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\begin{document}

\title[$N$-Widths of Holomorphic Functions]{On $N$-Widths of Holomorphic Functions of Several Variables}
\author{K. Yu.\ Osipenko}
\address{Department of Mathematics, Moscow Institute of Aviation Tech, Moscow 103767, Petrovka 27, Russia}

\begin{abstract}
We consider the classes of holomorphic functions whose radial derivative of order $r$ lies in the unit ball of the Hardy space $H_2(B_n)$ or the Bergman space $A_2(B_n)$. For these classes we calculate the linear and Gel'fand $N$-widths in $C(S_\rho)$, where $S_\rho$ is the sphere in $\mathbb C^n$ of radius $0<\rho<1$. Some results arc obtained for analogous problems in polydiscs and for $2\pi$-periodic functions of one variable holomorphic in a strip.
\end{abstract}


\maketitle


\section*{Introduction}

Let $A$ be a subset of a normed linear space $X$. The Kolmogorov $N$-width is defined by
$$d_N(A,X):=\infp_{X_N}\sup_{x\in A}\sup_{y\in X_N}\|x-y\|,$$
where $X_N$ runs over all $N$-dimensional subspaces of $X$. Denote by $\mathcal L(H,X)$ the class of all continuous linear operators from $H$ to $X$, where $H$ and $X$ are normed linear spaces. Let $BH$ be the closed unit ball of $H$. For $T\in\mathcal L(H,X)$ set
$$d_N(T):=d_N(T(BH),X).$$

The linear $N$-width is given by
$$\lambda_N(A,X):=\infp_{P_N}\sup_{x\in A}\|x-P_Nx\|,$$
where $P_N$ runs over all bounded linear operators mapping $X$ into $X$,
whose range has dimension $N$ or less. Assume that $0\in A$. The Gel'fand $N$-width is defined by
$$d^N(A,X):=\infp_{X_N}\sup_{x\in A\cap X^N}\|x\|,$$
where the infimum is taken over all subspaces $X^N$ of $X$ of codimension
$N$. Various properties of these $N$-widths (and others) may be found in \cite{1}.

Let $B_n$ be the unit ball of $\mathbb C^n$
$$B_n:=\biggl\{\,z:=(z_1,\ldots,z_n)\in\mathbb C^n:|z|^2:=\sum_{k=1}^n|z_k|^2<1\,\biggr\},$$
and $S_\rho$ the sphere of radius $\rho$
$$S_\rho:=\left\{\,z\in\mathbb C^n:|z|=\rho\,\right\}$$
(if $\rho=1$ we write $S$). The Hardy space $H_p(B_n)$ is the set of holomorphic functions in $B_n$ which satisfy
\begin{gather*}
\|f\|_{H_p(B_n)}:=\sup_{0<r<1}\biggl(\int_S|f(rz)|^p\,d\sigma(z)\biggr)^{1/p}<\infty,\quad1\le p<\infty,\\
\|f\|_{H_\infty(B_n)}:=\sup_{z\in B_n}|f(z)|,
\end{gather*}
where $\sigma$ is the probability measure on the sphere $S$ which is invariant with respect to the orthogonal group $O(2n)$. The Bergman space $A_p(B_n)$ is the set of holomorphic functions in $B_n$ which satisfy the condition
$$\|f\|_{A_p(B_n)}:=\biggl(\int_{B_n}|f(z)|^p\,d\nu(z)\biggr)^{1/p}<\infty,$$
where $\nu$ is the normalized Lebesgue measure in $B_n$ ($A_\infty(B_n)=H_\infty(B_n)$).

Let $f(z)$ be a holomorphic function in $B_n$ and
$$f(z)=\sum_{s=0}^\infty F_s(z)$$
be a homogeneous decomposition of $f$. The radial derivative of order $r$ is
defined by
$$\mathcal R^rf(z):=\sum_{s=r}^\infty\frac{s!}{(s-r)!} F_s(z)$$
(for $r=1$ see \cite[Chap.~6]{2}). Let $BX$ be the closed unit ball of a normed linear space $X$. We denote by $H\mathcal R_p^r(B_n)$ and $A\mathcal R_p^r(B_n)$ the classes of holomorphic functions in $B_n$ for which $\mathcal R^rf$ lie in $BH_p(B_n)$ and $BA_p(B_n)$, respectively.

The exact values of $d_N(H\mathcal R_p^r(B_n),L_p(S_\rho))$ were obtained in \cite{3}. When
$n=1$, $1\le q\le p\le\infty$ and $E$ is a compact subset of $B_1$, the values of
$d_N(BH_p(B_1),L_q(E))$ were determined in \cite{4} (for $E=S_\rho$ see also \cite{5}).

The first result for the classes of holomorphic functions concerning the case when $p<q$ appeared in \cite{6} where the values of $d^N(BH_2(B_n),C(S_\rho))$ and $\lambda_N(BH_2(B_n),C(S_\rho))$ were obtained (more precisely, for some subsequence of $\mathbb N$). The method of proof, as noted by V.~M.~Tikhomirov, was very similar to the one used in Ismagilov's Theorem \cite{7} (see also \cite{1}). In Section 1 we prove a theorem dual to the Ismagilov Theorem. Using this result, in Section 2 we obtain the values of the linear and Gel'fand $N$-widths of the classes $H\mathcal R_2^r(B_n)$ and $A\mathcal R_2^r(B_n)$ in $C(S_\rho)$.

Section 3 is devoted to analogous problems in polydiscs. Finally in
Section 4 we calculate the $N$-widths of holomorphic functions in the annulus
$$\Delta_R:=\{\,z\in\mathbb C:R^{-1}<|z|<R\,\},\quad R>1,$$
and $2\pi$-periodic functions holomorphic in the strip
$$D_H:=\{\,z\in\mathbb C:|\IM z|<H\,\}.$$

\section{A Theorem Dual to Ismagilov's Theorem}

Let $E$ be a compact set, $\mu$ a positive probability measure defined on $E$ and $T\in\mathcal L(H,C(E))$. Denote by $T_0$ the operator $T$ regarded as an operator from $H$ into $L_2(E,\mu)$. Assume that
$$T_0'T_0\phi_j=\lambda_j\phi_j,\quad j=1,2.\ldots,$$
where $\lambda_1\ge\lambda_2\ge\ldots>0$, and that $\phi_1,\phi_2,\ldots$ is a complete orthonormal basis for the range of $T_0'T_0$ (a sufficient condition is that $T_0$ be a compact operator).

\begin{theorem}\label{T1}
For $T$ as above
\begin{multline*}
\sqrt{\sum_{j=N+1}^\infty\lambda_j}\le d^N(T(BH),C(E))=\lambda_N(T(BH),C(E))\\
\le\sup_{z\in E}\sqrt{\sum_{j=N+1}^\infty|(T\phi_j)(z)|^2}.
\end{multline*}
\end{theorem}

\begin{proof}
Since $\Ker T_0'T_0=\Ker T_0=\Ker T$ we shall assume, without loss of generality, that $\phi_1,\phi_2,\ldots$ is a complete orthonormal basis for $H$. Set $\psi_j:=T\phi_j$. Let us show that for all $z\in E$
\begin{equation}\label{1}
\sum_{j=1}^\infty|\psi_j(z)|^2\le\|T\|^2:=\biggl(\sup_{\|h\|_H\le1}\|Th\|_\infty\biggr)^2
\end{equation}
(we denote by $\|\cdot\|_\infty$ the norm in $C(E)$ and by $\|\cdot\|_H$ the norm in $H$). Let
$z\in E$ and $m\in\mathbb N$. Then for $h:=\sum_{j=1}^m\overline{\psi_j(z)}\phi_j\in H$ we have
$$\|Th\|_\infty=\sup_{z\in E}\biggl|\sum_{j=1}^m\overline{\psi_j(z)}\psi_j(s)
\biggr|\ge\sum_{j=1}^m|\psi_j(z)|^2=\|h\|_H^2.$$
Thus for $h\ne0$
$$\|h\|_H\le\frac{\|Th\|_\infty}{\|h\|_H}\le\|T\|.$$
Consequently for all $z\in E$ and all $m\in\mathbb N$ the inequality
$$\sum_{j=1}^m|\psi_j(z)|^2\le\|T\|$$
holds. So \eqref{1} is proved.

Set
$$h_z:=\sum_{j=1}^\infty\overline{\psi_j(z)}\psi_j.$$
It is easy to check that for all $x\in H$ and all $z\in E$
$$(Tx)(z)=(x,h_z)_H.$$
Denote by $\varphi:E\to H$ the mapping
$$\varphi(z):=h_z.$$
Then
\begin{multline*}
\int_E(\varphi(z),\varphi(y))_H\overline{\psi_j(y)}\,d\mu(y)=
\int_E(Th_z)(y)\overline{\psi_j(y)}\,d\mu(y)\\
=(T_0h_z,T_0\psi_j)_{L_2(E,\mu)}=(h_z,T_0'T_0\psi_j)_H=\lambda_j\overline{\psi_j(z)}.
\end{multline*}
Furthermore
$$(\psi_j,\psi_k)_{L_2(E,\mu)}=\lambda_j\delta_{jk}.$$
By the Ismagilov Theorem we obtain
$$\sqrt{\sum_{j=N+1}^\infty\lambda_j}\le d_N(T')\le\sup_{z\in E}\sqrt{\sum_{j=N+1}^\infty|(T\phi_j)(z)|^2}.$$
From duality
$$d_N(T')=d^N(T):=\infp_{X^N}\sup_{h\in BH\cap X^N}\|Th\|_\infty,$$
where the infimum is taken over all subspaces $X^N$ of $H$ of codimension
$N$. Since $H$ is a Hilbert space
$$d_N(T')=d^N(T(BH),C(E))=\lambda_N(T(BH),C(E)).$$
The theorem is proved.
\end{proof}

\begin{corollary}\label{C1}
Assume that the conditions of Theorem~\ref{T1} hold and $X_r$ is any $r$-dimensional subspace of $C(E)$ such that $X_r\perp T_0(H)$ in $L_2(E,\mu)$. Then
\begin{multline*}
\sqrt{\sum_{j=N+1}^\infty\lambda_j}\le d^{N+r}(T(BH)+X_r,C(E))=\lambda_{N+r}(T(BH)+X_r,C(E))\\
\le\sup_{z\in E}\sqrt{\sum_{j=N+1}^\infty|(T\phi_j)(z)|^2}.
\end{multline*}
\end{corollary}

\begin{proof}
Let $e_1,\ldots,e_r$ be an orthonormal basis for $X_r$ in $L_2(E,\mu)$. Denote by $H_{r,\varepsilon}$ the Hilbert space of elements $\{f,g\}$, $f\in H$, $g\in X_r$ with inner product
$$(\{f_1,g_1\},\{f_2,g_2\})_{H_{r,\varepsilon}}:=(f_1,f_2)_H+
\varepsilon\sum_{j=1}^rc_j\overline d_j,\quad\varepsilon>0,$$
where
$$g_1=\sum_{j=1}^rc_je_j,\quad g_2=\sum_{j=1}^rd_je_j.$$
Put $L(f,g):=Tf+g$. Denote by $L_0$ the operator $L$ as an operator from $H_{r,\varepsilon}$	 into $L_2(E,\mu)$. Then
$$L_0'L_0\{f,g\}=\{T_0'T_0f,\varepsilon^{-1}g\}.$$
Set
$$\varphi_j:=\{0,\varepsilon^{-1/2}e_j\},\quad j=1,\ldots,r,\quad\varphi_j:=\{\phi_{j-r},0\},
\quad j=r+1,\ldots\,.$$
The elements $\varphi_1,\varphi_2,\ldots$ form a complete orthonormal basis for the range of $L_0'L_0$ and
$$L_0'L_0\varphi_j=\varepsilon^{-1}\varphi_j,\quad j=1,\ldots,r,\quad L_0'L_0\varphi_j=\lambda_{j-r}\varphi_j,\quad j=r+1,\ldots\,.$$
From Theorem~\ref{T1} for $\varepsilon\le\lambda_1^{-1}$ we have
$$d^{N+r}(L(BH_{r,\varepsilon}),C(E))\ge\sqrt{\sum_{j=N+1}^\infty\lambda_j}.$$
Since $T(BH)+X_r\subset L(BH_{r,\varepsilon})$
$$d^{N+r}(T(BH)+X_r,C(E))\ge d^{N+r}(L(BH_{r,\varepsilon}),C(E))\ge\sqrt{\sum_{j=N+1}^\infty\lambda_j}.$$

The equality
$$d^{N+r}(T(BH)+X_r,C(E))=\lambda_{N+r}(T(BH)+X_r,C(E))$$
follows from the fact that H is a Hilbert space (compare with Proposition~8.8 \cite[p.~33]{1}). It is easy to show that
$$\lambda_{N+r}(T(BH)+X_r,C(E))\le\lambda_N(T(BH),C(E)).$$
Now the upper bound follows directly from Theorem~\ref{T1}. The corollary 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 $w\in\Omega$ and for all $f\in H$
$$f(w)=(f(\cdot),K(\cdot,w))_H.$$
It is easily seen that
$$K(z,w)=\overline{K(w,z)}.$$

Let $E\subset\Omega$ be a compact with positive probability measure $\mu$. Suppose that $Tf:=f_{|E}$ is a bounded linear operator from $H$ to $C(E)$.

\begin{theorem}\label{T2}
Let $H$ and $E$ be as above. Assume that $\varphi_1,\varphi_2,\ldots$ is a complete orthonormal basis for $H$ and $X_r$ is any $r$-dimensional subspace of $C(E)$ such that $X_r\perp H$ in $L_2(E,\mu)$. If $\varphi_1,\varphi_2,\ldots$ is an orthogonal system in $L_2(E,\mu)$ and $\lambda_j:=\|\varphi_j\|^2_{L_2(E,\mu)}$ form a non-increasing sequence, then
\begin{multline*}
\sqrt{\sum_{j=N+1}^\infty\lambda_j}\le d^{N+r}(BH+X_r,C(E))=\lambda_{N+r}(BH+X_r,C(E))\\
\le\sup_{z\in E}\sqrt{\sum_{j=N+1}^\infty|\varphi_j(z)|^2}.
\end{multline*}
\end{theorem}

\begin{proof}
Put $T_0f:=f_{|E}$. Let us consider $T_0$ as an operator from $H$ into $L_2(E,\mu)$. For all $g\in L_2(E,\mu)$ we have
\begin{multline*}
(T_0'g)(w)=\left((T_0'g)(\cdot),K(\cdot,w)\right)_H=\left(g(\cdot),T_0K(\cdot,w)
\right)_{L_2(E,\mu)}\\
=\int_Eg(z)\overline{K(z,w)}\,d\mu(z)=\int_EK(w,z)g(z)\,d\mu(z).
\end{multline*}
Thus the eigenvalue-eigenfunction problem
$$T_0'T_0f=\lambda f$$
takes the form
\begin{equation}\label{2}
\int_EK(w,z)g(z)\,d\mu(z)=\lambda f(w).
\end{equation}
Since $\varphi_1,\varphi_2,\ldots$ is a complete orthonormal basis for $H$ the representation
$$K(z.w)=\sum_{j=1}^\infty\varphi_j(z)\overline{\varphi_j(w)}$$
holds. In view of the orthogonality of the system $\varphi_1,\varphi_2,\ldots$ in $L_2(E,\mu)$ we have
$$\int_EK(w,z)\varphi_j(z)\,d\mu(z)=\lambda_j\varphi_j(w).$$
Thus $\lambda_j$ is an eigenvalue and $\varphi_j$ is an eigenfunction for Eg.~\eqref{2}. Now the theorem follows from Corollary~\ref{C1}.
\end{proof}

\section{$N$-Widths of $H\mathcal R_2^r(B_n)$ and $A\mathcal R_2^r(B_n)$}

Set $N_m:=\displaystyle\sum_{s=0}^{m-1}\binom{n+s-1}{n-1}$. 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.

\newpage

\begin{theorem}\label{T3}
\
\begin{itemize}
\item[(i)] For all $0<\rho<1$ and all $m\ge	r\ge0$
\begin{multline*}
d^{N_m}\left(H\mathcal R_2^r(B_n),C(S_\rho)\right)=\lambda_{N_m}\left(H\mathcal R_2^r(B_n),C(S_\rho)\right)\\
=\rho^m\biggl(\frac1{(n-1)!}\sum_{s=0}^\infty\frac{((m-r+s)!)^2(n+m-1+s)!}
{((m+s)!)^3}\rho^{2s}\biggr)^{1/2}.
\end{multline*}
\item[(ii)] For all $0<\rho<1$ and all $m\ge r\ge1$
\begin{multline*}
d^{N_m}\left(A\mathcal R_2^r(B_n),C(S_\rho)\right)=\lambda_{N_m}\left(A\mathcal R_2^r(B_n),C(S_\rho)\right)\\
=\rho^m\biggl(\frac1{n!}\sum_{s=0}^\infty\frac{((m-r+s)!)^2(n+m+s)!}
{((m+s)!)^3}\rho^{2s}\biggr)^{1/2}.
\end{multline*}
\item[(iii)] For all
$$0<\rho\le\left(\frac n{n+m}\right)^{1/(2m)}$$
\begin{multline}\label{3}
d^{N_m}\left(BA_2(B_n),C(S_\rho)\right)=\lambda_{N_m}\left(BA_2(B_n),C(S_\rho)\right)\\
=\rho^m\biggl(\frac1{n!}\sum_{s=0}^\infty\frac{(n+m+s)!}{(m+s)!}\rho^{2s}\biggr)^{1/2}\\
=\frac{\rho^m}{(1-\rho^2)^{(n+1)/2}}\biggl(\binom{n+m}n\sum_{s=0}^n\frac{(-1)^s}{1+s/m}
\binom ns\rho^{2s}\biggr)^{1/2}.
\end{multline}
\end{itemize}
\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:=\partial/\partial z_j,\quad D^\alpha:=D_1^{\alpha_1}\ldots D_n^{\alpha_n}.
\end{gather*}
Denote by $\mathcal H_0$ the space of holomorphic functions in $B_n$ for which $(D^\alpha f)(0)=0$, $|\alpha|=0,\ldots,r-1$, and $\mathcal R^rf\in H_2(B_n)$. It is known (see~\cite{2}) that functions from $H_2(B_n)$ have finite boundary values almost everywhere. Moreover $H_2(B_n)$ can be considered as a Hilbert space with inner product
$$(f,g)_{H_2(B_n)}:=\int_Sf(z)\overline{g(z)}\,d\sigma(z).$$

Thus $\mathcal H_0$ is a Hilbert space with inner product
$$(f,g):=(\mathcal R^rf,\mathcal R^rg)_{H_2(B_n)}.$$
Let $f,g\in\mathcal H_0$ and
$$f(z)=\sum_{|\alpha|=r}^\infty c_\alpha z^\alpha,\quad g(z)=\sum_{|\alpha|=r}^\infty d_\alpha z^\alpha.$$
Since monomials are orthogonal in $H_2(B_n)$ and
$$\|z^\alpha\|_{H_2(B_n)}^2=\frac{(n-1)!\alpha!}{(n-1+|\alpha|)!}$$
we have
$$(f,g)=\sum_{|\alpha|=r}^\infty\left(\frac{|\alpha|!}{(|\alpha|-r)!}\right)^2
\frac{(n-1)!\alpha!}{(n-1+|\alpha|)!}c_\alpha\overline d_\alpha.$$
It is easily verified that
$$K(z,w)=\sum_{|\alpha|=r}^\infty\left(\frac{(|\alpha|-r)!}{|\alpha|!}\right)^2
\frac{(n-1+|\alpha|)!}{(n-1)!\alpha!}\overline w^\alpha z^\alpha$$
is the reproducing kernel of $\mathcal H_0$.

Let us consider the space $L_2(S_\rho,\sigma_\rho)$, where $\sigma_\rho$ is the probability measure on $S_\rho$, which is invariant with respect to the orthogonal group $O(2n)$. Set for $|\alpha|\ge r$
$$\varphi_\alpha(z):=\frac{(|\alpha|-r)!}{|\alpha|!}\left(\frac{(n-1+|\alpha|)!}{(n-1)!\alpha!}
\right)^{1/2}z^\alpha.$$
The functions $\varphi_\alpha(z)$ form a complete orthonormal basis for $\mathcal H_0$. Moreover these functions are orthogonal in $L_2(S_\rho,\sigma_\rho)$ and
\begin{multline*}
\|\varphi_\alpha\|_{L_2(S_\rho,\sigma_\rho)}^2=\int_{S_\rho}|\varphi_\alpha(z)|^2\,
d\sigma_\rho(z)=\int_S|\varphi_\alpha(\rho\xi)|^2\,d\sigma(\xi)\\
=\left(\frac{(|\alpha|-r)!}{|\alpha|!}\right)^2\rho^{2|\alpha|}.
\end{multline*}
The number of different monomials $z^\alpha$ with $|\alpha|=s$ is equal to $\displaystyle\binom{n+s-1}{n-1}$. As $H\mathcal R^r_2(B_n)=B\mathcal H_0+\mathcal P_r$, $\mathcal H_0\perp\mathcal P_r$ in $L_2(S_\rho,\sigma_\rho)$, and $\dim\mathcal P_r=N_r$, we have by Theorem~\ref{T2}
\begin{multline*}
\biggl(\sum_{s=m}^\infty\left(\frac{(s-r)!}{s!}\right)^2
\binom{n+s-1}{n-1}\rho^{2s}\biggr)^{1/2}\\
\le d^{N_m}\left(H\mathcal R^r_2(B_n),C(S_\rho)\right)=\lambda_{N_m}\left(H\mathcal R^r_2(B_n),C(S_\rho)\right)\\
\le\sup_{z\in S_\rho}\biggl(\sum_{|\alpha|\ge m}^\infty\left(\frac{(|\alpha|-r)!}{|\alpha|!}\right)^2
\frac{(n-1+|\alpha|)!}{(n-1)!\alpha!}\left|z^{2\alpha}\right|\biggr)^{1/2}.
\end{multline*}
Using the equation
$$\sum_{|\alpha|=s}\frac{s!}{\alpha!}\left|z^{2\alpha}\right|=|z|^{2s},$$
we obtain
\begin{multline*}
d^{N_m}\left(H\mathcal R^r_2(B_n),C(S_\rho)\right)=\lambda_{N_m}\left(H\mathcal R^r_2(B_n),C(S_\rho)\right)\\
=\biggl(\sum_{s=m}^\infty\left(\frac{(s-r)!}{s!}\right)^2
\binom{n+s-1}{n-1}\rho^{2s}\biggr)^{1/2}\\
=\rho^m\biggl(\frac1{(n-1)!}\sum_{s=m}^\infty\frac{((m-r+s)!)^2)(n+m-1+s)!}{((m+s)!)^3}
\rho^{2s}\biggr)^{1/2}.
\end{multline*}

To prove (ii) and (iii) we consider the space $\mathcal A_0$ of holomorphic functions in $B_n$ for which $(D^\alpha f)(0)=0$, $|\alpha|=0,\ldots,r-1$, and $\mathcal R^rf\in A_2(B_n)$. $\mathcal A_0$ is a Hilbert space with inner product
$$(f,g):=(\mathcal R^rf,\mathcal R^rg)_{A_2(B_n)}=\int_{B_n}\mathcal R^rf(z)\overline{\mathcal R^rg(z)}\,d\nu(z).$$
Analogous to the previous case, we can show that the functions
$$\psi_\alpha(z):=\sqrt{\frac{n+|\alpha|}n}\varphi_\alpha(z)$$
form a complete orthonormal basis for $\mathcal A_0$ in $L_2(S_\rho,\sigma_\rho)$. Furthermore
$$\|\psi_\alpha(z)\|_{L_2(S_\rho,\sigma_\rho)}^2=\frac{n+|\alpha|}n\left(\frac{(|\alpha|-r)!}
{|\alpha|!}\right)^2\rho^{2|\alpha|}=:\lambda_{|\alpha|}.$$
Let $r\ge1$ and $s\ge r$. Then
$$(n+s+1)\left(\frac{s+1-r}{s+1}\right)^2\le(n+s+1)\left(\frac s{s+1}\right)^2\le\frac{n+s+1}{s+1}s<n+s.$$
Thus
\begin{multline*}
\lambda_{s+1}=\frac{n+s+1}n\left(\frac{(s+1-r)!}{(s+1)!}\right)^2\rho^{2(s+1)}\le
\frac{n+s}n\left(\frac{(s-r)!}{s!}\right)^2\rho^{2s}\\
=\lambda_s.
\end{multline*}
If $r=0$ (in this case $A\mathcal R_2^r(B_n)=BA_2(B_n)$), then $\{\lambda_j\}$ is not in general a non-increasing sequence. But if $((n+m)/n)\rho^{2m}\le1$ then for all $s\ge m$ and all $q<m$, $\lambda_q\ge\lambda_s$. Now (ii) and the first two equations of \eqref{3} follow from Theorem~\ref{T2} in the same way as in the case of (i). Denote by
$$\Phi_n(m,\rho):=\sum_{s=0}^\infty\binom{n+m+s}n\rho^{2s}.$$
It easily verified that
$$\Phi_n(m,\rho)=\frac1{(1-\rho^2)^{n+1}}\binom{n+m}n\sum_{s=0}^n\frac{(-1)^s}{1+s/m}
\binom ns\rho^{2s}.$$
So (iii) is proved.
\end{proof}

{\it Remark.} The referee informed me that in the case $n=1$ the exact values of $N$-widths of the Bergman classes were obtained in \cite{8}.

For $n=1$ the class $H\mathcal R_2^r(B_1)$ coincides with the class $BH_2^r$, defined as the set of all holomorphic functions in $B_1$ for which $f^{(r)}(z)\in BH_2(B_1)$. The set of all holomorphic functions in $B_1$, for which $f^{(r)}(z)\in BA_2(B_1)$ we denote by $BA_2^r$. If $r\ge1$ the classes $BA_2^r$ and $A\mathcal R_2^r(B_1)$ are different. Nevertheless the method of Theorem~\ref{T2} can be applied. Thus we obtain the following result.

\begin{theorem}\label{T4}
Let $0<\rho<1$. Then:
\begin{itemize}
\item[(i)] for all $N\ge r\ge0$
\begin{multline*}
d^N\left(BH_2^r,C(S_\rho)\right)=\lambda_N\left(BH_2^r,C(S_\rho)\right)\\
=\rho^N\biggl(\sum_{s=0}^\infty\left(\frac{(N-r+s)!}{(N+s)!}\right)^2
\rho^{2s}\biggr)^{1/2};
\end{multline*}
\item[(ii)] for all $N\ge r\ge1$
\begin{multline*}
d^N\left(BA_2^r,C(S_\rho)\right)=\lambda_N\left(BA_2^r,C(S_\rho)\right)\\
=\rho^N\biggl(\sum_{s=0}^\infty\left(\frac{(N-r+s)!}{(N+s)!}\right)^2(N+s+1)
\rho^{2s}\biggr)^{1/2}.
\end{multline*}
\end{itemize}
\end{theorem}

\section{The $N$-Widths for Hardy and Bergman Classes in Polydiscs}

Set
\begin{align*}
U^n&:=\{\,z\in\mathbb C^n:|z_1|<1,\ldots,|z_n|<1\,\},\\
T^n&:=\{\,z\in\mathbb C^n:|z_1|=1,\ldots,|z_n|=1\,\},\\
T^n_\rho&:=\{\,z\in\mathbb C^n:|z_1|=\rho_1,\ldots,|z_n|=\rho_n\,\},
\end{align*}
where $\rho=(\rho_1,\ldots,\rho_n)$ and $0\le\rho_j<1$, $j=1,\ldots,n$. Denote by $H_2(U^n)$ the set of all holomorphic functions in $U^n$ for which
$$\|f\|_{H_2(U^n)}:=\sup_{0<r<1}\biggl(\int_{T^n}|f(rz)|^2\,d\mu(z)\biggl)^{1/2}<\infty,$$
where $\mu(z)$ is the normalized Lebesgue measure in $T^n$. We shall denote by $A_2(U^n)$ the set of all holomorphic functions in $U^n$ for which
$$\|f\|_{A_2(U^n)}:=\biggl(\int_{U^n}|f(z)|^2\,d\omega(z)\biggl)^{1/2}<\infty,$$
where $\omega(z)$ is the normalized Lebesgue measure in $U^n$. The spaces $H_2(U^n)$ and $A_2(U^n)$ are Hilbert spaces with the reproducing kernels
$$K_H(z,w):=\begin{cases}(1-z_1\overline w_1)^{-1}\ldots(1-z_n\overline w_n)^{-1},&H=H_2(U^n),\\
(1-z_1\overline w_1)^{-2}\ldots(1-z_n\overline w_n)^{-2},&H=A_2(U^n)\end{cases}$$
(the details can be found in \cite{9}).

\begin{theorem}\label{T5}
Let $\rho=(\rho_1,\ldots,\rho_n)$, $0\le\rho_j<1$.
\begin{itemize}
\item[(i)] Assume that $\alpha^{(1)},\ldots,\alpha^{(N)}$ are the N largest terms of the sequence $\{\rho^{2\alpha}\}$. Then
\begin{multline}\label{4}
d^N\left(BH_2(U^n),C(T_\rho^n)\right)=\lambda_N\left(BH_2(U^n),C(T_\rho^n)\right)\\
=\biggl((1-\rho_1^2)^{-1}\ldots(1-\rho_n^2)^{-1}-\sum_{s=1}^N\rho^{2\alpha^{(s)}}\biggr)^{1/2}.
\end{multline}
\item[(ii)] Assume that $\alpha^{(1)},\ldots,\alpha^{(N)}$ are the N largest terms of the sequence $\{k_\alpha\rho^{2\alpha}\}$, where $k_\alpha:=(\alpha_1+1)\ldots(\alpha_n+1)$. Then
\begin{multline*}
d^N\left(BA_2(U^n),C(T_\rho^n)\right)=\lambda_N\left(BA_2(U^n),C(T_\rho^n)\right)\\
=\biggl((1-\rho_1^2)^{-2}\ldots(1-\rho_n^2)^{-2}-\sum_{s=1}^Nk_{\alpha^{(s)}}
\rho^{2\alpha^{(s)}}\biggr)^{1/2}.
\end{multline*}
\end{itemize}
\end{theorem}

\begin{proof}
Let us prove (i). The monomials $z^\alpha$ form a complete orthonormal basis in $H_2(U^n)$. They are also an orthogonal system in $L_2(T_\rho^n,\mu_\rho)$, where $\mu_\rho$ is the normalized Lebesgue measure in $T_\rho^n$. Moreover
$$\|z^\alpha\|^2_{L_2(T_\rho^n,\mu_\rho)}=\rho^{2\alpha}$$
and for $z\in T_\rho^n$, $|z^\alpha|=\rho^{2\alpha}$. From Theorem~\ref{T2} we have
$$d^N\left(BH_2(U^n),C(T_\rho^n)\right)=\lambda_N\left(BH_2(U^n),C(T_\rho^n)\right)
=\biggl(\sum_{\alpha\notin\tau}\rho^{2\alpha}\biggr)^{1/2},$$
where $\tau:=\{\alpha^{(1)},\ldots,\alpha^{(N)}\}$. Now (i) follows from the representation
$$(1-\rho_1^2)^{-1}\ldots(1-\rho_n^2)^{-1}=\sum_{|\alpha|\ge0}\rho^{2\alpha}.$$
Using the representation
\begin{equation}\label{5}
(1-\rho_1^2)^{-2}\ldots(1-\rho_n^2)^{-2}=\sum_{|\alpha|\ge0}k_{\alpha}
\rho^{2\alpha},
\end{equation}
a similar argument proves (ii).
\end{proof}

We can obtain a more precise result in the case $\rho_1=\ldots=\rho_n$.

\begin{theorem}\label{T6}
Let $\rho_1=\ldots=\rho_n=\rho$ and $0<\rho<1$. Then:
\begin{itemize}
\item[(i)] for $N_{m-1}<N\le N_m$
\begin{multline*}
d^N\left(BH_2(U^n),C(T_\rho^n)\right)=\lambda_N\left(BH_2(U^n),C(T_\rho^n)\right)\\
=\rho^{m-1}\biggl(N_m-N+\binom{n+m-1}{n-1}(1-\rho^2)^{-n}\\
\times\sum_{s=0}^{n-1}
\frac{(-1)^s}{1+s/m}\binom{n-1}s\rho^{2(s+1)}\biggr)^{1/2};
\end{multline*}
\item[(ii)] for $n\ge2$ and
\begin{equation}\label{6}
0<\rho\le m^{1/2}(m/n+1)^{-n/2}
\end{equation}
\begin{multline*}
d^N\left(BA_2(U^n),C(T_\rho^n)\right)=\lambda_N\left(BA_2(U^n),C(T_\rho^n)\right)\\
=\frac{\rho^m}{(1-\rho^2)^n}\biggl(\binom{2n+m-1}{2n-1}\sum_{s=0}^{2n-1}\frac{(-1)^s}{1+s/m}
\binom{2n-1}s\rho^{2s}\biggr)^{1/2}.
\end{multline*}
\end{itemize}
\end{theorem}

\begin{proof}
The sequence $\rho^{2|\alpha|}$ is a non-increasing sequence for $|\alpha|\to\infty$. The number of different multiindexes $\alpha$ with $|\alpha|=s$ is equal to $\displaystyle\binom{n+s-1}{n-1}$. By \eqref{4} we have for $N_{m-1}<N\le N_m$
\begin{multline*}
d^N\left(BH_2(U^n),C(T_\rho^n)\right)=\lambda_N\left(BH_2(U^n),C(T_\rho^n)\right)\\
=\biggl((1-\rho^2)^{-n}-\sum_{s=0}^{m-2}\binom{n+s-1}{n-1}\rho^{2s}-
(N-N_{m-1})\rho^{2(m-1)}\biggr)^{1/2}\\
=\biggl((N_m-N)\rho^{2(m-1)}+\sum_{s=m}^\infty\binom{n+s-1}{n-1}\rho^{2s}\biggr)^{1/2}.
\end{multline*}
Now (i) follows from equations
\begin{multline*}
\sum_{s=m}^\infty\binom{n+s-1}{n-1}\rho^{2s}=\rho^{2m}\sum_{s=0}^\infty
\binom{n+m+s-1}{n-1}\rho^{2s}\\
=\rho^{2m}\Phi_{n-1}(m,\rho)\\
=\rho^{2m}(1-\rho^2)^{-n}\binom{n+m-1}{n-1}\sum_{s=0}^{n-1}\frac{(-1)^s}{1+s/m}
\binom{n-1}s\rho^{2s}.
\end{multline*}

To prove (ii) we will first prove that if the condition \eqref{6} holds, then for
all $|\beta|\ge m$ and all $|\alpha|<m$
\begin{equation}\label{7}
k_\beta\rho^{2|\beta|}\le k_\alpha\rho^{2|\alpha|}.
\end{equation}
In view of the monotone decreasing property of $y(x):=x(x/n+1)^{-n}$ for $x\ge2$ and $n\ge2$ we have
$$\rho^2\le\max\{y(1),y(2)\}\le1/2.$$
Consequently for all $s\ge1$
$$(s+1)\rho^{2s}\le s\rho^{2s-2}.$$
Thus for each $|\beta|\ge m$ choosing any $\beta_j\ge1$ we will have
$$k_\beta\rho^{2|\beta|}\le k_{\beta'}\rho^{2|\beta'|},$$
where $\beta'=(\beta_1,\ldots,\beta_j-1,\ldots,\beta_n)$. Continuing this process we will find
$\beta^*$ with $|\beta^*|=m$ for which
$$k_\beta\rho^{2|\beta|}\le k_{\beta^*}\rho^{2|\beta^*|}\le(m/n+1)^n\rho^{2m}.$$
On the other hand, if $|\alpha|<m$ then in view of the monotone decreasing of the sequence
$\{s\rho^{2s-2}\}_1^\infty$ and by \eqref{6} we obtain
$$k_\alpha\rho^{2|\alpha|}\ge(|\alpha|+1)\rho^{2|\alpha|}\ge m\rho^{2m-2}\ge(m/n+1)^n\rho^{2m}.$$
So \eqref{7} is proved.

From Theorem~\ref{T5} it follows that
\begin{multline*}
d^{N_m}\left(BA_2(U^n),C(T_\rho^n)\right)=\lambda_{N_m}\left(BA_2(U^n),C(T_\rho^n)\right)\\
=\biggl((1-\rho^2)^{-2n}-\sum_{|\alpha|=0}^{m-1}k_{\alpha}\rho^{2|\alpha|}\biggr)^{1/2}=:d.
\end{multline*}
By \eqref{5}
$$\sum_{|\alpha|=s}k_\alpha=\binom{2n+s-1}{2n-1}.$$
Therefore
\begin{multline*}
d^2=\sum_{s=m}^\infty\binom{2n+s-1}{2n-1}\rho^{2s}=\rho^{2m}\sum_{s=0}^\infty\binom{2n+s+m-1}{2n-1}
\rho^{2s}\\
=\rho^{2m}\Phi_{2n-1}(m,\rho)\\
=\rho^{2m}(1-\rho^2)^{-2n}\binom{2n+m-1}{2n-1}\sum_{s=0}^{2n-1}\frac{(-1)^s}{1+s/m}
\binom{2n-1}s\rho^{2s}.
\end{multline*}
\end{proof}

\section{$N$-Widths of Holomorphic Functions of One Variable}

Denote by $H_\gamma$ the space of holomorphic functions in $\Delta_R$
$$f(z)=\sum_{s=-\infty}^{+\infty}a_sz^s$$
which satisfy the condition
$$\sum_{s=-\infty}^{+\infty}\gamma_s|a_s|^2<\infty,$$
where $\{\gamma_s\}$ is a sequence of non-negative numbers such	that $\displaystyle\liminf_{s\to\mp\infty}\gamma_s^{1/|s|}\ge R^2$. Set $\Gamma:=\{s:\gamma_s=0\}$ and $r:=\card\Gamma$.

The space
$$H_\gamma^0:=\biggl\{\,f(z)=\sum_{s=-\infty}^{+\infty}a_sz^s\in H_\gamma:a_j=0,\ j\in\Gamma\,\biggr\}$$
is a Hilbert space with inner product
$$(f,g)=\sum_{s=-\infty}^{+\infty}\gamma_sa_s\overline b_s,$$
where
$$f(z)=\sum_{s=-\infty}^{+\infty}a_sz^s,\quad g(z)=\sum_{s=-\infty}^{+\infty}b_sz^s.$$
Moreover the space $H_\gamma^0$ has the reproducing kernel
$$K(z,w):=\sum_{s\notin\Gamma}\gamma_s^{-1}z^s\overline w^s.$$

Set $BH_\gamma:=BH_\gamma^0+\mathcal P_r$, where $\mathcal P_r:=\left\{\sum_{s\in\Gamma}a_sz^s\right\}$. This convenient form for generalization of certain classes in the case of the unit disk was proposed by Fisher and Micchelli \cite{10}.

For $1\le\rho<R$ and $k\ge r$ set $\sigma_k(\rho):=\{s_1,\ldots,s_{k-r}\}\cap\Gamma$, where
$\{s_1,\ldots,s_{k-r}\}$ are the $k-r$ largest terms of the sequence
$$\left\{\gamma_s^{-1}\frac{\rho^{2s}+\rho^{-2s}}2\right\}_{s\notin\Gamma}.$$

\begin{theorem}\label{T7}
Assume that for all $s\in\mathbb N$	$\gamma_s=\gamma_{-s}$.
\begin{itemize}
\item[(i)] If $N\ge(r+1)/2$ and $0\in\sigma_{2N-1}(\rho)$, then
\begin{multline*}
d^{2N-1}\left(BH_\gamma,C(\Delta_\rho)\right)=\lambda_{2N-1}\left(BH_\gamma,
C(\Delta_\rho)\right)\\
=\biggl(\sum_{s\notin\sigma_{2N-1}(\rho)}\gamma_s^{-1}\frac{\rho^{2s}+\rho^{-2s}}2\biggr)^{1/2}.
\end{multline*}
\item[(ii)] If $N\ge r/2$ and $0\notin\sigma_{2N}(\rho)$, then
\begin{multline*}
d^{2N}\left(BH_\gamma,C(\Delta_\rho)\right)=\lambda_{2N}\left(BH_\gamma,
C(\Delta_\rho)\right)\\
=\biggl(\gamma_0+\sum_{s\notin\sigma_{2N}(\rho)}\gamma_s^{-1}\frac{\rho^{2s}+\rho^{-2s}}2
\biggr)^{1/2}.
\end{multline*}
\end{itemize}
\end{theorem}

\begin{proof}
Let us prove (i). The functions
$$\varphi_s(z):=\gamma_s^{-1/2}z^s,\quad s\notin\Gamma,$$
form a complete orthonormal basis for $H_\gamma^0$. Denote by $L_2(\partial\Delta_\rho)$ a Hilbert space of functions defined on the boundary of $\Delta_\rho$ with inner product
$$(f,g):=\frac1{4\pi}\int_0^{2\pi}\left[f(\rho e^{i\theta})\overline{g(\rho e^{i\theta})}+
f(\rho^{-1}e^{i\theta})\overline{g(\rho^{-1}e^{i\theta})}\right]\,d\theta.$$
It is easily seen that $\varphi_s$ form an orthogonal system in $L_2(\partial\Delta_\rho)$ and
$$\|\varphi_s\|^2_{L_2(\partial\Delta_\rho)}=\gamma_s^{-1}\frac{\rho^{2s}+\rho^{-2s}}2.$$
From Theorem~\ref{T2} follows
\begin{multline*}
\biggl(\sum_{s\notin\sigma_{2N-1}(\rho)}\gamma_s^{-1}\frac{\rho^{2s}+\rho^{-2s}}2\biggr)^{1/2}
\le d^{2N-1}\left(BH_\gamma,C(\Delta_\rho)\right)\\
=\lambda_{2N-1}\left(BH_\gamma,
C(\Delta_\rho)\right)\le\sup_{z\in\partial\Delta_\rho}\biggl(\dfrac12\sum_{s\notin\sigma_{2N-1}(\rho)}
\gamma_s^{-1}(|z|^s+|z|^{-s})\biggr)^{1/2}\\
=\biggl(\sum_{s\notin\sigma_{2N-1}(\rho)}\gamma_s^{-1}\frac{\rho^{2s}+\rho^{-2s}}2\biggr)^{1/2}.
\end{multline*}

Part (ii) is proved in a similar way.
\end{proof}

For $\rho=1$ the analogous application of Theorem~\ref{T2} gives

\begin{theorem}\label{T8}
For all $N\ge r$
$$d^N\left(BH_\gamma,C(\Delta_1)\right)=\lambda_N\left(BH_\gamma,C(\Delta_1)\right)
=\biggl(\sum_{s\notin\sigma_N(1)}\gamma_s^{-1}\biggr)^{1/2}.$$
\end{theorem}

Now we consider some examples of the spaces $H_\gamma$. Denote by $H_2(\Delta_R)$ the class of holomorphic functions in $\Delta_R$ for which
$$\|f\|_{H_2(\Delta_R)}:=\sup_{1<\rho<R}\biggl(\frac1{4\pi}\int_0^{2\pi}\left[|f(\rho e^{i\theta})|^2+|f(\rho^{-1}e^{i\theta})|^2\right]\,d\theta\biggr)^{1/2}<\infty.$$
Let $A_2(\Delta_R)$ be the class of holomorphic functions in $\Delta_R$ for which
$$\|f\|_{A_2(\Delta_R)}:=\biggl(\int_{\Delta_R}|f(z)|^2\,d\eta(z)\biggr)^{1/2}<\infty,$$
where $\eta(z)$ is normalized Lebesgue measure in $\Delta_R$. Let us consider the classes $BH_2^r(\Delta_R)$ and $BA_2^r(\Delta_R)$, which are the sets of holomorphic functions in $\Delta_R$ such that $f^{(r)}(z)$ lies in $BH_2(\Delta_R)$ and $BA_2(\Delta_R)$, respectively.

It can be easily shown that the class $BH_2^r(\Delta_R)$ coincides with $BH_\gamma$	for
$$\gamma_s=(s(s-1)\ldots(s-r+1))^2\frac{R^{2(s-r)}+R^{-2(s-r)}}2,$$
and $BA_2^r(\Delta_R)$ coincides with $BH_\gamma$, where for $r\ge1$
$$\gamma_s=(s(s-1)\ldots(s-r+2))^2(s-r+1)\frac{R^{2(s-r+1)}+R^{-2(s-r+1)}}{R^2-R^{-2}}$$
and for $r=0$ (that is for $BA_2(\Delta_R)$)
$$\gamma_s=(s+1)^{-1}\frac{R^{2(s+1)}+R^{-2(s+1)}}{R^2-R^{-2}},\ s\ne-1,\quad \gamma_{-1}=\frac{4\log R}{R^2-R^{-2}}.$$

We give some more examples of the classes $BH_\gamma$. Let $H_2(D_H)$ and $A_2(D_H)$ be the sets of all $2\pi$-periodic holomorphic functions in $D_H$ which satisfy the conditions
$$\|f\|_{H_2(D_H)}:=\sup_{0<h<H}\biggl(\frac1{4\pi}\int_0^{2\pi}\left[|f(x+ih)|^2+|f(x-ih)|^2
\right]\,dx\biggr)^{1/2}<\infty$$
and
$$\|f\|_{A_2(D_H)}:=\biggl(\frac1{4\pi H}\int_0^{2\pi}\int_{-H}^H|f(x+iy)|^2\,dxdy\biggr)^{1/2}<\infty,$$
respectively. Denote by $BH_2^r(D_H)$ and $BA_2^r(D_H)$ the sets of all $2\pi$-periodic holomorphic functions in $D_H$ for which $f^{(r)}(z)$ lie in $BH_2(D_H)$ and $BA_2(D_H)$, respectively.

To find the linear and Gel'fand $N$-widths of $BH_2^r(D_H)$ and $BA_2^r(D_H)$ in
the space $C(D_h)$, $0\le h<H$, we use the map $z=(1/i)\log w$. Then the original problem reduces to the one for $BH_\gamma$ with $R=e^{i\theta}$	and the space $C(\Delta_\rho)$ with $\rho=e^h$,	 where
$$\gamma_s=s^{2r}\cosh(2sH)$$
in the case of $BH_2^r(D_H)$ and
$$\gamma_s=\frac1{2H}s^{2r-1}\sinh(2sH)$$
in the case of $BA_2^r(D_H)$.

By Theorems~\ref{T7} and \ref{T8} we obtain the following result.

\begin{theorem}\label{T9}
Let $r\ge0$.
\begin{itemize}
\item[(i)] For all $0\le h<H$
\begin{multline*}
d^{2N-1}\left(BH_2^r(D_H),C(D_h)\right)=\lambda_{2N-1}\left(BH_2^r(D_H),C(D_h)\right)\\
=\biggl(2\sum_{s=N}^\infty\frac{\cosh(2sh)}{s^{2r}\cosh(2sH)}\biggr)^{1/2},
\end{multline*}
\begin{multline*}
d^{2N-1}\left(BA_2^r(D_H),C(D_h)\right)=\lambda_{2N-1}\left(BA_2^r(D_H),C(D_h)\right)\\
=2H^{1/2}\biggl(\sum_{s=N}^\infty\frac{\cosh(2sh)}{s^{2r-1}\sinh(2sH)}\biggr)^{1/2}.
\end{multline*}
\item[(ii)] For all $H>0$
\begin{multline*}
d^{2N}\left(BH_2^r(D_H),C[0,2\pi]\right)=\lambda_{2N}\left(BH_2^r(D_H),C[0,2\pi]\right)\\
=\biggl(\frac1{N^{2r}\cosh(2NH)}+2\sum_{s=N+1}^\infty\frac1{s^{2r}\cosh(2sH)}
\biggr)^{1/2},
\end{multline*}
\begin{multline*}
d^{2N}\left(BA_2^r(D_H),C[0,2\pi]\right)=\lambda_{2N}\left(BA_2^r(D_H),C[0,2\pi]\right)\\
=H^{1/2}\biggl(\frac2{N^{2r-1}\sinh(2NH)}+4\sum_{s=N+1}^\infty\frac1{s^{2r-1}\sinh(2sH)}
\biggr)^{1/2}.
\end{multline*}
\end{itemize}
\end{theorem}


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\end{document}


































