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\begin{document}
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\begin{center}
USSR ACADEMY OF SCIENCE\\
Scientific Council for Cybernetics
\end{center}
\vspace{50pt}
\vspace{60pt}
\begin{center}
{\bf On $N$-Width of One Class of Holomorphic Functions}\\

\vspace{50pt}
K.Yu.~Osipenko and M.I.~Stessin
\vspace{50pt}

\vspace{100pt}

Preprint\\

\vspace{200pt}

Moscow 1990
\end{center}

\newpage


1. \underline{Introduction}. Let
$$B_n=\biggl\{\,z=(z_1,\ldots,z_n)\in\mathbb C^n:|z|^2=\sum_{l=1}^n|z_l|^2<1\,\biggr\}$$
be the unit ball of $\mathbb C^n$. Remind that $H_2(B_n)$ is a space of holomorphic in $B_n$ functions which satisfy the inequality
$$\|f\|_2=\sup_{0<r<1}\biggl(\int_{S^{2n-1}}|f(rz)|^2\,d\sigma_n(z)\biggr)^{1/2}<\infty$$
where $\sigma_n$ is a positive normalized measure on $S^{2n-1}=\partial B_n$ which is invariant with respect to orthogonal group $O^{2n}$ (see Rudin \cite{7}). The unit ball of the Hilbert space $H_2(B_n)$ we denote $BH_2(B_n)$. Let $L_{\infty,r}(B_n)$ ($0<r<1$) be the space $H_2(B_n)$ supplied with the following norm
$$\|f\|_{\infty,r}=\max_{|z|=r}|f(z)|.$$

The purpose of this work is finding the precise values of Gelfand and linear $N$-width of the class $BH_2(B_n)$ in $L_{\infty,r}(B_n)$ metrics for some values of $N$. In the case $n=1$ we shall find $N$-width for arbitrary $N\in\mathbb Z_+$. There are a number of works devoted to calculating of precise values of $N$-width of Hardy classes $BH_p$ in integral $q$-metrics (see, for example, \cite{1,2,3,4}). All of them considered the case $p\ge q$ only and in all these works $n=1$. In our case the situation is opposite.

2. \underline{Main results}. Remind definitions of Gelfand and linear $N$-width. Let $A$ be the subset of Banach space $X$.
$$d^N(A,X)=\infp_{L^N}\sup_{x\in A\cap L^N}\|x\|$$
where $L^N$ is a subspace of $X$, $\codim L^N=N$, is the Gelfand $N$-width of $A$.
$$\lambda_N(A,X)=\infp_{\Lambda_N}\sup_{x\in A}\|x-\Lambda_Nx\|,$$
where $\Lambda_N$ is a linear continuous operator mapping $X$ into it's $N$-dimensional subspace, is the linear $N$-width of $A$. It is easy to verify that
\begin{equation}\label{1}
d^N(A,X)\le\lambda_N(A,X).
\end{equation}
For $m\in\mathbb Z_+$ put
$$N_m=\begin{cases}0,&m=0,\\
\displaystyle\sum_{l=0}^{m-1}\binom{n+l-1}{n-1},&m>0.\end{cases}$$
Note that $N_m$ is equal to dimension of the space of polynomials of $n$ variables which degree is no more than $m-1$.

\begin{theorem}
\begin{multline*}
d^{N_m}(BH_2(B_n),L_{\infty,r}(B_n))=\lambda_{N_m}(BH_2(B_n),L_{\infty,r}(B_n))\\
=\begin{cases}\dfrac1{(1-r^2)^{n/2}},&m=0,\\[10pt]
\displaystyle\sqrt{\frac1{(1-r^2)^n}-\sum_{l=0}^{m-1}\binom{n+l-1}{n-1}r^{2l}},&m>0.\end{cases}
\end{multline*}
\end{theorem}

Before proving this theorem we shall prove two lemmas.

\begin{lemma}\label{L1}
Let $f\in BH_2(B_n)$, $w\in B_n$, $f=\displaystyle\sum_{l=0}^\infty F_l$ is a homogeneous expansion of $f$. Then
$$\biggl|f(w)-\sum_{l=0}^s F_l(w)\biggr|\le\sqrt{\frac1{(1-|w|^2)^n}-\sum_{l=0}^s\binom{n+l-1}{n-1}|w|^{2l}}.$$
\end{lemma}

\begin{proof}
Recall that that the Caushy's kernel for $B_n$ is
$$K(w,z)=\frac1{(1-\la w,z\ra)^n}$$
where $\la w,z\ra=\displaystyle\sum_{l=1}^nw_l\ov z_l$ is the hermitian binary product. Put
$$K_s(w,z)=\frac1{(1-\la w,z\ra)^n}-\sum_{|p|\le s}\frac{(n+|p|-1)!}{(n-1)!p!}w^p\ov z^p$$
(for multyindex $p=(p_1,\ldots,p_n)$ \ $|p|=p_1+\ldots+p_n$, $a^p=a_1^{p_1}\ldots a_n^{p_n}$, $p!=p_1!\ldots p_n!$).

For $f\in BH_2(B_n)$ we have
\begin{equation}\label{2}
\int_{S^{2n-1}}K_s(w,z)f(z)\,d\sigma_n(z)=f(w)-\sum_{l=0}^s F_l(w).
\end{equation}
We use that monomials are orthogonal in $H_2(B_n)$ and
$$\|z^p\|_2^2=\int_{S^{2n-1}}|z_1|^{p_1}\ldots|z_n|^{p_n}\,d\sigma_n(z)=\frac{(n-1)!p!}
{(n+|p|-1)!}$$
(see Rudin \cite{7}). Now it follows from \eqref{2} and H\"older inequality that for $f\in BH_2(B_n)$
\begin{multline*}
\biggl|f(w)-\sum_{l=0}^s F_l(w)\biggr|\le\biggl(\int_{S^{2n-1}}|K_s(w,z)|^2\,d\sigma_n(z)\biggr)^2\\
=\sqrt{\frac1{(1-|w|^2)^n}-\sum_{|p|\le s}\frac{(n+|p|-1)!}{(n-1)!p!}|w_1|^{2p_1}\ldots|w_n|^{2p_n}}\\
=\sqrt{\frac1{(1-|w|^2)^n}-\sum_{|p|\le s}\binom{n+|p|-1}{n-1}|w|^{2|p|}}.
\end{multline*}
The lemma is proved.
\end{proof}

\underline{Remark}. In fact, it follows from \eqref{2}and Theorem~1 from \cite{5} that the Taylor's series is an optimal recovery algorithm for functions from $BH_2(B_n)$ if the Taylor's information in the origin is at the disposal. In case $n=1$ this fact is well-known (see \cite{6}).

\begin{lemma}\label{L2}
Let $s\in\mathbb N$, $N_m\le s<N_{m+1}$, $f_1,\ldots,f_s$ be the orthonormal set of functions of $H_2(B_n)$, $0<r<1$. Then
\begin{equation}\label{3}
\int_{S^{2n-1}}\sum_{l=1}^s|f_l(rz)|^2\,d\sigma_n(z)\le
\sum_{l=0}^{m-1}\binom{n+l-1}{n-1}r^{2l}+(s-N_m)r^{2m}.
\end{equation}
\end{lemma}

\begin{proof}
Let $F_l$ be the homogeneous polynomial of degree $l$. It is obvious that
\begin{equation}\label{4}
\int_{S^{2n-1}}|F_l(rz)|^2\,d\sigma_n(z)=r^{2l}\|F_l\|_2^2.
\end{equation}
If $f\in H_2(B_n)$ and $f(z)=\displaystyle\sum_{l=k}^\infty F_l(z)$ is a homogeneous decomposition of $f$, then it follows from \eqref{4} and the orthogonality of monomials that
\begin{equation}\label{5}
\int_{S^{2n-1}}|f(rz)|^2\,d\sigma_n(z)\le r^{2k}\|f\|_2^2.
\end{equation}
Let $U$ be a unitary transformation of $\mathbb C^s$ and
$$\begin{pmatrix}
u_{11}&\ldots&u_{1s}\\
\hdotsfor{3}\\
u_{s1}&\ldots&u_{ss}\end{pmatrix}$$
is the matrix of this transformation. The mapping $U$ transforms the orthonormal set $\{f_1,\ldots,f_s\}$ into the set $\{g_1,\ldots,g_s\}$, $g_l=\displaystyle\sum_{k=1}^su_{lk}f_k$ which is the orthonormal one (in $H_2(B_n)$) too. Besides that for every $w\in B_n$ we have $\displaystyle\sum_{l=1}^s|g_l(w)|^2=\sum_{l=1}^s|f_l(w)|^2$ because $U$ is the unitary transformation. Hence
$$\int_{S^{2n-1}}\sum_{l=1}^s|f_l(rz)|^2\,d\sigma_n(z)=
\int_{S^{2n-1}}\sum_{l=1}^s|g_l(rz)|^2\,d\sigma_n(z).$$
It is easy to check that there exists the unitary transformation $U$ such that for $N_q<k\le N_{q+1}$ the homogeneous expansion of $g_k$ has no polynomials of degree less than $q+1$. Indeed, let $U_1$ be the unitary transformation of $\mathbb C^s$ which transforms $(f_1(0),\ldots,f_s(0))$ into $(a_1,0,\ldots,0)$ where $a_1=\left(\displaystyle\sum_{l=1}^s|f_l(0)|^2\right)^{1/2}$. Transformation $U_1$ maps $(f_1,\ldots,f_s)$ into $(g_1^1,\ldots,g_s^1)$ and $g_2^1(0)=\ldots=g_s^1(0)=0$. Let $\widetilde U_2$ be the unitary transformation of $\mathbb C^{s-1}$ which maps $\left\{\dfrac{\partial g_2^1}{\partial z_1}\bigg|_0,\ldots,\dfrac{\partial g_s^1}{\partial z_1}\bigg|_0\right\}$ into $(a_2,0,\ldots,0)$, and
$$U_2=\begin{pmatrix}1&0&\ldots&0\\
0&&&\\
\vdots&&\widetilde U_2&\\
0&&&\end{pmatrix},\quad U_2\colon\{g_1^1,\ldots,g_s^1\}\to\{g_1^1,g_2^2\ldots,g_s^2\}.$$
Then $\dfrac{\partial g_3^2}{\partial z_1}\bigg|_0=\ldots=\dfrac{\partial g_s^2}{\partial z_1}\bigg|_0=0$. Iterating this process we shall construct the sequence of transformations $U_3,\ldots,U_s$. The transformation $U=U_s\circ U_{s-1}\circ\ldots\circ U_1$ satisfies our requirement. We have $U\colon\{f_1,\ldots,f_s\}\to\{g_1^1,\ldots,g_s^s\}$, for every $N_q<k\le N_{q+1}$
$$\int_{S^{2n-1}}|g_k^k(rz)|^2\,d\sigma_n(z)\le r^{2q}.$$
The lemma is proved.
\end{proof}

\begin{proof}[Proof of the Theorem]
In view of \eqref{1} it is sufficient to prove that
\begin{gather}\label{6}
\lambda_{N_k}(BH_2(B_n),L_{\infty,r}(B_n))\le\sqrt{\frac1{(1-r^2)^n}-
\sum_{l=0}^{k-1}\binom{n+l-1}{n-1}r^{2l}},\\
d^{N_k}(BH_2(B_n),L_{\infty,r}(B_n))\ge\sqrt{\frac1{(1-r^2)^n}-
\sum_{l=0}^{k-1}\binom{n+l-1}{n-1}r^{2l}}.\label{7}
\end{gather}

The upper evaluation \eqref{6} follows from Lemma~\ref{L1}. Operator $\Lambda_{N_k}$ is the one which maps a function of $H_2(B_n)$ into its Taylor's series of order $N_k$.

The lower bound \eqref{7} will be proved by using Lemma~\ref{L2}. Consider some subspace $L^{N_k}$ of $H_2(B_n)$ which codimension is equal to $N_k$. Let $f_1,\ldots,f_{N_k}$ be an orthonormal basis in $(L^{N_k})^\perp$. For $w\in B_n$ define the following function
$$h_w(z)=\frac1{(1-\la z,w\ra)^n}-\sum_{l=1}^{N_k}\ov{f_l(w)}f_l(z).$$

It is easy to check that for every $w\in B_n$
$$h_w\in L^{N_k}.$$

Therefore
\begin{multline*}
\sup_{g\in BH_2(B_n)\cap L^{N_k}}\|g\|_{\infty,r}\ge\sup_{|w|=r}\frac{\|h_w\cd\|_{\infty,r}}
{\|h_w\cd\|_2}\ge\sup_{|w|=r}\frac{|h_w(w)|}{\|h_w\cd\|_2}\\
=\sup_{|w|=r}\frac1{\|h_w\cd\|_2}\left(\frac1{(1-r^2)^n}-\sum_{l=1}^{N_k}|f_l(w)|^2\right).
\end{multline*}

The direct computation shows that
$$\|h_w\cd\|_2=\left(\frac1{(1-|w|^2)^n}-\sum_{l=1}^{N_k}|f_l(w)|^2\right)^{1/2}.$$

Hence
\begin{multline*}
\sup_{g\in BH_2(B_n)\cap L^{N_k}}\|g\|_{\infty,r}\ge\sup_{|w|=r}\sqrt{\frac1{(1-r^2)^n}-\sum_{l=1}^{N_k}|f_l(w)|^2}\\
=\sqrt{\frac1{(1-r^2)^n}-\inf_{|w|=r}\sum_{l=1}^{N_k}|f_l(w)|^2}.
\end{multline*}

In accordance with Lemma~\ref{L2}
$$\inf_{|w|=r}\sum_{l=1}^{N_k}|f_l(w)|^2\le\int_{S^{2n-1}}\sum_{l=1}^{N_k}|f_l(rw)|^2\,d\sigma_n(w)
\le\sum_{l=0}^{k-1}\binom{n+l-1}{n-1}r^{2l}.$$

Theorem is proved.
\end{proof}

If $n=1$ then $N_k=k$, the series $\displaystyle\sum_{l=0}^{k-1}\binom{n+l-1}{n-1}r^{2l}$ converts into $\displaystyle\sum_{l=0}^{k-1}r^{2l}$ and we obtain the following corollary.

\begin{corollary}
$$d^k(BH_2,L_{\infty,r})=\lambda_k(BH_2,L_{\infty,r})=\frac{r^k}{\sqrt{1-r^2}}.$$
\end{corollary}

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\bibitem{7}	W.~Rudin, Function Theory in the Unit Ball of $\mathbb C^n$. Springer-Verlag. Berlin, Heidelberg, New York, 1980.

\bibitem{1} V.M.~Tikhomirov, Approximation Theory, Itogi Nauki i Techniki, Ser. Sovr. Probl. Math., Fund. Napravlenia, 1987 (in Russian).

\bibitem{2} A.~Pinkus, $N$-Width in Approximation Theory, Berlin, Springer, 1985.

\bibitem{3} O.G.~Parfenov, Gelfand's $n$-width of the unit ball of Hardy class in the weighted space, Math. Zametki, V.~39 (1985), \No.~2, 171--175.

\bibitem{4} J.F.~Fisher and C.A.~Micchelli, The $n$-width of sets of analytic functions, Duke Math. J., 47 (1980), 789--801.

\bibitem{5} K.Yu.~Osipenko and M.I.~Stessin, On recovery problems in the Hardy and Bergman spaces, Math. Zametki, to appear (in Russian).

\bibitem{6}	T.J.~Rivlin, The optimal recovery of functions, Contemp. Math., 9 (1982), 121--151.



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