\documentclass[12pt,draft,a4paper]{amsart} \usepackage{amsmath,amsthm} \usepackage[T2A]{fontenc} \usepackage[cp1251]{inputenc} \usepackage[english]{babel} \usepackage{amsfonts} \usepackage{latexsym} %\usepackage{srctex} \tolerance 1950 \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{corollary}{Corollary} \DeclareMathOperator*{\infp}{inf\vphantom p} \DeclareMathOperator*{\codim}{codim} \DeclareMathOperator*{\spa}{span} \DeclareMathOperator*{\Ker}{Ker} \DeclareMathOperator*{\supp}{supp} \DeclareMathOperator*{\IM}{Im} \begin{document} \title[Ismagilov Type Theorems for $n$-Widths]{Ismagilov Type Theorems for Linear, Gel'fand and Bernstein $n$-Widths} \author{K. Yu.\ Osipenko and O. G. Parfenov} \thanks{The research of the first author was supported in part by Russian Foundation of Fundamental Research and by the International Scientific Fund} \thanks{The research of the second author was supported in part by the International Scientific Fund and by the Natural Sciences Academy of Russian Federation} \begin{abstract} Using a variational principle for $s$-numbers, we obtain estimates for the linear, Gel'fand and Bernstein $n$-widths. A simple proof of some results concerned with the exact values of $n$-widths of diagonal operators is given. We also calculate the exact values of the Bernstein $n $-widths for the Hardy--Sobolev classes. \end{abstract} \maketitle \section{Introduction} Let $X$, $Y$ be normed linear spaces and $T\colon X\to Y$ be a bounded linear operator. The linear $\lambda_n$, Gel'fand $d^n$ and Bernstein $b_n$ $n$-widths of the operator $T$ are defined by \begin{gather*} \lambda_n(T):=\infp_{P_n}\,\sup_{x\in BX}\|Tx-P_nx\|_Y,\quad d^n(T):=\infp_{X^n}\,\sup_{x\in BX^n}\|Tx\|_Y,\\ b_n(T):=\sup_{X_{n+1}}\,\infp_{\substack{x\in X_{n+1}\\x\ne0}}\frac{\|Tx\|_Y}{ \|x\|_X}, \end{gather*} where $P_n$ is any linear operator mapping $X$ into $Y$ of rank at most $n$, $BX$ is the closed unit ball of $X$, $X^n$ runs over all $n$-codimensional subspaces of $X$ and $X_{n+1}$ runs over all $(n+1)$-dimensional subspaces of $X$. In Osipenko and Stessin \cite{1} the exact values of the linear and Gel'fand $n$-widths of the Hardy classes $H_2$ were obtained. A method of the proof was very close to the one from Ismagilov's Theorem \cite{2} (see also \cite[p.~93]{3}). After the paper \cite{1} several results were obtained for $\lambda_n(T)$ and $d^n(T)$ where $T$ is a map from a Hilbert space $H$ into $C(E)$ (see \cite{4}--\cite{6}). Parfenov \cite{7} solved an analogous problem for the Bernstein $n$-widths $b_n(T)$ where $T\colon L_\infty(E,\nu)\to H$ and $\nu$ is a probability measure on $E$. In this paper we show that many of these results can be obtained, using a general principle concerned with extremal properties of $s$-numbers. Section~2 is devoted to this principle. In Section~3 we prove the estimates of the linear, Gel'fand and Bernstein $n$-widths. In Section~4 we give a simple proof of two results about the exact values of $n$-widths for diagonal operators in the discrete case. Finally, in Section~5 we calculate the Bernstein $n$-widths of the Hardy--Sobolev classes. \section{Variational Principle for $s$-Numbers} Let $H$ and $H_1$ be Hilbert spaces and $T\colon H\to H_1$ a bounded linear operator. Suppose that $$T'T\varphi_k=\lambda_k\varphi_k,\quad k=1,2,\ldots,$$ where $\lambda_1\ge\lambda_2\ge\dots>0$ and $\{\varphi_k\}$ form a complete orthonormal basis for the range of $T'T$ (a sufficient condition is that $T$ be a compact operator). The values $s_k(T)=\sqrt{\lambda_k}$ are called the $s$-numbers of $T$. Set $\psi_k:=s_k^{-1}(T)T\varphi_k$. Note that $\{\psi_k\}$ is an orthonormal system in $H_1$. Then there exists the Schmidt decomposition of $T$ (see, for example, \cite{8}) which is given by $$T=\sum_{k=1}^\infty s_k(T)(\cdot,\varphi_k)\psi_k.$$ \begin{theorem}\label{T1} Let $T$ be as above. Then \begin{equation}\label{1} \sum_{k=1}^ns_k^2(T)=\max_{\{e_k\}_1^n}\sum_{k=1}^n\|Te_k\|_{H_1}^2, \end{equation} where the maximum is taken over all orthonormal systems $\{e_k\}_1^n$ in $H$. Furthermore, \begin{equation}\label{2} \sum_{k=n+1}^\infty s_k^2(T)=\min_{\{e_k\}_1^\infty}\sum_{k=1}^\infty\|Te_k\|_{H_1}^2, \end{equation} where the minimum is taken over all orthonormal systems $\{e_k\}_1^\infty$ in $H$ such that $\codim\spa\{e_k\}_1^\infty\le n$. \end{theorem} For a compact operator $T$ this theorem was proved in Parfenov \cite{6}. In our case the proof is almost the same because it does not so much depend on the compactness of $T$ as on the fact that the eigenvectors $\{\varphi_k\}$ of $T'T$ form an orthonormal basis for the range of $T'T$. We remark that both parts in \eqref{2} are finite iff $$\sum_{k=1}^\infty s_k^2(T)<\infty.$$ In Parfenov \cite{6} Theorem~\ref{T1} was the basic tool in calculating the Gel'fand $n$-widths of operators $T\colon H\to L_\infty(E,\nu)$. Similar results were obtained in Osipenko \cite{5}, using Ismagilov's Theorem and the duality between the Kolmogorov and Gel'fand $n$-widths. In order to estimate the Bernstein $n$-widths we need the following properties of $s$-numbers. \begin{theorem}\label{T2} Let $T$ be as in Theorem~\ref{T1} and $\Ker T=0$. Then \begin{equation}\label{3} \sum_{k=1}^n s_k^{-2}(T)=\min_{\{f_k\}_1^n}\sum_{k=1}^n\|T^{-1}f_k\|_H^2, \end{equation} where the minimum is taken over all orthonormal systems $\{f_k\}_1^n$ in $T(H)$. Furthermore, if $\dim H=N<\infty$, then \begin{equation}\label{4} \sum_{k=n+1}^N s_k^{-2}(T)=\max_{\{f_k\}_1^{N-n}}\sum_{k=1}^{N-n}\|T^{-1} f_k\|_H^2, \end{equation} where the maximum is taken over all orthonormal systems $\{f_k\}_1^{N-n}$ in $T(H)$. \end{theorem} \begin{proof} Let $\{f_k\}_1^n$ be any orthonormal system in $T(H)$. Set $L_n:=\spa\{T^{-1}f_k\}_1^n$, $\widetilde L_n:=\spa\{f_k\}_1^n$, and $T_n:=T_{\big|L_n}$. Suppose that the Schmidt decomposition of $T_n$ has the form $$T_n=\sum_{k=1}^n s_k(T_n)(\cdot,\varphi_k')\psi_k'.$$ The value $$\left(\sum_{k=1}^n\|T^{-1}_nf_k\|_H^2\right)^{1/2}$$ is the Hilbert--Schmidt norm of $T_n^{-1}$ and does not depend on the choice of the orthonormal basis in $\widetilde L_n$. Therefore, $$\sum_{k=1}^n\|T^{-1}f_k\|_H^2=\sum_{k=1}^n\|T^{-1}_nf_k\|_H^2=\sum_{k=1}^ n\|T^{-1}_n\psi_k'\|_H^2=\sum_{k=1}^ns^{-2}_k(T_n).$$ Let $P_n$ be an orthoprojector in $H$ onto $L_n$. Using the properties of $s$-numbers of bounded linear operators (see \cite[p.~82]{8}) we have $$s_k(T_n)=s_k(T\circ P_n)\le\|P_n\|s_k(T)=s_k(T).$$ Thus $$\sum_{k=1}^n\|T^{-1}f_k\|_H^2\ge\sum_{k=1}^ns^{-2}_k(T).$$ If $f_k=\psi_k$, $k=1,\ldots,n$, then $$\sum_{k=1}^n\|T^{-1}\psi_k\|_H^2=\sum_{k=1}^ns^{-2}_k(T).$$ The equality \eqref{3} is proved. To prove \eqref{4} note that $$s_k(T^{-1})=s_{N-k+1}^{-1}(T),\quad k=1,\ldots,N.$$ Now \eqref{4} follows from \eqref{1}. The theorem is proved. \end{proof} \section{Estimates of Linear, Gel'fand and Bernstein $n$-Widths} To obtain estimates of $n$-widths we need the following simple result. \begin{lemma}\label{L1} Let $H$ be a Hilbert space, $\omega:=\dim H$ and $T\colon H\to C(E)$ a bounded linear operator. Then $$\|T\|_{H\to C(E)}=\sup_{z\in E}\left(\sum_{k=1}^\omega|(Te_k)(z)|^2\right)^{1/2}$$ for any orthonormal basis $\{e_k\}_1^\omega$ in $H$. \end{lemma} \begin{proof} We have \begin{multline*} \|T\|_{H\to C(E)}=\sup_{h\in BH}\,\sup_{z\in E}|(Th)(z)|=\sup_{z \in E}\,\sup_{h\in BH}|(Th)(z)|\\ =\sup_{z\in E\vphantom{\}^\omega}}\,\sup_{\{c_k\}_1^\omega\in Bl_2}\Big| \sum_{k=1}^\omega c_k(Te_k)(z)\Big|=\sup_{z\in E}\left(\sum_{k=1}^\omega|(T e_k)(z)|^2\right)^{1/2}. \end{multline*} The lemma is proved. \end{proof} Let $H$ be a Hilbert space of functions defined on some set $\Omega$. A function $K(z,w)$ defined on $\Omega\times\Omega$ is called a reproducing kernel of $H$ if for each $w\in\Omega$, \ $K(z,w)\in H$, and for all $f\in H$ $$f(w)=(f(\cdot),K(\cdot,w))_H.$$ It is a well-known fact that if the $\{\varphi_k\}_1^\omega$ form an orthonormal basis in $H$, then $$K(z,w)=\sum_{k=1}^\omega\varphi_k(z)\overline{\varphi_k(w)}.$$ Suppose that $\Omega$ is a topological space, $E\subset\Omega$ and $Tf:=f_{\big|E}$ is a bounded linear operator from $H$ into $C(E)$. Then from Lemma~\ref{L1} we obtain $$\|T\|_{H\to C(E)}=\sup_{z\in E}(K(z,z))^{1/2}.$$ \begin{theorem}[\cite{5}, \cite{6}]\label{T3} Let $H$ be a Hilbert space, $E$ a topological space with probability measure $\nu$ such that $\supp\nu=E$, and $T\colon H\to C(E)$ a bounded linear operator. Define $T_0\colon H\to L_2(E,\nu)$ by the equation $T_0h:=Th$. Assume that \begin{equation}\label{5} T_0'T_0\varphi_k=s_k^2\varphi_k,\quad k=1,2,\ldots\,, \end{equation} where $s_1\ge s_2\ge\ldots>0$ and $\{\varphi_k\}$ is an orthonormal basis for the range of $T_0'T_0$. Then $$\left(\sum_{k=n+1}^\infty s_k^2\right)^{1/2}\le\lambda_n(T)=d^n(T)\le\sup _{z\in E}\left(\sum_{k=n+1}^\infty|(T\varphi_k)(z)|^2\right)^{1/2}.$$ \end{theorem} \begin{proof} Since $\supp\nu=E$, $\Ker T_0'T_0=\Ker T_0=\Ker T$ and we can assume, without loss of generality, that $\{\varphi_k\}$ is an orthonormal basis in $H$. From the definition of the Gel'fand $n$-width it follows that $$d^n(T)=\inf_{H^n}\|T\|_{H^n\to C(E)},$$ where $H^n$ runs over all $n$-codimensional subspaces of $H$. Consider $H^n=\{\varphi_k\}_{n+1}^\infty$. We obtain from Lemma~\ref{L1} $$d^n(T)\le\sup_{z\in E}\left(\sum_{k=n+1}^\infty|(T\varphi_k)(z)|^2 \right)^{1/2}.$$ Let $H^n$ be any $n$-codimensional subspace of $H$. Suppose that $\{\varphi_k'\}$ is an orthonormal basis in $H^n$. Using Lemma~\ref{L1} and \eqref{2}, we have \begin{multline*} \|T\|^2_{H^n\to C(E)}=\sup_{z\in E}\sum_{k=1}^\infty|(T\varphi_k ')(z)|^2\ge\int_E\sum_{k=1}^\infty|(T\varphi_k')(z)|^2\,d\nu(z)\\ =\sum_{k=1}^\infty\|T_0\varphi_k'\|^2_{L_2(E,\nu)}\ge\sum_{k=n+1}^\infty s_ k^2. \end{multline*} Thus, $$d^n(T)\ge\left(\sum_{k=n+1}^\infty s_k^2\right)^{1/2}.$$ The equality $\lambda_n(T)=d^n(T)$ is the well-known fact for operators defined on Hilbert spaces (see, \cite[p.~33]{3}). The theorem is proved. \end{proof} Now we will obtain the similar estimates for the Bernstein $n$-widths. \begin{theorem}\label{T4} Let $H$, $E$, and $\nu$ be as above, $H_1$ be a subspace of $L_2(E,\nu)$, and $X_E\subset H_1$ be a subspace of $C(E)$. Assume that a bounded linear operator $T_0\colon H_1\to H$ satisfies the conditions \eqref{5} where $s_1\ge s_2\ge\ldots>0$, $\{\varphi_k\}$ is an orthonormal basis for the range of $T_0'T_0$, and $\varphi_k\in X_E$, $k=1,2,\ldots\,\,$. Define $T\colon X_E\to H$ by the equation $Tf:=T_0f$. Then \begin{equation}\label{6} \left(\sup_{z\in E}\sum_{k=1}^{n+1}s_k^{-2}|\varphi_k(z)|^2\right)^{-1/2} \le b_n(T)\le\left(\sum_{k=1}^{n+1}s_k^{-2}\right)^{-1/2}. \end{equation} \end{theorem} \begin{proof} Let $L_{n+1}\subset X_E$ and $\dim L_{n+1}=n+1$. Consider the operator $T_{n+1}:=T_{\big|L_{n+1}}$. If $\Ker T_{n+1}\ne0$, then $$\inf_{\substack{f\in L_{n+1}\\f\ne0}}\frac{\|Tf\|_H}{\|f\|_{C(E)}}=0.$$ Suppose that $\Ker T_{n+1}=0$. Then we can define the operator $T_{n+1}^{-1}\colon T(L_{n+1})\to L_{n+1}$. Using Lemma~\ref{L1} and \eqref{4}, for any orthonormal system $\{e_k\}_1^{n+1}$ in $T(L_{n+1})$ we have \begin{multline*} \inf_{\substack{f\in L_{n+1}\\f\ne0}}\frac{\|Tf\|_H}{\|f\|_{C(E)}}= \inf_{\substack{g\in T(L_{n+1})\\g\ne0}}\frac{\|g\|_H}{\|T_{n+1}^{-1}g\|_{C(E)} }=\|T_{n+1}^{-1}\|^{-1}_{T(L_{n+1})\to C(E)}\\ =\left(\sup_{z\in E}\sum_{k=1}^{n+1}|(T_{n+1}^{-1}e_k)(z)|^2\right)^{-1/2} \le\left(\sum_{k=1}^{n+1}\int_E|(T_{n+1}^{-1}e_k)(z)|^2\,d\nu(z)\right)^{-1 /2}\\ =\left(\sum_{k=1}^{n+1}\|T_{n+1}^{-1}e_k\|^2_{H_1}\right)^{-1/2}\le\left( \sum_{k=1}^{n+1}s_k^{-2}(T_{n+1})\right)^{-1/2}. \end{multline*} Since $s_k(T_{n+1})\le s_k(T_0)=s_k$, $k=1,\ldots,n+1$, we obtain $$b_n(T)\le\left(\sum_{k=1}^{n+1}s_k^{-2}\right)^{-1/2}.$$ Let $L_{n+1}=\spa\{\varphi_k\}_1^{n+1}$. Then $\psi_k:=s_k^{-1}T\varphi_k$, $k=1,\ldots,n+1$, form an orthonormal system in $T(L_{n+1})$. Thus, \begin{multline*} b_n(T)\ge\inf_{\substack{f\in L_{n+1}\\f\ne0}}\frac{\|Tf\|_H}{\|f\| _{C(E)}}=\left(\sup_{z\in E}\sum_{k=1}^{n+1}|(T_{n+1}^{-1}\psi_k)(z)|^2 \right)^{-1/2}\\ =\left(\sup_{z\in E}\sum_{k=1}^{n+1}s_k^{-2}|\varphi_k(z)|^2\right)^{-1/2}. \end{multline*} The theorem is proved. \end{proof} \section{$n$-Widths of Diagonal Operators} Let $T\colon l_2\to l_\infty$ be the diagonal operator \begin{equation}\label{7} T\left(\{x_k\}_1^\infty\right):=\{\lambda_kx_k\}_1^\infty, \end{equation} where $\lambda_1\ge\lambda_2\ge\ldots>0$. Smolyak \cite{9} (in the finite-dimensional case) proved that \begin{equation}\label{8} d^n(T)=\sup_{m>n}\left(\frac{m-n}{\sum_{k=1}^m\lambda_k^{-2}}\right)^{1/2}. \end{equation} In dual terms this result was obtained by Sofman \cite{10} (see also \cite{11}). We will show that the lower bound in \eqref{8} easily follows from Theorem~\ref{T3}. Denote by $\{e_k\}_1^\infty$ the standard basis of $l_2$. Fix any $m>n$. Let $T_m\colon l_2^m\to l_\infty^m$ be the operator defined by $$T_m\left(\{x_k\}_1^m\right):=\{\lambda_kx_k\}_1^m.$$ It is easy to see that $d^n(T)\ge d^n(T_m)$. Define the probability measure $\nu_m$ on the set $\{1,2,\ldots,m\}$ as $$\nu_m(\{j\}):=\lambda_j^{-2}\left(\sum_{k=1}^m\lambda_k^{-2}\right)^{-1}.$$ Denote by $T_{m0}$ the operator $T_m$ regarded as an operator from $l_2^m$ into $l_2^m(\nu_m)$. Then $$T_{m0}'T_{m0}e_j=\left(\sum_{k=1}^m\lambda_k^{-2}\right)^{-1}e_j,\quad j=1,\ldots,m.$$ From Theorem~\ref{T3} $$d^n(T_m)\ge\left(\frac{m-n}{\sum_{k=1}^m\lambda_k^{-2}}\right)^{1/2}.$$ Thus $$\lambda_n(T)=d^n(T)\ge\sup_{m>n}d^n(T_m)\ge\left(\frac{m-n}{\sum_{k=1}^m \lambda_k^{-2}}\right)^{1/2}.$$ Note that the values \eqref{8} are also related to linear stochastic $n$-widths (see \cite{12}). Consider the operator \eqref{7} as an operator from $l_\infty$ into $l_2$. Here we assume that $\{\lambda_k\}_1^\infty\in l_2$. Galeev \cite{13} proved the equality $$b_n(T)=\min_{0\le mm,\\ \dfrac\varepsilon m,&j\le m.\end{cases}$$ Denote $T_0\colon l_2(\nu_m)\to l_2$ by the equation $T_0x:=Tx$. It is easy to obtain that $$T_0'T_0e_j=s_j^2e_j,\quad j=1,2,\ldots,$$ where $$s_j^2=\begin{cases}(1-\varepsilon)^{-1}\sum_{k=m+1}^\infty\lambda_k^2,&j>m,\\ \varepsilon^{-1}\lambda_j^2m,&j\le m.\end{cases}$$ From Theorem~\ref{T4} $$b_n(T)\le\left((1-\varepsilon)\frac{n-m+1}{\sum_{k=m+1}^\infty\lambda_k^2 }+\frac\varepsilon m\sum_{j=1}^m\lambda_j^{-2}\right)^{-1/2}.$$ Letting $\varepsilon\to0$, we obtain $$b_n(T)\le\min_{0\le m0.$$ Put $H_{n+r+1}:=H_{n+1}+H_r$. Since $A\perp H_r$ it follows that $H_{n+1}\perp H_r$ and $\dim H_{n+r+1}=n+r+1$. If $x\in H_{n+r+1}$, $\|x\|_H\le\mu$, then $x=x_1+x_2$, where $x_1\in H_{n+1}$, $x_2\in H_r$, and $$\|x_1\|_H^2\le\|x_1\|_H^2+\|x_2\|_H^2=\|x\|_H^2\le\mu^2.$$ Thus $x_1\in A$. Consequently, $x\in A+H_r$. We have $$\sup\{\,\lambda:\lambda BH_{n+r+1}\subset A+H_r\,\}\ge\mu.$$ So we proved that \begin{equation}\label{10} b_{n+r}(A+H_r,H)\ge b_n(A,H). \end{equation} If $b_{n+r}(A+H_r,H)=0$, then \eqref{9} follows from \eqref{10}. Suppose that $b_{n+r}(A+H_r,H)>0$. Let $H_{n+r+1}\subset H$, $\dim H_{n+r+1}=n+r+1$, and $$\sup\{\,\lambda:\lambda BH_{n+r+1}\subset A+H_r\,\}=\mu>0.$$ Since $H_{n+r+1}\subset\spa A+H_r$, \ $\dim(H_{n+r+1}\cap\spa A)\ge n+1$. Hence there exists a subspace $H_{n+1}\subset H_{n+r+1}\cap\spa A$ with $\dim H_{n+1}=n+1$. Let $x\in H_{n+1}$ and $\|x\|_H\le\mu$. Then $x\in A+H_r$. In addition $x\in\spa A$. Thus $x\in A$ and $$\sup\{\,\lambda:\lambda BH_{n+1}\subset A\,\}\ge\mu.$$ Therefore, $$b_n(A,H)\ge b_{n+r}(A+H_r,H).$$ The lemma is proved. \end{proof} \begin{theorem}\label{T5} Let $H$, $E$, $\nu$, $H_1$, and $X_E$ be as in Theorem~\ref{T4}. Suppose that the Hilbert space $H_1$ has a reproducing kernel. Let $\{\varphi_k\}$ be an orthonormal basis in $H_1$ and $T_0\colon H_1\to H$ a bounded linear operator. Assume that $\{T_0\varphi_k\}$ is an orthogonal system in $H$, $\varphi_k\in X_E$, $k=1,2,\ldots\,\,$ and $s_k:=\|T_0\varphi_k\|_H$ form a non-increasing sequence. Define $T\colon X_E\to H$ by the equation $Tf:=T_0f$. Then the inequalities \eqref{6} hold. \end{theorem} \begin{proof} Denote by $K(z,w)$ the reproducing kernel of $H_1$. Since $\{\varphi_k\}$ is an orthonormal basis in $H_1$ the representation $$K(z,w)=\sum_{k=1}^\infty\varphi_k(z)\overline{\varphi_k(w)}$$ holds. We have $$(T_0'T_0f)(w)=\bigl((T_0'T_0f)(\cdot),K(\cdot,w)\bigr)_{H_1}=\bigl((T_0f) (\cdot),(T_0K)(\cdot,w)\bigr)_H.$$ Thus $$(T_0'T_0\varphi_j)(w)=\bigl((T_0\varphi_j)(\cdot),\sum_{k=1}^\infty \overline{\varphi_k(w)}(T_0\varphi_k)(\cdot)\bigr)_H=s_j^2\varphi_j(w).$$ Now it suffices to apply Theorem~\ref{T4}. The theorem is proved. \end{proof} Let $B_\rho$ be the ball of $\mathbb C^n$ of radius $\rho$ $$B_\rho:=\{\,z:=(z_1,\ldots,z_n)\in\mathbb C^n:|z|^2:=\sum_{k=1}^n|z_k|^2<\rho \,\},$$ $S_\rho:=\partial B_\rho$, $\sigma_\rho$ the probability measure on the sphere $S_\rho$ which is invariant with respect to the orthogonal group $O(2n)$, and $\nu_\rho$ the normalized Lebesgue measure in $B_\rho$ (if $\rho=1$ we will write $B$, $S$, $\sigma$, and $\nu$). The Hardy space $H_p(B)$ ($H_p$) is the set of holomorphic functions in $B$ which satisfy \begin{gather*} \|f\|_{H_p}:=\sup_{0<\rho<1}\left(\int_S|f(z)|^p\,d\sigma(z)\right)^{1/p}<\infty,\quad1\le p<\infty,\\ \|f\|_{H_\infty}:=\sup_{z\in B}|f(z)|. \end{gather*} Let $f(z)$ be a holomorphic function in $B$ and let $$f(z)=\sum_{k=0}^\infty F_k(z)$$ be a homogeneous decomposition of $f$. The radial derivative of order $r$ is defined by $$\mathcal R^rf(z):=\sum_{k=r}^\infty\frac{k!}{(k-r)!}F_k(z)$$ (for $r=1$ see \cite[Chap.~6]{14}). Denote by $H\mathcal R^r_\infty(B)$ ($H\mathcal R^r_\infty$) the class of holomorphic functions in $B$ for which $\mathcal R^rf\in BH_\infty$. Set $$N_m:=\sum_{k=0}^{m-1}\binom{n+k-1}{n-1}.$$ Note that $N_m=\dim\mathcal P_{m-1}^n$, where $\mathcal P_m^n$ is the space of $n$-variable polynomials of degree $m$ or less. \begin{theorem}\label{T6} For all $0<\rho\le1$ and all $m\ge r+1$ \begin{align}\label{11} b_{N_m-1}\bigl(H\mathcal R_\infty^r,L_2(S_\rho,\sigma_\rho)\bigr)&= \left(\frac1{(n-1)!}\sum_{k=r}^{m-1}\frac{k!(n+k-1)!}{((k-r)!)^2}\rho^{-2k} \right)^{-1/2},\\ b_{N_m-1}\bigl(H\mathcal R_\infty^r,L_2(B_\rho,\nu_\rho)\bigr)&=\left(\frac1{n! }\sum_{k=r}^{m-1}\frac{k!(n+k)!}{((k-r)!)^2}\rho^{-2k}\right)^{-1/2}.\label{12} \end{align} \end{theorem} \begin{proof} For multiindex $\alpha:=(\alpha_1,\ldots,\alpha_n)$ and $z\in\mathbb C^n$ set \begin{gather*} z^\alpha:=z_1^{\alpha_1}\ldots z_n^{\alpha_n},\quad|\alpha|:= \alpha_1+\ldots+\alpha_n,\quad\alpha!:=\alpha_1!\ldots\alpha_n!,\\ D_j:=\frac\partial{\partial z_j},\quad D^\alpha:=D_1^{\alpha_1}\ldots D_n^{\alpha_n}. \end{gather*} Denote by $H_p^0$ the space of all functions $f\in H_p$ for which $(D^\alpha f)(0)=0$, $|\alpha|=0,\ldots,r-1$. It is known (see \cite{14}) that functions from $H_p$, $1\le p\le\infty$, have finite boundary values almost everywhere. Moreover, $H_2$ is a Hilbert space with the inner product $$(f,g)_{H_2}:=\int_Sf(z)\overline{g(z)}\,d\sigma(z).$$ The space $H_2$ has the reproducing kernel $$K(z,w)=\left(1-\sum_{k=1}^nz_k\overline w_k\right)^{-n}.$$ Define $T_0\colon H_2^0\to L_2(S_\rho,\sigma_\rho)$ and $T\colon H_\infty^0\to L_2(S_\rho,\sigma_\rho)$ by the equations \begin{equation}\label{13} (T_0f)(z):=\sum_{k=r}^\infty\frac{(k-r)!}{k!}F_k(z),\quad Tf:=T_0f, \end{equation} where $$f(z)=\sum_{k=r}^\infty F_k(z).$$ It is easy to see that $$H\mathcal R_\infty^r=T(BH_\infty^0)+\mathcal P_{r-1}^n.$$ Since monomials $z^\alpha$ are orthogonal in $L_2(S_\rho,\sigma_\rho)$ we obtain from Lemma~\ref{L2} \begin{equation}\label{14} b_{N_m-1}\bigl(H\mathcal R_\infty^r,L_2(S_\rho,\sigma_\rho)\bigr)=b_{N_m-N_r-1}(T). \end{equation} For every $0<\rho\le1$ (see \cite{14}) $$\|z^\alpha\|_{L_2(S_\rho,\sigma_\rho)}^2=\frac{(n-1)!\alpha!}{(n+|\alpha| -1)!}\rho^{2|\alpha|}.$$ Thus the functions $$\varphi_\alpha(z):=\left(\frac{(n+|\alpha|-1)!}{(n-1)!\alpha!}\right)^{1/ 2}z^\alpha,\quad|\alpha|\ge r,$$ form a complete orthonormal basis in $H_2^0$. We have $$\|T_0\varphi_\alpha\|^2_{L_2(S_\rho,\sigma_\rho)}=\left(\frac{(|\alpha|-r )!}{|\alpha|!}\right)^2\rho^{2|\alpha|}.$$ The number of different monomials $z^\alpha$ with $|\alpha|=k$ equals $\dbinom{n+k-1}{n-1}$. By Theorem~\ref{T5} \begin{multline*} \left(\sup_{z\in S}\sum_{|\alpha|=r}^{m-1}\left(\frac{|\alpha|!} {(|\alpha|-r)!}\right)^2\rho^{-2|\alpha|}|\varphi_\alpha(z)|^2\right)^{-1/2 }\le b_{N_m-N_r-1}(T)\\ \le\left(\sum_{k=r}^{m-1}\left(\frac{k!}{(k-r)!}\right)^2\binom{n+k-1}{n-1} \rho^{-2k}\right)^{-1/2}. \end{multline*} Using the equation $$\sum_{|\alpha|=k}\frac{|z^{2\alpha}|}{\alpha!}=\frac{|z|^{2k}}{k!},$$ we obtain $$b_{N_m-N_r-1}(T)=\left(\frac1{(n-1)!}\sum_{k=r}^{m-1}\frac{k!(n+k-1)!}{(( k-r)!)^2}\rho^{-2k}\right)^{-1/2}.$$ Now \eqref{11} follows from \eqref{14}. The proof of \eqref{12} is almost the same. The difference is that we have to consider the operators $T_0\colon H_2^0\to L_2(B_\rho,\nu_\rho)$ and $T\colon H_ \infty^0\to L_2(B_\rho,\nu_\rho)$ defined by \eqref{13}. Then we use $$\|T_0\varphi_\alpha\|^2_{L_2(B_\rho,\nu_\rho)}=\left(\frac{(|\alpha|-r)!} {|\alpha|!}\right)^2\frac n{n+|\alpha|}\rho^{2|\alpha|}.$$ The theorem is proved. \end{proof} For $n=1$ the class $H\mathcal R_\infty^r$ coincides with the class $BH_\infty^ r$ defined as the set of all holomorphic functions in $B$ for which $f^{(r) }(z)\in BH_\infty$. From Theorem~\ref{T6} we obtain the following result. \begin{corollary}\label{C1} For all $0<\rho\le1$ and all $m\ge r$ \begin{align*} b_m\bigl(BH_\infty^r,L_2(S_\rho,\sigma_\rho)\bigr)&=\left(\sum_{k= r}^m\left(\frac{k!}{(k-r)!}\right)^2\rho^{-2k}\right)^{-1/2},\\ b_m\bigl(BH_\infty^r,L_2(B_\rho,\nu_\rho)\bigr)&=\left(\sum_{k=r}^m\frac{k! (k+1)!}{((k-r)!)^2}\rho^{-2k}\right)^{-1/2}. \end{align*} \end{corollary} In particular, we have for $r=0$, $m\ge0$, and $0<\rho<1$ \begin{align}\label{15} b_m\bigl(BH_\infty,L_2(S_\rho,\sigma_\rho)\bigr)&=\rho^m\left( \frac{1-\rho^2}{1-\rho^{2m+2}}\right)^{1/2},\\ b_m\bigl(BH_\infty,L_2(B_\rho,\nu_\rho)\bigr)&=\rho^m \frac{1-\rho^2}{\sqrt{(m+1)(1-\rho^2)-\rho^2(1-\rho^{2m+2})}}.\label{16} \end{align} The values \eqref{11} for $r=0$ and \eqref{15} were calculated in \cite{7}. Let us compare \eqref{15} and \eqref{16} with the exact values of the Kolmogorov, linear, and Gel'fand $n$-widths. From \cite{15} and \cite{16} it follows that \begin{multline*} d_m(BH_\infty,X)=\lambda_m(BH_\infty,X)=d^m(BH_\infty,X)\\ =\begin{cases}\rho^m,&X=L_2(S_\rho,\sigma_\rho),\\ \dfrac{\rho^m}{\sqrt{m+1}},&X=L_2(B_\rho,\nu_\rho).\end{cases} \end{multline*} Finally, we will determine exact values of the Bernstein $n$-widths for some classes of periodic holomorphic functions. Let $D_\beta:=\{\,z\in\mathbb C:|\IM z|<\beta\,\}$. Denote by $\widetilde H_{p,\beta}$ the set of all $2\pi$-periodic holomorphic functions in $D_\beta$ which satisfy the conditions \begin{gather*} \|f\|_{\widetilde H_{p,\beta}}:=\sup_{0