}}+r_{1}}^{3N+r_{1}}\sqrt{k+1/2}\Big{)}$
$\times\mathop{{\sum}^{*}}\limits_{l_{1}=l_{r_{1}}+r_{1}-1}^{k-1}(l_{1}+1/2)\mathop{{\sum}^{*}}\limits_{l_{2}=l_{r_{1}}+r_{1}-2}^{l_{1}-1}(l_{2}+1/2)\ldots\mathop{{\sum}^{*}}\limits_{l_{r_{1}-1}=l_{r_{1}}+1}^{l_{r_{1}-2}-1}(l_{r_{1}-1}+1/2).$

Let us introduce the following notation

$c^{\prime}=\frac{2^{2(r_{1}-r_{2})+1}}{r_{2}^{2\mu}}\,\widetilde{c}^{2}(r_{2}+1/2)\frac{((2r_{2})!)^{2}}{(r_{2}!)^{2}}.$

Then it follows that

$\|f_{1}^{(r_{1},r_{2})}-f_{2}^{(r_{1},r_{2})}\|_{L_{2}}^{2}=\|f_{1}^{(r_{1},r_{2})}\|_{L_{2}}^{2}\geq
c^{\prime}\,{N^{-2\mu-2/s}}\,\mathop{{\sum}^{*}}\limits_{l_{r_{1}}=0}^{N}(l_{r_{1}}+1/2)$
$\times\left(\mathop{{\sum}\,^{\prime}}\limits_{k=N+r_{1}}^{3N+r_{1}}\sqrt{k+1/2}\mathop{{\sum}^{*}}\limits_{l_{1}=l_{r_{1}}+r_{1}-1}^{k-1}(l_{1}+1/2)\mathop{{\sum}^{*}}\limits_{l_{2}=l_{r_{1}}+r_{1}-2}^{l_{1}-1}(l_{2}+1/2)\ldots\mathop{{\sum}^{*}}\limits_{l_{r_{1}-1}=l_{r_{1}}+1}^{l_{r_{1}-2}-1}(l_{r_{1}-1}+1/2)\right)^{2}$
$\geq c^{\prime}\,\frac{N^{-2\mu-2/s}\cdot
N^{4r_{1}-1}}{4^{4(r_{1}-1)}((r_{1}-1)!)^{2}}\mathop{{\sum}^{*}}\limits_{l_{r_{1}}=0}^{N/2}(l_{r_{1}}+1/2).$

That is, the estimate holds

$\|f_{1}^{(r_{1},r_{2})}-f_{2}^{(r_{1},r_{2})}\|_{L_{2}}\geq\frac{\sqrt{c^{\prime}}}{4^{2r_{1}-1}(r_{1}-1)!}N^{-\mu+2r_{1}-1/s+1/2}.$

Whence we obtain that the relation

$\varepsilon_{\delta}(L^{\mu}_{s,2},\psi^{(r_{1},r_{2})}(\hat{\Omega}),L_{2},\ell_{p})\geq\overline{\overline{c}}\,N^{-\mu+2r_{1}-1/s+1/2}$

is true for any
$N\geq\Big{(}r_{2}^{\mu}\delta/\widetilde{c}\Big{)}^{-1/(\mu+1/s-1/p)}$. From
the fact that the domain $\hat{\Omega}$ and the algorithm
$\psi^{(r_{1},r_{2})}(\hat{\Omega})\in\Psi(\hat{\Omega})$ are arbitrary, it
follows that

$R_{N,\delta}^{(r_{1},r_{2})}(L^{\mu}_{s,2},L_{2},\ell_{p})\geq\overline{\overline{c}}\,N^{-\mu+2r_{1}-1/s+1/2}.$

Thus, the proof of Theorem 4.3 has been complied.

$\Box$

The following assertion contains sharp estimates (in the power scale) for the
minimal radius in the integral metric.

###### Theorem 4.4

Let $f\in L^{\mu}_{s,2}$, $1\leq s<\infty$, $r_{1}\geq r_{2}$,
$\mu>2r_{1}+1/2-1/s$.

I) If $r_{1}=r_{2}$, then for
$N\asymp\left(\delta/\ln^{\mu}\frac{1}{\delta}\right)^{-\frac{1}{\mu-1/p+1/s}}$
it holds true

$N^{-\mu+2r_{1}-1/s+1/2}\preceq
R_{N,\delta}^{(r_{1},r_{1})}(L^{\mu}_{s,2},L_{2},\ell_{p})\preceq
N^{-\mu+2r_{1}-1/s+1/2}\ln^{\mu-2r_{1}+1}N,$
$\left(\delta/\ln^{\mu}\frac{1}{\delta}\right)^{\frac{\mu-2r_{1}+1/s-1/2}{\mu-1/p+1/s}}\preceq
R_{N,\delta}^{(r_{1},r_{1})}(L^{\mu}_{s,2},L_{2},\ell_{p})\preceq\left(\delta\ln^{1/s-1/p}\frac{1}{\delta}\right)^{\frac{\mu-2r_{1}+1/s-1/2}{\mu-1/p+1/s}}\ln^{3/2-1/s}\frac{1}{\delta}.$

The upper bounds are implemented by the method
$\mathcal{D}^{(r_{1},r_{1})}_{n,\gamma}$ (2.1) with
$n\asymp\left(\delta^{-1}\ln^{1/p-1/s}\frac{1}{\delta}\right)^{\frac{1}{\mu-1/p+1/s}}$
and $\gamma=1$.

II) If $r_{1}>r_{2}$, then for $N\asymp\delta^{-\frac{1}{\mu-1/p+1/s}}$ it
holds true

$R_{N,\delta}^{(r_{1},r_{2})}(L^{\mu}_{s,2},L_{2},\ell_{p})\asymp
N^{-\mu+2r_{1}-1/s+1/2}\asymp\delta^{\frac{\mu-2r_{1}+1/s-1/2}{\mu-1/p+1/s}}.$

The order-optimal bounds are implemented by the method
$\mathcal{D}^{(r_{1},r_{2})}_{n,\gamma}$ (2.1) with
$n\asymp\delta^{-\frac{1}{\mu-1/p+1/s}}$ and any
$\gamma\in\left(1,\frac{\mu-2r_{2}+1/s-1/2}{\mu-2r_{1}+1/s+1/2}\right)\bigcup\left(\frac{\mu-2r_{2}+1/s-1/2}{\mu-2r_{1}+1/s+1/2},\frac{\mu-2r_{2}+1/s+1/2}{\mu-2r_{1}+1/s+1/2}\right)\bigcup\left(\frac{\mu-2r_{2}+1/s+1/2}{\mu-2r_{1}+1/s+1/2},\frac{\mu-2r_{2}+1/s-1/2}{\mu-2r_{1}+1/s-1/2}\right)$
.

Proof. The upper bounds for
$R_{N,\delta}^{(r_{1},r_{2})}(L^{\mu}_{s,2},L_{2},\ell_{p})$ follow from
Theorem 2.3. The lower bound is found in Theorem 4.3.

$\Box$

##  5\.  Acknowledgements

This project has received funding through the MSCA4Ukraine project, which is
funded by the European Union. In addition, the first named author is supported
by the Volkswagen Foundation project ”From Modeling and Analysis to
Approximation”. Also, the authors acknowledge partial financial support due to
the project ”Mathematical modelling of complex dynamical systems and processes
caused by the state security” (Reg. No. 0123U100853).

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Institute of Mathematics, National Academy of Sciences of Ukraine, 3,
Tereschenkivska Str., 01024, Kiev, Ukraine
E-mail address<EMAIL_ADDRESS><EMAIL_ADDRESS>
Received 11.02.2009; revised 23.03.2009


# The Stochastic Guitar

Andreas Eggenberger  Department of Physics, ETH Zurich, Otto-Stern-Weg 1, 8093
Zurich, Switzerland  Alexander Eichler  Laboratory for Solid State Physics,
ETH Zürich, CH-8093 Zürich, Switzerland.  Quantum Center, ETH Zurich, CH-8093
Zurich, Switzerland

( )

###### Abstract

Stochastic physics is a central pillar of modern research in many fields, but
is rarely presented to undergrad students in a hands-on experiment. Here, we
demonstrate how a human-scale, simple, and affordable experimental setup can
be used to fill this gap, and to illustrate many advanced concepts in a step-
by-step approach. Based on a metal wire (such as a guitar string), our setup
facilitates the observation of fluctuating dynamics in the time domain, the
frequency spectrum, and in the rotating phase space. The latter allows
introducing time-dependent cross-correlations between the sine and cosine
quadratures of the stochastic motion, which feature deterministic order even
in the absence of any deterministic forces.

††preprint: APS/123-QED

##  I Introduction

The driven and damped harmonic oscillator is a textbook model equation that
every physics student is familiar with. Application of this model can be found
in every field of physics, from quantum optics to particle physics,
biophysics, and astronomy. For that reason, the harmonic oscillator is also a
popular experiment on display in many student labs and practical lecture
experiments. Students typically learn about the oscillator’s resonance
frequency $\omega_{0}=2\pi f_{0}$, the rate of energy loss to the environment
$\Gamma$, and the corresponding quality factor $Q=\omega_{0}/\Gamma$. In the
vast majority of student experiments, the harmonic oscillator is treated as a
deterministic system without any noise.

The physics of the harmonic oscillator becomes much richer when a stochastic
driving force is added. In this case, the oscillation has no definite
amplitude $X$ and phase $\phi$, but explores different values thereof as a
function of time in a stochastic random walk. It is therefore generally
impossible to predict $X$ and $\phi$ at any point in time in the future, and
only statistical probabilities can be calculated. Such a scenario is immensely
difficult to grasp for many undergraduate students for several reasons: (i)
Dealing with probabilities instead of deterministic equations of motion
requires very abstract thinking. (ii) Stochastic physics is not common, or at
least not obvious, in our everyday lives. The systems that offer themselves
for such studies, such as nanomechanical resonators, usually operate on scales
of amplitudes and frequencies that are far from our intuition and hard to
measure. (iii) Due to the stringent requirements for observing e.g. the
Brownian motion of molecules, student labs rarely offer experiments to explore
stochastic physics. Most students therefore only come into contact with this
important aspect of physics when they perform projects in research groups.
This is deplorable, as stochastic physics is not only a crucial element in
many contemporary fields of physics, but also offers an excellent preparation
for the stochastic phenomena encountered in quantum mechanics.

In this work, we describe a student lab experiment that allows observing the
harmonic oscillator with a stochastic driving force. The experiment is based
on a metal string whose fundamental resonance frequency is around
$30\text{\,}\mathrm{Hz}$ and whose oscillation can be visible to the naked
eye. In spite of these “human” scales, the string can be driven in a
stochastic manner, and its displacement can be measured digitally with high
precision. This results in rich measurement data and complex concepts, such as
the rotating phase space, the power spectral density of different types of
noise, the Parseval theorem, and the equivalent of under- or overdamped
oscillations around the stationary solutions of an out-of-equilibrium system.
The experiment offers motivated students a hands-on approach to many
fascinating concepts, and a bridge to advanced experiments in research labs.
In addition, the experiment is ideally suited for public demonstrations thanks
to its accessible character: everybody loves a guitar!

##  II Experimental Setup

The heart of our experiment is a metal string with a cross section of
$0.15\times$0.9\text{\,}\mathrm{mm}$$ made out of stainless steel, as inspired
by previous experiments Leuch  _et al._ (2016); Eichler  _et al._ (2018). The
string is suspended between two clamping points with a separation of
$275\text{\,}\mathrm{mm}$, see Fig. 1. The tension applied to the string can
be used to tune its resonance frequencies. For our demonstration, we selected
a relatively low tension, which results in low resonance frequencies and low
frequency drift in response to temperature changes in the room. In the center
of the string, we place a neodymium magnet. This allows us to apply a force to
the string by running a current through a coil below the magnet. The
displacement of the string is recorded through a piezoelectric crystal
embedded in one of the clamping points. In this crystal, the pressure that the
string exerts onto the clamping point is transduced into a voltage that we
measure with a Zurich Instruments MFLI lock-in amplifier (any analog-digital
converter would do).

Figure 1: Sketch of the setup Figure 2: Amplitude and phase of a measured
frequency sweep (light blue) plotted versus the results of Eq. (2) and (3)
(dark blue), using
$f_{0}=\omega_{0}/2\pi=$28.52\text{\,}\mathrm{H}\mathrm{z}$$,
$\Gamma=$0.209\text{\,}\mathrm{Hz}$$, and a global phase offset of
$50\text{\,}\mathrm{\SIUnitSymbolDegree}$ as free parameters. The measurement
used $15\text{\,}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}$
waiting time between each point and steps of $10\text{\,}\mathrm{mHz}$.

##  III Deterministic Response

To probe the basic properties of our string, we apply an oscillating current
to the coil. The current generates a force $F(t)=F_{0}\cos(\omega t+\theta)$
with amplitude $F_{0}$ and frequency $f=\omega/2\pi$. For fixed forcing
parameters, the string “rings up” until it responds with stable oscillations
at the driving frequency. This behaviour can be understood from the well-know
equation of motion of a single harmonic resonator

$\displaystyle\ddot{x}+\omega_{0}^{2}x+\Gamma\dot{x}=\frac{F(t)}{m}\,,$ (1)

where $x$ is the displacement of the string and $m$ is its effective mass.
With the Ansatz $x(\omega)=X(\omega)e^{i(\omega t+\phi(\omega))}$, we can
solve Eq. (1) and obtain for the long-time limit the amplitude

$\displaystyle\left|{x}(\omega)\right|=X(\omega)=\frac{F_{0}/m}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+\omega^{2}\Gamma^{2}}}$
(2)

and the phase

$\displaystyle\phi(\omega)=\tan^{-1}\left(\frac{\omega\Gamma}{\omega_{0}^{2}-\omega^{2}}\right)\,.$
(3)

In Fig. 2, we show the result of such a measurement together with those of
analytical expressions in Eqs. (2) and (3) for the fit values
$f_{0}=\omega_{0}/2\pi=$28.52\text{\,}\mathrm{H}\mathrm{z}$$ and
$\Gamma=$0.209\text{\,}\mathrm{Hz}$$.

In the following, we will characterize and analyze our resonator in terms of
the phase space quadratures $u$ and $v$, which fulfill $X^{2}=u^{2}+v^{2}$ and
$\phi=\tan^{-1}(v/u)$ Eichler and Zilberberg (2023). These quadratures are the
amplitudes of the two independent, fundamental solutions of the harmonic
oscillator, $x=u\cos(\omega t)-v\sin(\omega t)$. Via the averaging method
Holmes and Holmes (1981); Papariello  _et al._ (2016), we can transform Eq.
(1) into so-called slow-flow equations, which are coupled first-order
differential equations of the form

$\displaystyle\dot{u}=-u\frac{\Gamma}{2}+v\frac{\omega^{2}-\omega_{0}^{2}}{2\omega}+F_{0}\frac{\sin(\theta)}{2m\omega}\,,$
(4)
$\displaystyle\dot{v}=-v\frac{\Gamma}{2}-u\frac{\omega^{2}-\omega_{0}^{2}}{2\omega}-F_{0}\frac{\cos(\theta)}{2m\omega}\,.$
(5)

The slow-flow variables $u(t)$ and $v(t)$ correspond to the in-phase and out-
of-phase channels of a lock-in amplifier and are therefore directly accessible
in many experiments.

##  IV Stochastic Response

In order to observe stochastic physics, we replace the sinusoidal oscillating
force by a fluctuating force of the form $F(t)=\xi(t)$, which has the
properties

$\displaystyle\left\langle\xi(t)\right\rangle$ $\displaystyle=0\,,$ (6)
$\displaystyle\left\langle\xi(t)\xi(t+\tau)\right\rangle$
$\displaystyle=\varsigma^{2}\delta(\tau)\,,$ (7)

where $\left\langle...\right\rangle$ denotes the average in the long-time
limit, $\tau$ is a time delay, $\delta(...)$ is the delta function that is $1$
for $\tau=0$ and $0$ otherwise, and $\varsigma^{2}$ has the meaning of a force
noise power spectral density Eichler and Zilberberg (2023).

The slow-flow Eqs. (4) and (5) are correspondingly modified to read

$\displaystyle\dot{u}=-u\frac{\Gamma}{2}+v\frac{\omega^{2}-\omega_{0}^{2}}{2\omega}+\frac{\Xi_{u}}{m}\,,$
(8)
$\displaystyle\dot{v}=-v\frac{\Gamma}{2}-u\frac{\omega^{2}-\omega_{0}^{2}}{2\omega}+\frac{\Xi_{v}}{m}\,,$
(9)

where the $\Xi_{i}$ are fluctuating force terms whose standard deviation can
be calculated from an integration of $\xi$ over one oscillation period Eichler
and Zilberberg (2023). These fluctuating force terms are still white and
random on timescales $\tau>1/f_{0}$, which is sufficient for all practical
purposes.

Figure 3: Frequency spectrum of the string driven by a fluctuating force with
a standard deviation of $500\text{\,}\mathrm{m}\mathrm{V}$ over a bandwidth of
$30\text{\,}\mathrm{MHz}$. We clearly see the resonance frequency $f_{0}$ at
$28.5\text{\,}\mathrm{H}\mathrm{z}$, the $50\text{\,}\mathrm{H}\mathrm{z}$
noise of the power grid, and a higher mode at roughly $2f_{0}$. A dark blue
line is shows the result of Eq. (2) with the same parameters as in Fig. 2.

The white force effectively drives the string at all frequencies
simultaneously, producing the oscillation spectrum shown in Fig. 3. In the
frequency range between $0$ and $100\text{\,}\mathrm{Hz}$, we observe four
peaks. The mode at $50\text{\,}\mathrm{Hz}$ is assigned to purely electrical
noise induced by the local power grid, while a peak below
$60\text{\,}\mathrm{Hz}$ may belong to the second vibrational mode of the
string. The lowest mode at $28.5\text{\,}\mathrm{Hz}$ is our vibration mode of
interest. Using the known value for $f_{0}$ and $\Gamma$, we can reproduce its
spectral signature with Eq. (2), using only the effective amplitude
$\varsigma/m$ as a free parameter.

Figure 4: The two quadrature signals in response to a fluctuating force term,
measured as a function of time with a lock-in amplifier whose local oscillator
matches $f_{0}$.

The time response of the lowest oscillator mode to a stochastic force is shown
in Fig. 4. In this measurement, the quadratures $u$ and $v$ are directly
measured with a lock-in amplifier with a local oscillator (a reference clock)
at $28.5\text{\,}\mathrm{H}\mathrm{z}$. Selecting a bandpass filter of
$4.5\text{\,}\mathrm{H}\mathrm{z}$ ensures that signals from higher modes are
rejected, even though they are driven by the white noise as well. We find that
the resonator no longer rings up to a fixed solution as in the deterministic
case discussed earlier. Instead, the quadratures slowly fluctuate in time
around the mean values $\left\langle u\right\rangle=\left\langle
v\right\rangle=0$ with a Gaussian profile.

Figure 5: Phase space representation of the string oscillations driven by
fluctuating force terms. The quadratures $u$ and $v$ are measured with in a
frame rotating a the local oscillator frequency of the lock-in, set as (a)
$28.5\text{\,}\mathrm{H}\mathrm{z}$ and (b)
$28.4\text{\,}\mathrm{H}\mathrm{z}$ (b). The starting point of the measurement
is marked as a red dot, and the first 30 points are connected with a line to
demonstrate a random walk for $\omega=\omega_{0}$ and counter-clockwise
rotation for $\omega<\omega_{0}$.

To study the dynamics of the stochastically driven oscillator in more detail,
we plot $v$ as a function of $u$ in Fig. 5. This representation is known as
“phase space” and plays an important role in the visualization of classical
and quantum oscillation states. In our classical case, the system can be
located at a single point in phase space at any given time, and a stable
solution appears as a single fixed point. For instance, setting
$\dot{u}=\dot{v}=0$ and $\omega=\omega_{0}$ in Eqs. (4) and (5) results in two
simple equations $u\propto F_{0}\sin(\theta)$ and $v\propto-
F_{0}\cos(\theta)$.

We plot the stochastic data similar to that in Fig. 4 in phase space, see Fig.
5(a). The system explores phase space in a random walk over time, leading to
Gaussian probability distributions in both coordinates. As we selected
$\omega=\omega_{0}$ for this measurement, the terms
$\propto\omega^{2}-\omega_{0}^{2}$ become zero in Eqs. (8) and (9) and the
random trajectories in both coordinates are decoupled. Setting
$\dot{u}=\dot{v}=0$, the equations can be individually simplified to
$u\propto\Xi_{u}$ and $v\propto\Xi_{v}$. The trajectory of the system over
time follows a random walk in both $u$ and $v$ independently.

Our observations become more intricate when we introduce a detuning of the
measurement clock from the resonator’s eigenfrequency, $\omega\neq\omega_{0}$.
At first glance, the phase space representation in Fig. 5(b) closely resembles
the one in Fig. 5(a). However, when we follow a short part of the trajectory,
we find it to revolve around the origin with a defined orientation. This
feature, which is absent in Fig. 5(a), clearly stems from the terms
$\propto\omega^{2}-\omega_{0}^{2}$ in Eqs. (8) and (9), which create a
correlation between $u$ and $v$ with a time delay. We can directly see this
delayed correlation in Eqs. (8) and (9) by inserting $\Gamma=\Xi_{u,v}=0$ and
$\omega^{2}-\omega_{0}^{2}=(\omega-\omega_{0})(\omega+\omega_{0})\approx(\omega-\omega_{0})2\omega$,
which leads to the simple equations $\dot{u}=\Delta v$ and $\dot{v}=\Delta u$
with $\Delta\equiv\omega-\omega_{0}$. When initialized at $u_{i}=1$ and
$v_{i}=0$, this system rotates around the origin with an angular frequency
$\Delta=(\omega-\omega_{0})$, such that a correlation between $u$ and $v$
appears after a quarter period (when $u=0$ and $v=\pm 1$). The sign of the
correlation at this delay depends on the sign of $\Delta$, i.e., whether the
rotation proceeds in clockwise or anti-clockwise direction.

Figure 6: Time-dependent covariance $Cxx$ measured as a function of the time
shift $\tau$ for different detuning $\Delta$. The data sets are offset
vertically for better visibility, with dashed horizontal lines indicating
$Cxx=0$ in all cases. Solid lines correspond to the function
$e^{-\Gamma\tau/2}\sin(\Delta\tau)$ with $\Delta$ fitted for each curve
individually.

To quantitatively analyze the effect of a detuned rotating frame in our data,
we employ the delay-dependent covariance

$\displaystyle C_{uv}(n)=\frac{1}{N-n}\sum_{i=1}^{N-n}\left[u(i)-\left\langle
u\right\rangle\right]\left[v(i+n)-\left\langle v\right\rangle\right]$ (10)

where $N$ is the total number of data points in a measurement, $i$ runs over
all measurement points that are separated in time by the increment
$\Delta_{t}$, and $\tau=n\Delta_{t}$ is a time delay. Equation (10) is a
covariance between $u$ and $v$ with a variable time delay $\tau$. We expect to
see maxima in $C_{uv}$ whenever $u$ and $v$ values are correlated over a well-
defined timescale $\tau$, indicating a rotation with a well-defined period.

The results of Eq. (10) for measurements with different $\Delta$ are shown in
Fig. 6, together with a heuristic theory line corresponding to
$C_{uv}=e^{-\Gamma\tau/2}\sin(\Delta\tau)$, that is, an exponentially decaying
oscillating correlation. Even though the full dependency of the covariance on
$\tau$ is more complicated, this simple form reproduces our measurements
qualitatively: detuning the measurement clock $\omega$ from the eigenfrequency
$\omega_{0}$ leads to a periodic pattern in the measured time-dependent
covariance $C_{uv}$, which decays under the influence of the dissipation term
$\Gamma$. The detuning $\Delta$ therefore acts like a adjustable oscillation
frequency in the rotating frame by generating correlations between $u$ and
$v$. For a given value of $\Gamma$, we observe the rotating-frame analogue to
an underdamped-to-overdamped transition between $\Delta>\Gamma$ (red and blue
traces in Fig. 6) and $\Delta\ll\Gamma$ (light orange to light blue traces).
We have thus identified deterministic correlations in a system driven purely
by fluctuating force terms.

##  V Summary and outlook

In this work, we show how a simple and affordable guitar string setup can be
used to demonstrate intricate physical phenomena. We start from the driven and
damped harmonic oscillator, the most widely used model in physics, to describe
the string oscillation to an external force. Building on this model, we
introduce stochastic driving terms and observe the string’s spectral response.
We also study the stochastic dynamics of the string as a function of time and
in phase space, which allows us to appreciate the notion of a Gaussian
amplitude distribution in both quadratures and the influence of the selected
rotating frame. Finally, we manipulate the rotating frame rate to obtain a
rotating-frame analogue to an underdamped-to-overdamped transition in the
quadrature correlations.

In our opinion, such a demonstration has high didactic value, as it provides
hands-on experience with the driven and damped harmonic oscillator,
deterministic versus stochastic physics, temporal versus spectral information,
the rotating frame, and phase space representation. These topics are relevant
for many fields of modern physics, such as quantum optics, nanomechanics,
radio-frequency and microwave electronics, and gravitation wave experiments.
From a teaching perspective, introducing these concepts with an experiment is
a valuable supplement to traditional lectures. In particular, the intuitive
mechanics and macroscopic scale of the string resonator is beneficial to
illustrate complex concepts in an accessible way.

Looking forward, we propose that the same string setup is also suitable to
instruct students in nonlinear and parametric dynamics, especially the Duffing
resonator, as well as parametric driving and squeezing Leuch  _et al._ (2016).
These complex topics are often neglected in the experimental education of
undergraduate students for lack of an affordable and easily maintainable
experimental setup. Our guitar string offers a solution for this problem, and
can serve as a catalyst to introduce students to a range of fascinating
experimental techniques at an early stage in their curriculum.

List of student tasks:

1\. Apply an oscillating drive voltage to the coil and measure the response
voltage from the piezo with an oscilloscope. Look at the frequencies within
the response by Fourier transforming it.

2\. Change the frequency $f$ of the drive voltage and observe the change in
the amplitude when this drive frequency is close to $f_{0}$.

3\. Measure both the drive voltage and the response voltage at the same time
with the oscilloscope. How can you measure the phase in this way?

4\. Measure the amplitude and the phase with a lock-in amplifier and plot it
as a function of the driving frequency $f$ (a free software lock-in module is
available here). Add matching lines corresponding to Eq. (2) and (3) and
extract $\omega_{0}=2\pi f_{0}$ and $\Gamma$ in this way.

5\. Replace the oscillating driving voltage with a white noise voltage and
check that your noise is indeed white and has a Gaussian distribution in
voltage. Measure the response of the resonator as a frequency spectrum. Fit
Eq. (2) to your spectrum and extract $\omega_{0}$ and $\Gamma$. Compare the
spectrum of the resonator response to that of the white noise itself.

6\. Measure $u$ and $v$ with a lock-in amplifier for different local
oscillator frequencies, that is, detunings $\Delta$. Plot the results in a
rotating phase space as in Fig. 5 and determine the histograms in both
quadratures. Pay attention to the sampling rate relative to $1/\Gamma$. What
is an ideal sampling rate, and why? Does $\Delta$ impact the histograms?

7\. Use Eq. (10) to obtain a delay-dependent covariance for every measurement
and extract the true $\Delta$ in each case. What is the origin of the
deterministic order you find here?

The authors have no conflicts to disclose.

Acknowledgements: the authors thank Oded Zilberberg for useful discussions.
Oliver Schwager and the mechanical workshop at the Physics Department of ETH
Zurich contributed to the development of the setup.

## References

  * Leuch  _et al._ (2016) Anina Leuch, Luca Papariello, Oded Zilberberg, Christian L. Degen, R. Chitra, and Alexander Eichler, “Parametric symmetry breaking in a nonlinear resonator,” Phys. Rev. Lett. 117, 214101 (2016).
  * Eichler  _et al._ (2018) Alexander Eichler, Toni L Heugel, Anina Leuch, Christian L Degen, Ramasubramanian Chitra, and Oded Zilberberg, “A parametric symmetry breaking transducer,” Applied Physics Letters 112, 233105 (2018).
  * Eichler and Zilberberg (2023) Alexander Eichler and Oded Zilberberg,  _Classical and Quantum Parametric Phenomena_ (Oxford University Press, 2023).
  * Holmes and Holmes (1981) C. Holmes and P. Holmes, “Second order averaging and bifurcations to subharmonics in duffing’s equation,” Journal of Sound and Vibration 78, 161 – 174 (1981).
  * Papariello  _et al._ (2016) Luca Papariello, Oded Zilberberg, Alexander Eichler, and R. Chitra, “Ultrasensitive hysteretic force sensing with parametric nonlinear oscillators,” Phys. Rev. E 94, 022201 (2016).