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\newtheorem{theorem}{Theorem}
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\newcommand{\hh}{H^r_{\infty,\beta}}
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\begin{document}

\title[Optimal Information for Approximating]{Optimal Information for Approximating Periodic Analytic Functions}
\author{Konstantin Yu.~Osipenko\ and Klaus Wilderotter}
\thanks{This research was supported in
part by the Russian Foundation of Fundamental Research (Grant No\
93--01--00237) and by the ISF \& RG (Grant MP 1300).\hfill\break
1991 {\em Mathematics Subject Classification.} Primary 65E05, 41A46; Secondary
30E10. \hfill\break
{\em Key words and phrases.} Optimal recovery, Optimal information, Periodic
Blaschke products.}

\begin{abstract}
Let $S_\beta:=\{z\in\mathbb C:|\IM z|<\beta\}$ be a strip in the complex plane.
For fixed integer $r\ge0$ let $\hh$ denote the class of $2\pi$-periodic
functions $f$, which are analytic in $S_\beta$ and satisfy $|f^{(r)}(z)|\le1$ in $S_\beta$. Denote by $\hr$ the subset of functions from $\hh$ that are
real-valued on the real axis. Given a function $f\in\hh$, we try to recover
$f(\zeta)$ at a fixed point $\zeta\in\mathbb R$ by an algorithm $A$ on the
basis of the information
$$If=(a_0(f),a_1(f),\ldots,a_{n-1}(f),b_1(f),\ldots,b_{n-1}(f)),$$
where $a_j(f)$, $b_j(f)$ are the Fourier coefficients of $f$. We find the
intrinsic error of recovery
$$E(\hh,I):=\infp_{A\colon\mathbb C^{2n-1}\to\mathbb C}\,\sup_{f\in\hh}|f(\zeta)-A(
If)|.$$
Furthermore the $(2n-1)$-dimensional optimal information error, optimal
sampling error and $n$-widths of $\hr$ in $C$, the space of continuous
functions on $[0,2\pi]$, are determined. The optimal sampling error
turns out to be strictly greater than the optimal information error.
Finally the same problems are investigated for the class $\hp$, consisting
of all $2\pi$-periodic functions, which are analytic in $S_\beta$  with $p$-integrable boundary values. In the case $p=2$ sampling fails to yield optimal information as well in odd as in even dimensions.
\end{abstract}
\maketitle

\section*{Introduction}

Let $W$ be a class of $2\pi$-periodic, real-valued (or complex-valued)
functions. Suppose that $W\subset C$, where $C$ is the space of continuous
functions on $[0,2\pi]$ endowed with the supremum norm. Consider the
problem of optimal recovery of the linear functional $U$ on $W$ given by $U
f=f(\zeta)$, i.e.\ point evaluation in $\zeta$, on the basis of the
information
$$If=(L_1f,\ldots,L_nf),$$
where $L_1,\ldots,L_n$ are continuous linear functionals on $W$.

By an algorithm we mean any map (not necessarily linear or continuous) $A
\colon Z^n\to Z$, where $Z=\mathbb R$ or $\mathbb C$ depending on whether $W$ is a set of real-valued or complex-valued functions.

$$\begin{picture}(180,50)
\put(60,37){$W$}
\put(75,41){\vector(1,0){63}}
\put(143,37){$Z$}
\put(103,49){$U$}
\put(70,32){\vector(1,-1){26.5}}
\put(98,0){$Z^n$}
\put(114,5){\vector(1,1){26.5}}
\put(70,12){$I$}
\put(131,12){$A$}
\end{picture}$$

The algorithm $A$ produces the error
$$E_A(W,I):=\sup_{f\in W}|Uf-A(If)|.$$
The value
$$E(W,I):=\inf_{A\colon Z^n\to Z}E_A(W,I)$$
is called the intrinsic error of the problem. An algorithm $A^*$, for which
$$E_{A^*}(W,I)=E(W,I)$$
is said to be an optimal algorithm.

The optimal information error for estimating $W$ in $C$ by $n$ linear
observations is defined as follows:
\begin{equation}\label{i}
i_n(W,C):=\infp_{L_1,\ldots,L_n}\,\infp_{A\colon Z^n\to C}\,\sup_{f\in W}\|
f-A(If)\|_C\,.
\end{equation}
Any continuous linear functionals $L_1^*,\ldots,L_n^*$ for which the
infimum is attained are called optimal.

If we restrict the class of admissible linear observations to function
values, then we have the value
$$s_n(W,C):=\infp_{z_1,\ldots,z_n\in[0,2\pi)}\,\infp_{A\colon Z^n\to C}\,
\sup_{f\in W}\|f-A(f(z_1),\ldots,f(z_n))\|_C\,,$$
which is called the optimal sampling error. If the infimum is attained at the
points $z_1^*,\ldots,z_n^*$, then these points are said to be optimal.

The study of optimal recovery problems has received much attention in the
last years. For a detailed survey we refer to the papers of Micchelli and
Rivlin \cite{mr7} and \cite{mr4} as well as to the book of Traub and
Wozniakowski \cite{tw}. The values $i_n$ and $s_n$ were considered by
Fisher and Micchelli \cite{FM1} and \cite{FM2} for the unit balls of
Hilbert spaces of nonperiodic functions with simply connected domain of
holomorphy.

Let $S_\beta:=\{z\in\mathbb C:|\IM z|<\beta\}$ be a strip in the complex plane.
For fixed integer $r\ge0$ let $\hh$ denote the Hardy--Sobolev class of
functions $f$, which are $2\pi$-periodic, analytic in $S_\beta$, and
satisfy $|f^{(r)}(z)|\le1$ in $S_\beta$. Denote by $\hr$ the subset of
functions from $\hh$ that are real-valued on the real axis. In the case $r=
0$ we will omit the upper index $r$. The Fourier coefficients of $f$ are
given by
\begin{align*}
a_k(f)&:=\frac1\pi\int_0^{2\pi}f(x)\cos kx\,dx,\quad k=0,1,
\ldots\,\,,\\
b_k(f)&:=\frac1\pi\int_0^{2\pi}f(x)\sin kx\,dx,\quad k=1,2,\ldots\,\,.
\end{align*}

In Section 1 we find an optimal algorithm for approximating $f(\zeta)$,
$\zeta\in[0,2\pi)$, on the basis of the information
\begin{equation}\label{I}
If=(a_0(f),a_1(f),\ldots,a_{n-1}(f),b_1(f),\ldots,b_{n-1}(f)),
\end{equation}
uniformly for all $f\in\hh$. We show that the error $E(\hh,I)$ of an
optimal algorithm is given by $\|\ef\|_C$, where $\ef$ is the $r$-th
indefinite integral of a periodic Blaschke product with $2n$ equidistant,
real zeros.

In Section 2 this result is applied to determine the optimal information
error $i_{2n-1}(\hr,C)$. We show that the Fourier coefficients are optimal
information and that
\begin{multline*}
i_{2n-1}(\hr,C)=d_{2n-1}(\hr,C)=d^{2n-1}(\hr,C)\\
=\delta_{2n-1}(\hr,C)=\|\ef\|_C,
\end{multline*}
where $d_{2n-1}$, $d^{2n-1}$ and $\delta_{2n-1}$ denote the Kolmogorov,
Gel'fand and linear widths, respectively. Osipenko \cite{osib} proved that
a corresponding equation is valid in the even dimensional case. Thus $i_{2n
-1}(\hr,C)=i_{2n}(\hr,C)$ and all three widths of order $2n-1$ and $2n$
coincide and are equal to $\|\ef\|_C$.

In the case $r=0$ we find in addition the optimal error $s_{2n-1}(\hi,C)$,
which coincides with $s_{2n-1}(\hir,C)$. It turns out that equidistant
nodes are optimal. However, $s_{2n-1}(\hir,C)$ is strictly greater than $i_
{2n-1}(\hir,C)$, i.e.\ sampling in optimal nodes does not yield optimal
information. In particular, we calculate the value
$$\frac{s_{2n-1}(\hir,C)}{i_{2n-1}(\hir,C)},$$
which gives a quantitative measure, how much sampling fails to be optimal.
This situation is in a sharp contrast to the even dimensional case, where
it is known that sampling in equidistant nodes is optimal information (cf.
Osipenko \cite{osia} and Wilderotter \cite{wil95a}). Moreover, we recall
that Fisher and Micchelli \cite{fm} proved that for a simply connected
domain of holomorphy sampling always yields optimal information.

In Section 3 we consider the problem of optimal recovery and optimal
information for the class $\hp$, $1\le p<\infty$. Here $\hp$ denotes the
set of all functions $f$, which are $2\pi$--periodic, analytic in $S_\beta
$, and satisfy
$$\sup_{0\le\eta<\beta}\left(\frac1{4\pi}\int_0^{2\pi}\left(|f(t+i\eta)|^p+
|f(t-i\eta)|^p\right)\,dt\right)^{1/p}\le1.$$

For fixed points $z_1,\ldots,z_n\in[0,2\pi)$ with multiplicities $\nu_1,
\ldots,\nu_n\in\mathbb N$ and $\zeta\in[0,2\pi)$ we find an optimal algorithm
and the intrinsic error for approximating $f(\zeta)$, $f\in\hp$, on the
basis of the Hermite information
$$If=\left(f(z_1),\ldots,f^{(\nu_1-1)}(z_1),\ldots,f(z_n),\ldots,f^{(\nu_n-
1)}( z_n)\right).$$
We also find the optimal sampling error $s_n(\hp,C)$. It turns out that
sampling in equidistant nodes is optimal for all $p$ and all $n$. Moreover,
for $p=2$ we compare $s_n(\htw,C)$ with the optimal information error $i_n(
\htw,C)$. We show that these quantities do not coincide and calculate the
ratios
$$\frac{s_{2n-1}(\htw,C)}{i_{2n-1}(\htw,C)},\qquad\frac{s_{2n}(\htw,C)}{i_{
2n}(\htw,C)}.$$
The nonoptimality of sampling in the even dimensional case is quite
remarkable. In all examples studied so far for the imbedding of $\hp$ in $L
_q$ with $p\ge q$ (see Osipenko \cite{osia}, Wilderotter \cite{wil95b}) we
found that sampling in $2n$ equidistant nodes yields optimal information
for $i_{2n}$. The present paper shows that this fails to be valid for the
imbedding of $\htw$ in $C$.

Throughout the paper we use substantially elliptic function techniques. We
emphasize that pretty optimal elliptic function bounds date back already to the
classical work of N. I. Achieser \cite{achia}, which influenced and stimulated
the present article.

\section{Optimal Recovery from Fourier Coefficients}

This section deals with the optimal recovery of the linear functional $Uf=f
(\zeta)$, $\zeta\in[0,2\pi)$, on $\hh$, using the information (\ref I). Of
central importance for our considerations is the following well known
general duality formula due to Smolyak (we use here the complex version of
Smolyak's result proved by Osipenko \cite{Os1}):
\begin{equation}\label{d}
E(\hh,I)=\sup_{\substack{f\in\hh\\If=0}}|f(\zeta)|.
\end{equation}
Moreover, the minimal error is achieved by a linear method of the form
\begin{equation}\label{A}
A^*(If)=c_0a_0(f)+\sum_{j=1}^{n-1}(c_ja_j(f)+d_jb_j(f)).
\end{equation}
By an extremal function we mean any function $f_0\in\hh$ with $If_0=0$ and
$|f_0(\zeta)|=E(\hh,I)$.

Our further strategy will be to determine explicitely an extremal function
$f_0$. For this purpose we need some auxiliary facts about periodic
Blaschke products.

In order to introduce periodic Blaschke products, we transfer the
analysis from the strip $S_\beta$ to the annulus $\Omega:=\{w\in\mathbb C:R<|w|
<R^{-1}\}$, where $R=e^{-\beta}$. The universal covering transformation $w=
e^{iz}$ maps $S_\beta$ onto $\Omega$ and induces a correspondence $f(z)\to
g(w)=f\oi$ between analytic periodic functions in $S_\beta$ and analytic
functions in $\Omega$.

A Blaschke product $B$ of degree $m$ on $\Omega$ is a function of the form
$$B(w)=\exp\biggl(-\sum_{j=1}^m(g(w,\alpha_j)+ih(w,\alpha_j))\biggr).$$
Here $\alpha_1,\ldots,\alpha_m$ are points in $\Omega$, $g(w,\alpha_j)$ is
the Green's function for $\Omega$ with singularity at $\alpha_j$ and $h(w,
\alpha_j)$ is the harmonic conjugate of $g(w,\alpha_j)$. In general $B$ is
multiple valued. However, if we choose $m=2n$ and locate all points $\alpha
_1,\ldots,\alpha_{2n}$ on the unit circle $\{w:|w|=1\}$, it turns out that
$B$ is single valued. For a proof of the last fact and further details on
Blaschke products we refer to Fisher \cite{fis} and Wilderotter \cite
{wil95a}.

In particular we may choose the $2n$ zeros on the unit circle to be
equidistant. Let $\alpha^*_j=\exp\left(i(j-1)\dfrac\pi n\right)$ for $j=
1,\ldots ,2n$ and
$$B_{2n}(w)=\exp\biggl(-\sum^{2n}_{j=1}(g(w,\alpha^*_j)+ih(w,\alpha^*_j))
\biggr).$$

Finally we go back again from the annulus to our original setting of the
strip and introduce the periodic Blaschke product $\wt B_{2n}$ on $S_\beta$
by defining $\wt B_{2n}(z):=B_{2n}(e^{iz})$.

Blaschke products are closely related to elliptic functions. Throughout the
present paper we will use the following terminology (see for example
Achieser \cite{achib}, Bateman \cite{bat}):  $\sn(z,k)$, $\cn(z,k)$, and
$\dn(z,k)$ denote the Jacobi elliptic functions with modulus $k$ (further we
will note the dependence of the Jacobi elliptic functions on the modulus only in
case the modulus is different from $k$); the complementary modulus is given by $
k'=\sqrt{1-k^2}$ and the complete elliptic integrals of the first kind with
moduli $k$ and $k'$ are denoted by $K$ and $K'$, respectively. We always
suppose that $K$ and $ K'$ satisfy the equation
$$\frac{\pi K'}{2K}=\beta.$$

With this notation $\wt B_{2n}$ can be written in the form (see Osipenko
\cite{osia}):
$$\wt B_{2n}(z)=k^n\prod_{j=1}^{2n}\sn\left(\frac K\pi z-(j-1)\frac Kn\right).$$
Using the first fundamental transformation of elliptic functions of degree
$2n$ one can show that
$$\wt B_{2n}(z)=-\sqrt\lambda\sn\left(\frac{2n\Lambda}\pi z,\lambda\right).$$
Here $\Lambda$ is the complete elliptic integral of the first kind with
modulus $\lambda$ determined by the equation
$$\frac{\Lambda'}\Lambda=2n\frac{K'}K.$$

In order to cope with the optimal recovery problem, we introduce the $r$-th
indefinite integral $\ef$ of $-\wt B_{2n}$ defined by
$$\Phi_{n,0}^\beta:=-\wt B_{2n},\qquad\ef:=D_r*\Phi_{n,0}^\beta,\quad r\ge1
.$$
Here
$$D_r(t)=2\sum_{k=1}^{\infty}\frac{\cos(kt-\pi r/2)}{k^r},\quad r=1,2,
\ldots,$$
is the Bernouilli Monospline, while
$$(f*g)(z)=\frac1{2\pi}\int_0^{2\pi}f(z-t)g(t)\,dt$$
denotes the convolution of two periodic functions.

Osipenko \cite{osib} gave the following explicit representation for $\ef$
and $\|\ef\|_C$:
$$
\begin{aligned}
\ef(z)&=\frac\pi{\sqrt\lambda\Lambda n^r}\sum_{k=0}^\infty\frac{
\sin((2k+1 )nz-\pi r/2)}{(2k+1)^r\sinh((2k+1)2n\beta)},\\
\|\ef\|_C&=\frac\pi{\sqrt\lambda\Lambda n^r}\sum_{k=0}^\infty\frac{(-1)^{k(
r+1)}}{(2k+1)^r\sinh((2k+1)2n\beta)},
\end{aligned}\quad r=0,1,\ldots\,\,.$$
From this one can read off that $I\ef=0$.

We now are ready to formulate our first main result.

\begin{theorem}
For all integers $r\ge0$ and with $I$ defined by {\rm(\ref I)},
$$E(\hh,I)=\|\ef\|_C.$$
\end{theorem}

\begin{proof}
We can assume without loss of generality that the fixed evaluation point in the
problem (\ref d) is equal to $\zeta=0$. Put
$$\varphi(z):=\begin{cases}\ef\left(z+\dfrac\pi{2n}\right),&r=2k,\\
\ef(z),&r=2k+1.\end{cases}$$
We wish to show that $\varphi$ is an extremal function of the problem (\ref
d). Note that $I\varphi=0$, $|\varphi(0)|=\|\ef\|_C$, and $\varphi$ is an
even function. Suppose there exists a function $f_0\in\hh$ with $If_0=0$
and $|f_0(0)|>|\varphi(0)|$. After scaling $f_0$ with the factor $\exp(-i
\arg f_0(0))$, we may assume $f_0(0)$ to be real and positive. Let us
define
$$f_1(z):=\frac{f_0(z)+\overline{f_0(\overline z)}}2,\qquad f_2(z):=\frac{f
_1(z)+f_1(-z)}2.$$
Then $f_2\in\hr$, $If_2=0$, and $f_2(0)=f_0(0)$. Moreover, $f_2$ is an even
function. Set
$$\rho:=\varphi(0)/f_2(0)\qquad F:=\varphi-\rho f_2.$$
We claim that the function $F$ has at least $2n+1$ distinct zeros in $[-\pi
,\pi)$. Clearly $F(0)=0$. Moreover, since both $\varphi$ and $f_2$ are even
functions, $F$ does not change its sign in $\zeta=0$. On the other side we
have $IF=0$, since $I\varphi=If_2=0$. The condition $IF=0$ means that
\begin{align*}
\int_0^{2\pi}F(x)\cos kx\,dx&=0,\quad k=0,1,\ldots,n-1,\\
\int_0^{2\pi}F(x)\sin kx\,dx&=0,\quad k=1,2,\ldots,n-1.
\end{align*}
Since the trigonometric polynomials of degree at most $n-1$ are a
Tchebycheff system of dimension $2n-1$, it follows from Pinkus \cite
[Chap.\ III, Prop. 1.4]{pin}, that $F$ has at least $2n$ cyclic sign
changes. In addition $F$ has a zero in $\zeta=0$ without sign change. Hence
$F$ has altogether at least $2n+1$ zeros in $[-\pi,\pi)$. By Rolle's
theorem the same conclusion remains valid for the $r$-th derivative $F^{(r)
}=\varphi^{(r)}-\rho f_2^{(r)}$.

Transferring this result from the strip to the annulus, we see that the
function $F^{(r)}\oi$ is single valued and analytic in $\Omega$ and has at
least $2n+1$ zeros in $\Omega$. By the definition of $\ef$ we have
$$\varphi^{(r)}\oi=\begin{cases}-B_{2n}\left(w\exp\left(i\dfrac\pi{2n}\right)
\right),&r=2k,\\
-B_{2n}(w),&r=2k+1.\end{cases}$$
The boundary values of the Blaschke product $B_{2n}$ satisfy identically $|
B_{2n}(w)|\equiv1$ on $\partial\Omega$. Consequently we have for $w\in
\partial\Omega$
\begin{multline*}
\left|\varphi^{(r)}\oi-F^{(r)}\oi\right|=\left|\rho f_2^{(r)}\oi\right|
\le|\rho|<1\\=\left|\varphi^{(r)}\oi\right|.
\end{multline*}
Since $B_{2n}$ has $2n$ zeros in $\Omega$, Rouche's theorem implies that $F
^{(r)}\oi$ has exactly $2n$ zeros in $\Omega$. This is a contradiction and
the proof of Theorem 1 is complete.
\end{proof}

\section{Optimal Information and $n$-Widths of $\hr$}

In this section Theorem 1 is applied to determine the optimal information
error $i_{2n-1}(\hr,C)$. It turns out that $i_{2n-1}(\hr,C)$ coincides with
certain odd dimensional $n$-widths. Therefore we start by recalling the
definition of the various $n$-widths.

Let $V$ be a subset of a normed linear space $X$. The Kolmogorov $n$-widths
are defined by
$$d_n(V,X):=\infp_{X_n}\,\sup_{x\in V}\,\infp_{y\in X_n}\|x-y\|_X,$$
where $X_n$ runs over all subspaces of $X$ of dimension $n$ or less.

The Gel'fand $n$-widths are defined by
$$d^n(V,X):=\infp_{X^n}\,\sup_{x\in X^n\cap V}\|x\|_X,$$
where $X^n$ runs over all subspaces of codimension at most $n$ (here we
assume that $0\in V$).

The linear $n$-widths are given by
$$\delta_n(V,X):=\infp_{P_n}\,\sup_{x\in V}\|x-P_nx\|_X,$$
where $P_n$ is any linear operator of $X$ into $X$ of rank at most $n$.

Much information on $n$-widths can be found in the book of A. Pinkus \cite
{pin}. In particular, the following fundamental inequality holds:
\begin{equation}\label{3}
d_n(V,X)\,,\,d^n(V,X)\,\le\delta_n(V,X).
\end{equation}

Analogously to \eqref{i} we can define the optimal information error $i_n(V,
X)$ for estimating $V$ in $X$ by $n$ linear observations.

\begin{lemma}
Assume that $V$ is a centrally symmetric set and $0\in V$. Then
\begin{equation}\label{4}
d^n(V,X)\le i_n(V,X)\le\delta_n(V,X).
\end{equation}
\end{lemma}

\begin{proof}
The inequality
$$i_n(V,X)\le\delta_n(V,X)$$
evidently follows from the definition. To prove the lower bound consider
any continuous linear functionals $L_1,\ldots,L_n$. For each $\varepsilon>0
$ there exists $x_\varepsilon\in V$ such that $L_1x_\varepsilon=\ldots=L_nx
_\varepsilon=0$ and
$$\sup_{\substack{x\in V\\L_1x=\ldots=L_nx=0}}\|x\|_X\le\|x_\varepsilon\|_X+
\varepsilon.$$
For all algorithms $A$ we have
$$\|x_\varepsilon-A(0,\ldots,0)\|_X+\|-x_\varepsilon-A(0,\ldots,0)\|_X\ge2
\|x_\varepsilon\|_X.$$
Therefore,
\begin{multline*}
\sup_{x\in V}\|x-A(L_1x,\ldots,L_nx)\|_X\ge\|x_\varepsilon\|_X\\
\ge\sup_{\substack
{x\in V\\L_1x=\ldots=L_nx=0}}\|x\|_X-\varepsilon\ge d^n(V,X)-\varepsilon.
\end{multline*}
Taking the infimum over $A$ and $L_1\ldots,L_n$ we obtain
$$i_n(V,X)\ge d^n(V,X).$$
\end{proof}

Our result reads now as follows:

\begin{theorem}
For all integer $r\ge0$
\begin{multline*}
i_{2n-1}(\hr,C)=d_{2n-1}(\hr,C)=d^{2n-1}(\hr,C)\\
=\delta_{2n-1}(\hr,C)=\|\ef\|_C.
\end{multline*}
\end{theorem}

\begin{proof}
In view of \eqref{3} and \eqref{4} to establish upper bounds we may restrict
ourselves to $\delta(\hr,C)$. It follows from Theorem 1 that there exists
an optimal method \eqref{A} such that
$$|f(0)-A^*(If)|\le\|\ef\|_C$$
for all $f\in\hr$. Let now $\eta$ be an arbitrary fixed point in the
interval $[0,2\pi)$ and set $(T_\eta f)(z):=f(z+\eta)$. Since
\begin{align*}
a_j(T_\eta f)&=\phantom-a_j(f)\cos j\eta+b_j(f)\sin j\eta,\\
b_j(T_\eta f)&=-a_j(f)\sin j\eta+b_j(f)\cos j\eta,
\end{align*}
we obtain that
\begin{multline*}
\biggl|f(\eta)-c_0a_0(f)-\sum_{j=1}^{n-1}\Bigl((c_j\cos j\eta-d_j
\sin j\eta)a_j (f)\\
+(c_j\sin j\eta+d_j\cos j\eta)b_j(f)\Bigr)\biggr|\le\|\ef\|_C.
\end{multline*}
This pointwise estimate holds uniformly in $[0,2\pi)$. Thus we have
$$\delta_{2n-1}(\hr,C)\le\|\ef\|_C.$$

As mentioned in the introduction, Osipenko \cite{osib} proved that
\begin{equation}\label{O}
d_{2n}(\hr,C)=d^{2n}(\hr,C)=\delta_{2n}(\hr,C)=\|\ef\|_C.
\end{equation}
The lower bounds follow now from the monotonicity of the $n$-widths.
\end{proof}

Combining \eqref{O} with Theorem 2, we get in view of \eqref{4} that $i_{2n-1}
(\hr,C)$ and $i_{2n}(\hr,C)$ as well as all three kinds of widths of order
$2n-1$ and $2n$ coincide and are equal to $\|\ef\|_C$.

The preceding analysis may arise the impression that the situation in odd
and even dimensions is identical. This is definitely not true. Although the
different values of the widths are all the same, the properties of optimal
information are substantially different in odd and even dimensions. In the
sequel we will restrict ourselves to the case $r=0$. Our course of proof
showed that the Fourier coefficients $(a_0(f),a_1(f),\ldots,a_{n-1}(f),b_1(
f),\ldots,b_{n-1}(f))$ are optimal information for $i_{2n-1}(\hir,C)$ and
consequently also for $i_{2n}(\hir,C)$. However, Osipenko \cite{osia} and
Wilderotter \cite{wil95a} proved that in the even dimensional case sampling
in $2n$ equidistant nodes yields optimal information as well, that is $s_{2
n}(\hir,C)=i_{2n-1}(\hir,C)$. We now try to find the optimal sampling error
$s_{2n-1}(\hir,C)$.

For this purpose we consider in a first step fixed sampling points $z_1,
\ldots,z_{2n-1}\in[0,2\pi)$. From the results of Ovchincev \cite{ovc}
and Wilderotter \cite{wil92} it follows that
\begin{multline*}
\infp_{A\colon\mathbb R^{2n-1}\to C}\,\sup_{f\in\hir}\|f-A(f(z_1),\ldots,f
(z_{2n - 1}))\|_C\\
=k^n\biggl\|\prod_{j=1}^{2n-1}\sn\left(\frac K\pi(\cdot-z_j)\right)\biggr\|_C.
\end{multline*}

In a second step we minimize the righthand side of the last equation over
all possible choices of sampling points. Osipenko \cite{osia} showed in a
different context that
\begin{equation}\label{ex}
\inf_{z_1,\ldots,z_n\in[0,2\pi)}k^{n/2}\biggl\|\prod_{j=1}^n\sn\left(\frac
K\pi(\cdot-z_j)\right)\biggr\|_C=\sqrt{\lambda_n},
\end{equation}
where
\begin{equation}\label{l}
\lambda_n=4e^{-\beta n}\left(\frac{\sum_{m=0}^\infty e^{-2\beta nm(m+1)}}{1
+2\sum_{m=1}^\infty e^{-2\beta nm^2}}\right)^2=4e^{-\beta n}+O(e^{-3\beta n
})
\end{equation}
($\lambda_n$ can be also defined as a solution of the equation $\Lambda'/
\Lambda=nK'/K$). Moreover, equidistant nodes are the unique nodes (up to a
shift), for which the infimum in \eqref{ex} is attained. Thus
$$s_{2n-1}(\hir,C)=\sqrt{k\lambda_{2n-1}}.$$

On the other side we have
$$i_{2n-1}(\hir,C)=i_{2n}(\hir,C)=s_{2n}(\hir,C)=\|\Phi_{n,0}\|_C=\sqrt{
\lambda_{2n}}.$$
Set $z_j^*:=(j-1)\dfrac{2\pi}{2n-1}$, $j=1,\ldots,2n-1$,
$$b_1(z):=\sqrt k\sn\left(\frac K\pi(z-z_{n+1}^*)\right),\ b_2(z):=k^{
n-1/2}\prod_{j=1}^{2n-1}\sn\left(\frac K\pi(z-z_j^*)\right).$$
Using the first fundamental transformation of elliptic functions of degree
$2n-1$ it can be shown that
$$b_2(z)=\sqrt{\lambda_{2n-1}}\sn\left(\frac{(2n-1)\Lambda_{2n-1}}\pi z,
\lambda_{2n-1}\right),$$
where $\Lambda_{2n-1}$ is the complete elliptic integral of the first kind
with modulus $\lambda_{2n-1}$. It is easy to check that
$$\|b_1\|_C=-b_1\left(\frac\pi{2n-1}\right)=\sqrt k,\quad\|b_2\|_C=b_2
\left(\frac\pi{2n-1}\right)=\sqrt{\lambda_{2n-1}}.$$
Consequently
$$\|b_1b_2\|_C=\sqrt{k\lambda_{2n-1}}.$$
Since equidistant nodes are unique optimal nodes in the extremal problem
\eqref{ex} we obtain that $\sqrt{\lambda_{2n}}<\sqrt{k\lambda_{2n-1}}$.
Thus
$$s_{2n-1}(\hir,C)>i_{2n-1}(\hir,C),$$
i.e. sampling does not yield optimal information in odd dimensions. More
precisely we may calculate the following ratio, which gives a quantitative
measure, how much sampling fails to be optimal:
$$\frac{s_{2n-1}(\hir,C)}{i_{2n-1}(\hir,C)}=\frac{\sqrt{k\lambda_{2n-1}}}{
\sqrt{\lambda_{2n}}}=\sqrt ke^{\beta/2}+O(e^{-4\beta n}).$$
For $n=1$ from \eqref{l} it follows that
$$k=4e^{-\beta}\left(\frac{\sum_{m=0}^\infty e^{-2\beta m(m+1)}}{1+2\sum_{m
=1}^\infty e^{-2\beta m^2}}\right)^2.$$
Using this equality it is easy to show that
$$e^{-\beta/2}<\sqrt k<2e^{-\beta/2}.$$
Thus $1<\sqrt ke^{\beta/2}<2$ for all $\beta\in(0,+\infty)$.

\section{Optimal Sampling and Information in $\hp$}

Denote by $\Hp$, $1\le p\le\infty$, the space of all $2\pi$-periodic
functions $f$, which are analytic in $S_\beta$ and satisfy
\begin{align*}
\|f\|_{\Hp}&:=\sup_{0\le\eta<\beta}\left(\frac1{4\pi}\int_0^{2
\pi}(|f(t+i\eta)|^p+|f(t-i\eta)|^p)\,dt\right)^{1/p}<\infty,\\
&\hskip240pt1\le p<\infty,\\
\|f\|_{\Hi}&:=\sup_{z\in S_\beta}|f(z)|<\infty.
\end{align*}
Let $\hp$ be the closed unit ball of $\Hp$. Given an evaluation point $
\zeta\in[0,2\pi)$ consider the problem of optimal recovery of $f(\zeta)$, $
f\in\hp$, on the basis of the Hermite information
$$If=(f(z_1),\ldots,f^{(\nu_1-1)}(z_1),\ldots,f(z_n),\ldots,f^{(\nu_n-1)}(z
_n)),\quad N:=\sum_{j=1}^n\nu_j,$$
where $z_1,\ldots,z_n\in[0,2\pi)$. The case $p=\infty$ was obtained by
Ovchincev \cite{ovc} and Wilderotter \cite{wil92}. The solution of this
recovery problem for $1\le p<\infty$ reads as follows.

\begin{theorem}
Set
$$W(z)=k^{N/2}\prod_{j=1}^n\sn^{\nu_j}\left(\frac K\pi(z-z_j)\right).$$
Then
$$E(\hp,I)=\begin{cases}\left(\dfrac{2K}\pi\right)^{1/p}|W(\zeta)|,&N\mbox{ even},\\
\left(\dfrac{2K}\pi\right)^{1/p}\sqrt k|W(\zeta)|,&N\mbox{ odd}.\end{cases}$$
An optimal method of recovery is given by
\begin{equation}\label{m}
f(\zeta)\approx\sum_{j=1}^n\sum_{\nu=0}^{\nu_j-1}c_{j\nu}(\zeta,p)f^{(\nu)}
(z_j),
\end{equation}
where
\begin{multline*}
c_{j\nu}(\zeta,p)=\frac K\pi\,\frac{W(\zeta)}{\nu!(\nu_j-
\nu- 1)!}\\
\times\lim_{z\to z_j}\left(\frac{(z-z_j)^{\nu_j}\gamma_N(z)\dn^{\frac{p-
2}p}\kp}{W(z)\sn\kp}\right)^{(\nu_j-\nu-1)},
\end{multline*}
$$\gamma_N(z)=\begin{cases}\cn\kp,&N\mbox{ even},\\
\dn\kp,&N\mbox{ odd}.\end{cases}$$
\end{theorem}

\begin{proof}
The function
$$b(z)=\sqrt k\sn\frac K\pi z$$
is analytic in $S_\beta$. Moreover, $b(z+2\pi)=-b(z)$ and $|b(x+i\beta)|
\equiv1$ for all $x\in\mathbb R$. Thus $\overline{W(z)}=W^{-1}(z)$ for $z\in
\partial S_\beta$.

Suppose $N$ is an even number. Consider the function
$$g(z)=W(z)\dn^{2/p}\kp.$$
Since $\dn\dfrac K\pi z$ is $2\pi$-periodic and does not vanish in the
strip $S_\beta$, $g\in\Hi$. Set
$$\alpha:=\frac{2K}\pi g(\zeta).$$
For $f\in\Hp$ consider the integral
$$Jf:=\frac\alpha{4\pi}\int_{\Gamma_0}\overline{g(z)}|g(z)|^{p-2}f(z)\,dz,$$
where $\Gamma_0:=[-i\beta,2\pi-i\beta]\cup[i\beta,2\pi+i\beta]$. Using the
properties of elliptic functions, we have
$$\overline{\dn\left(\frac K\pi(x\pm i\beta)\right)}=\pm i\frac{\cn\left(
\dfrac K \pi(x\pm i\beta)\right)}{\sn\left(\dfrac K\pi(x\pm i\beta)\right)}.$$
The element of integration in $Jf$ is $2\pi$-periodic. So we can rewrite $Jf$ in the following form
$$Jf=\frac{KW(\zeta)}\pi\,\frac1{2\pi i}\int_{\Gamma_\varepsilon}\frac{\cn\kp
\dn^{\frac{p-2}p}\kp}{W(z)\sn\kp}f(z)\,dz,$$
where $\Gamma_\varepsilon$ is the boundary of rectangle $-\varepsilon\le\RE
z\le2\pi-\varepsilon$, $|\IM z|\le\beta$, and $\varepsilon$ such that $
\zeta,z_ 1,\dots,z_n$ lie inside this rectangle. By the residue theorem
\begin{equation}\label{r}
Jf=f(\zeta)-\sum_{j=1}^n\sum_{\nu=0}^{\nu_j-1}c_{j\nu}(\zeta,p)f^{(\nu)}(z_j).
\end{equation}
For $f(z)=g(z)$ this equality gives
$$\|g\|_{\Hp}=\left(\frac\pi{2K}\right)^{1/p}.$$
If $f\in\hp$, then by H\"older inequality we obtain
$$|Jf|\le|\alpha|\|g\|_{\Hp}^{p/q}\|f\|_{\Hp}\le\left(\frac{2K}\pi\right)^{1/
p}|g(\zeta)|,\qquad\frac1p+\frac1q=1.$$
In view of \eqref{r} we have
$$E(\hp,I)\le\left(\frac{2K}\pi\right)^{1/p}|g(\zeta)|.$$
On the other hand, $g_0:=g/\|g\|_{\Hp}\in\hp$ and $Ig_0=0$. Consequently,
$$E(\hp,I)=\sup_{\substack{f\in\hp\\If=0}}|f(\zeta)|\ge|g_0(\zeta)|=\left(\frac{2K}
\pi\right)^{1/p}|g(\zeta)|.$$
Hence
$$E(\hp,I)=\left(\frac{2K}\pi\right)^{1/p}|g(\zeta)|$$
and \eqref{m} is an optimal method of recovery.

For odd $N$ the same scheme of proof is applied to $Jf$ with
$$g(z)=\sqrt k\sn\left(\frac K\pi(z-\zeta+\pi)\right)W(z)\dn^{2/p}\kp$$
(here we use that $\sn(u+K)=\cn u/\dn u$).
\end{proof}

Taking in account the equality \eqref{ex}, we have

\begin{corollary}
For all $1\le p<\infty$ and $n\in\mathbb N$
$$s_n(\hp,C)=\begin{cases}\left(\dfrac{2K}\pi\right)^{1/p}\sqrt{\lambda_n},&n\mbox{
even},\\
\left(\dfrac{2K}\pi\right)^{1/p}\sqrt{k\lambda_n},&n\mbox{ odd},\end{cases}$$
where $\lambda_n$ is defined by \eqref{l}. Moreover, equidistant nodes
are optimal.
\end{corollary}

Finally we compare in the case $p=2$ the optimal sampling error $s_n(\htw,C
)$ with the optimal information error $i_n(\htw,C)$. Osipenko \cite{osic}
proved that
\begin{multline*}
\delta_{2n-1}(\htw,C)=d^{2n-1}(\htw,C)=\left(2\sum_{j=
n}^\infty\frac1{\cosh2j\beta}\right)^{1/2}\\
=\frac2{\sqrt{1-e^{-2\beta}}}e^{-\beta n}+O(e^{-5\beta n}),
\end{multline*}
\begin{multline*}
\delta_{2n}(\htw,C)=d^{2n}(\htw,C)=\left(\frac1{\cosh2n\beta}+2\sum_{j
=n+1}^\infty\frac1{\cosh2j\beta}\right)^{1/2}\\
=\sqrt{2\frac{1+e^{-2\beta}}{1-e^{-2\beta}}}\,e^{-\beta n}+O(e^{-5
\beta n}).
\end{multline*}
In view of \eqref{4} the same equalities hold for $i_n(\htw,C)$. Thus we
obtain
\begin{align*}
\frac{s_{2n-1}(\htw,C)}{i_{2n-1}(\htw,C)}&=2\sqrt{\frac{kK}\pi
\sinh\beta}\,+O(e^{-4\beta n}),\\
\frac{s_{2n}(\htw,C)}{i_{2n}(\htw,C)}&=2\sqrt{\frac
K\pi\tanh\beta}\,+O(e^{-4\beta n}).
\end{align*}
The last result is very interesting, inasmuch as it is the first example
known so far of a periodic Hardy space imbedding, for which sampling in
equidistant nodes does not yield optimal information in even dimensions.



\begin{thebibliography}{99}

\bibitem{achia}{\sc N. I. Achieser}: Vorlesungen \"uber Approximationstheorie.
Akademie--Verlag, Berlin, 1953

\bibitem{achib}{\sc N. I. Achieser}: Elements of the Theory of Elliptic
Functions. Nauka, Moscow, 1970

\bibitem{bat}{\sc Bateman Manuscript Project}: Higher Transcendental
Functions, Volume II. McGraw--Hill, New York, 1953

\bibitem{fis}{\sc S. D. Fisher}: Function Theory on Planar Domains: A
Second Course in Complex Analysis. Wiley--Interscience, New York, 1983

\bibitem{fm}{\sc S. D. Fisher, C. A. Micchelli}: The $n$-Widths of Sets of
Analytic Functions. Duke Math. J. {\bf 47}, (1980), 789--801

\bibitem{FM1}{\sc S. D. Fisher, C. A. Micchelli}: Optimal Sampling of
Holomorphic Functions. Amer. J. Math. {\bf106}, (1984), 593--609

\bibitem{FM2}{\sc S. D. Fisher, C. A. Micchelli}: Optimal Sampling of
Holomorphic Functions. II. Math. Ann. {\bf273}, (1985), 131--147

\bibitem{mr7}{\sc C. A. Micchelli, T. J. Rivlin}: A Survey of Optimal
Recovery, in `Optimal Estimation in Approximation Theory' (eds.
C.~A.~Micchelli and T.~J.~Rivlin), Plenum Press, New York, (1977), 1--54

\bibitem{mr4}{\sc C. A. Micchelli, T. J. Rivlin}: Lectures on Optimal
Recovery, in `Numerical Analysis Lancaster, Proceedings', Lecture Notes in
Mathematics {\bf 1129}, Springer--Verlag, Berlin, (1984), 21--93

\bibitem{Os1}{\sc K. Yu. Osipenko}: Best Approximation of Analytic
Functions from Information about Their Values at a Finite Number of Points.
Mat. Zametki {\bf19}, (1976), 29--40; Math. Notes {\bf19}, (1976), 17--23.

\bibitem{osia}{\sc K. Yu. Osipenko}: On $n$-Widths, Optimal Quadrature
Formulas and Optimal Recovery of Functions Analytic in a Strip. Izv. Ross.
Akad. Nauk, Ser. Mat. {\bf58}, (1994), 55--79

\bibitem{osic}{\sc K. Yu. Osipenko}: On $n$-Widths of Holomorphic Functions
of Several Variables. J. Approx. Theory {\bf 82}, (1995), 135-155

\bibitem{osib}{\sc K. Yu. Osipenko}: Exact $n$-Widths of Hardy--Sobolev
Classes, to appear in Constr. Approx.

\bibitem{ovc}{\sc M. P. Ovchincev}: Best Methods for Approximating Regular
Bounded Functions in an Annulus by Their Values in Given Points. Izv.
Vuzov. Mat., (1989), No. 5, 32--39

\bibitem{pin}{\sc A. Pinkus}: $n$-Widths in Approximation Theory.
Springer--Verlag, Berlin, 1985

\bibitem{tw}{\sc J. F. Traub, H. Wozniakowski}: A General Theory of Optimal
Algorithms. Academic Press, New York, 1980

\bibitem{wil92}{\sc K. Wilderotter}: Optimal Recovery of Bounded Analytic
Functions in an Annulus, in `Optimal Recovery' (eds. B.~Bojanov and
H.~Wozniakowski), Nova Science Publishers, New York, (1992), 311--323

\bibitem{wil95a}{\sc K. Wilderotter}: On $n$-Widths of Bounded Periodic
Holomorphic Functions. Ukrainian Math. J. {\bf 47}, (1995), 1170--1175

\bibitem{wil95b}{\sc K. Wilderotter}: Optimal Sampling of Periodic Analytic
Functions. J. Approx. Theory {\bf 82}, (1995), 304--316

\end{thebibliography}

\vspace*{5mm}
\parbox{5cm}{\raggedright\small\it Department of Mathematics\\
Moscow State University\\of Aviation Technology\\
Petrovka 27\\Moscow 103767\\Russia}
\hfill
\parbox{6cm}{\raggedright\small\it Mathematisches Seminar der
\\Landwirtschaftlichen Fakult\"at \\ Rheinische
Friedrich--Wilhelms--\\Universit\"at Bonn\\Nu{\ss}allee 15\\53115
Bonn\\Germany}




\end{document} 