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\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
$$b_n(T)=\min_{0\le m<n+1}\left(\frac{\sum_{k=m+1}^\infty\lambda_k^2}{n-m+1
}\right)^{1/2}.$$
We will show how the upper bound can be obtained from Theorem~\ref{4}.

Let $0<\varepsilon<1$ and $0\le m<n+1$. Define the probability measure $\nu_m $ on $\mathbb N$ as
$$\nu_m(\{j\}):=\begin{cases}(1-\varepsilon)\dfrac{\lambda_j^2}{\sum_{k=m+1}^
\infty\lambda_k^2},&j>m,\\
\dfrac\varepsilon m,&j\le m.\end{cases}$$
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
$$b_n(T)\le\min_{0\le m<n+1}\left(\frac{\sum_{k=m+1}^\infty\lambda_k^2}{n-m
+1}\right)^{1/2}.$$

\section{Bernstein $n$-Widths of Hardy--Sobolev Classes}

Let $A$ be a closed, convex, centrally symmetric subset of a normed linear
space $Y$. The Bernstein $n$-width of $A$ is defined by
$$b_n(A,Y):=\sup_{Y_{n+1}}\sup\{\,\lambda:\lambda BY_{n+1}\subset A\,\},$$
where $Y_{n+1}$ is any $(n+1)$-dimensional subspace of $Y$. If $\Ker T=0$, then it is easily shown that
$$b_n(T)=b_n(T(BX),Y).$$

We need the following simple property of $b_n$.

\begin{lemma}\label{L2}
Let $H$ be a Hilbert space, $A$ a closed, convex, centrally symmetric subset of $H$, and $H_r$ an $r$-dimensional subspace of $H$ such that $A\perp H_r$. Then
\begin{equation}\label{9}
b_{n+r}(A+H_r,H)=b_n(A,H).
\end{equation}
\end{lemma}

\begin{proof}
Assume that $H_{n+1}\subset H$, $\dim H_{n+1}=n+1$, and
$$\sup\{\,\lambda:\lambda BH_{n+1}\subset A\,\}=\mu>0.$$
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<h<\beta}\left(\frac1{4\pi}
\int_0^{2\pi}\left(|f(x+ih)|^p+|f(x-ih)|^p\right)\,dx\right)^{1/p}<\infty,\\
\hspace{290pt}1\le p<\infty,\\
\|f\|_{\widetilde H_{\infty,\beta}}:=\sup_{z\in D_\beta}|f(z)|<\infty.
\end{gather*}
Let $B\widetilde H_{p,\beta}^r$ be the set of all $2\pi$-periodic holomorphic functions in $D_\beta$ for which $f^{(r)}(z)\in B\widetilde H_{p,\beta}$. Denote by $L_2$ the periodic complex-valued Lebesgue space on the real axis with the norm
$$\|f\|_{L_2}:=\left(\frac1{2\pi}\int_0^{2\pi}|f(x)|^2\,dx\right)^{1/2}.$$
\newpage
\begin{theorem}\label{T7}
\begin{enumerate}
\item For all $n\ge1$ and $r\ge1$
$$b_{2n}(B\widetilde H_{\infty,\beta}^r,L_2)=\left(2\sum_{k=1}^nk^{2r}\cosh
2k\beta\right)^{-1/2}.$$
\item For all $n\ge0$
\begin{equation}\label{17}
b_{2n}(B\widetilde H_{\infty,\beta},L_2)=\left(\frac{\sinh\beta}{\sinh(2n
+1)\beta}\right)^{1/2}.
\end{equation}
\end{enumerate}
\end{theorem}

\begin{proof}
The space $\widetilde H_{2,\beta}$ is the Hilbert space with the inner product
$$(f,g)_{\widetilde H_{2,\beta}}:=\frac1{4\pi}\int_0^{2\pi}\left(f(x+i\beta
)\overline{g(x+i\beta)}+f(x-i\beta)\overline{g(x-i\beta)}\right)\,dx.$$
The functions
$$\varphi_k(z):=\frac{e^{ikz}}{\sqrt{\cosh2k\beta}},\quad k\in\mathbb Z,$$
form an orthonormal basis in $\widetilde H_{2,\beta}$. The space $\widetilde H_{2,\beta}$ has the reproducing kernel
$$K(z,w)=\sum_{k\in\mathbb Z}\varphi_k(z)\overline{\varphi_k(w)}=1+2\sum_{k=1}
^\infty\frac{\cos k(z-\overline w)}{\cosh2k\beta}.$$
Let $r\ge1$. Denote by $\widetilde H_{p,\beta}^0$ the space of functions $f\in\widetilde H_{p,\beta}$ for which
$$\int_0^{2\pi}f(x)\,dx=0.$$
Define $T_0\colon\widetilde H_{2,\beta}^0\to L_2$ and $T\colon\widetilde H_{\infty,\beta}^0\to L_2$ by the equations
$$(T_0f)(z):=\sum_{\substack{k\in\mathbb Z\\k\ne0}}\frac{c_k}{(ik)^r}e^{ikz},\quad Tf:=T_0f,$$
where
$$f(z)=\sum_{\substack{k\in\mathbb Z\\k\ne0}}c_ke^{ikz}.$$
It is easily seen that
$$B\widetilde H_{\infty,\beta}^r=T(B\widetilde H_{\infty,\beta}^0)+\mathbb C.$$
By Lemma~\ref{L2} we obtain
$$b_{2n}(B\widetilde H_{\infty,\beta}^r,L_2)=b_{2n-1}(T).$$
The functions $\varphi_k(z)$, $k=\pm1,\pm2,\ldots$, form a complete orthonormal basis in $\widetilde H_{2,\beta}^0$ and
$$\|T_0\varphi_k\|_{L_2}^2=\frac1{k^{2r}\cosh2k\beta}.$$
Since for all $z\in\partial D_\beta$
$$|\varphi_k(z)|^2+|\varphi_{-k}(z)|^2=2$$
we have by Theorem~\ref{T5}
$$b_{2n-1}(T)=\left(2\sum_{k=1}^nk^{2r}\cosh2k\beta\right)^{-1/2}.$$

To obtain \eqref{17} we use the same scheme and the equality
$$1+2\sum_{k=1}^n\cosh2k\beta=\frac{\sinh(2n+1)\beta}{\sinh\beta}.$$
The theorem is proved.
\end{proof}

Denote by $B\widetilde H_{\infty,\beta}^{\mathbb R}$ the set of functions from
$B\widetilde H_{\infty,\beta}$ that are real-valued on $\mathbb R$. For even $n$ the exact values of the Kolmogorov, linear, and Gel'fand $n$-widths of $B\widetilde H_{\infty,\beta}^{\mathbb R}$ in $L_q$, $1\le q\le\infty$, were determined in \cite{17}. In particular, for $q=2$
\begin{multline*}
d_{2n}(B\widetilde H_{\infty,\beta}^{\mathbb R},L_2)=\lambda_{2n}(
B\widetilde H_{\infty,\beta}^{\mathbb R},L_2)=d^{2n}(B\widetilde H_{\infty,
\beta}^{\mathbb R},L_2)\\
=\left(\frac\lambda\Lambda\int_0^1\frac{t^2\,dt}{\sqrt{(1-t^2)(1-\lambda^2t
^2)}}\right)^{1/2}=\sqrt2e^{-\beta n}+O(e^{-5\beta n}),
\end{multline*}
where $\Lambda$ is the complete elliptic integral of the first kind with modulus
$$\lambda=4e^{-2\beta n}\left(\sum_{k=0}^\infty e^{-4\beta nk(k+1)}\right)^
2\left(1+2\sum_{k=1}^\infty e^{-4\beta nk^2}\right)^{-2}.$$
By Theorem~\ref{T5} it can be shown that \eqref{17} also holds in the real case
for the class $B\widetilde H_{\infty,\beta}^{\mathbb R}$. Thus,
$$b_{2n}(B\widetilde H_{\infty,\beta}^{\mathbb R},L_2)=\sqrt{1-e^{-2\beta}}e^{
-\beta n}+O(e^{-5\beta n}).$$


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\bigskip

\sc Department of Mathematics, Moscow State University of Aviation
Technology, Moscow, Russia, 103767

\bigskip

Department of Mathematics, Pskov Pedagogical Institute, Pskov,
Russia, 180000
\end{document}
