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\newcommand{\h}{\tilde h^r_{\infty,\beta}}
\newcommand{\hh}{\tilde h_{\infty,\beta}}
\newcommand{\HH}{\widetilde H^r_{\infty,\beta}}
\newcommand{\Ht}{\widetilde H_{\infty,\beta}}
\newcommand{\W}{\widetilde W^r_\infty}
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\begin{document}

\title{Exact $n$-Widths of Hardy--Sobolev Classes}
\author{K. Yu. Osipenko}
\thanks{{\em \hskip-11pt AMS Classification:} 41A46, 30D55\\
{\em Key words and phrases:} Hardy--Sobolev class, $n$-width, bounded analytic functions}

\begin{abstract}
Let $\h$ and $\HH$ denote those $2\pi$-periodic, real-valued functions on
$\mathbb R$ that are analytic in the strip $|\IM z|<\beta$ and satisfy the
restrictions $|\RE f^{(r)}(z)|\le1$ and $|f^{(r)}(z)|\le1$, respectively.
We determine the Kolmogorov, linear, and Gel'fand widths of $\h$ in
$L_q[0,2\pi]$, $1\le q\le\infty$, and $\HH$ in $L_\infty[0,2\pi]$.
\end{abstract}

\maketitle



\section*{Introduction}

The Kolmogorov $n$-width of a subset $A$ of a normed linear space $X$ is
defined by
$$d_n(A,X):=\infp_{X_n}\sup_{x\in A}\infp_{y\in X_n}\|x-y\|,$$
where $X_n$ runs over all $n$-dimensional subspaces of $X$. The linear
$n$-width of $A$ in $X$ is defined by
$$\lambda_n(A,X):=\infp_{P_n}\sup_{x\in A}\|x-P_nx\|,$$
where $P_n$ varies over all bounded linear operators mapping $X$ into
itself whose range has dimension $n$ or less. The Gel'fand $n$-width is
given 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
$n$ (here we assume that $0\in A$).

Set $S_\beta:=\{z\in\mathbb C:|\IM z|<\beta\}$. For integer $r\ge0$ denote by
$\h$ ($\HH$) the set of $2\pi$-periodic functions, real-valued on $\mathbb R$
and analytic in the strip $S_\beta$ which satisfy the conditions
$|\RE f^{(r)}(z)|\le1$ ($|f^{(r)}(z)|\le1$). For $r=0$ we shall omit the
upper index in the notation of these classes. In this paper we determine
the exact values of the Kolmogorov, linear, and Gel'fand widths of $\h$ in
$L_q:=L_q[0,2\pi]$, $1\le q\le\infty$, and $\HH$ in $L_\infty$. We also show
that the $n$-widths of the subset of analytic functions on $[0,2\pi)$ from
the Sobolev class $\W$ are the same as for $\W$.

For $q=\infty$ the $n$-widths of the class $\hh$ were obtained by
V.~M.~Tikhomirov \cite{Ti}. For $1\le q\le\infty$ the exact values of the
even $n$-widths of the classes $\hh$ and $\Ht$ were determined in
\cite{Os1} and \cite{Os2}. In the nonperiodic case for the functions which
are analytic on the open unit disk, real-valued on $(-1,1)$, and which
satisfy the restriction $|f^{(r)}(z)|\le1$, the $n$-widths were obtained in
\cite{Fi}. The results concerning the $n$-widths of the classes of analytic
functions of several variables whose $r$th radial derivative is bounded may
be found in \cite{Fa} and \cite{Os3}.

\section{Exact $n$-Widths of $\widetilde h^r_{\infty,\beta}$}

If $f\in\hh$, then (see \cite[p.~269]{Ak})
$$f(z)=\frac1{2\pi}\int_0^{2\pi}K_\beta(z-t)\RE f(t+i\beta)\,dt,$$
where
$$K_\beta(z)=1+2\sum_{k=1}^\infty\frac{\cos kz}{\cosh k\beta}.$$
Set
\begin{align*}
(f*g)(z)&:=\frac1{2\pi}\int_0^{2\pi}f(z-t)g(t)\,dt,\\
D_r(t)&:=2\sum_{k=1}^\infty\frac{\cos(kt-\pi r/2)}{k^r},\quad
r=1,2,\ldots\,\,.
\end{align*}
Using the representation
$$f(z)=\frac1{2\pi}\int_0^{2\pi}f(t)\,dt+(D_r*f^{(r)})(z),$$
for $r\ge1$ we have
$$\h=\left\{\,a+G_{r,\beta}*h:\|h\|_\infty\le1,\ h\perp1,\ a\in\mathbb R\,\right\},$$
where $G_{r,\beta}:=D_r*K_\beta$.

We say that a real, $2\pi$-periodic, continuous function $G$ satisfies
Property~B (cf.~\cite[p.~129]{Pi}) if for every choice of $0\le t_1<\ldots<t_m<2\pi$ and each $m$, the subspace
$$X_m:=\left\{\,b+\sum_{j=1}^mb_jG(\cdot-t_j):\sum_{j=1}^mb_j=0\,\right\}$$
is of dimension $m$, and for every $f\in X_{2m+1}$, $f\not\equiv0$, the
number of cyclic sign changes $S_c(f)\le2m$.

Set
$$\Lambda_{2n}:=\left\{\,\xi:\xi=(\xi_1,\ldots,\xi_{2n}),\ 0\le\xi_1<\ldots<
\xi_{2n}<2\pi\,\right\}.$$
For each $\xi\in\Lambda_{2n}$ we define
$$h_\xi(t):=(-1)^j,\quad t\in[\xi_{j-1},\xi_j),\quad j=1,\ldots,2n+1,$$
where $\xi_0:=0$, $\xi_{2n+1}:=2\pi$. Denote by $h_n(t)$ the function
$h_\xi$ for $\xi_j=(j-1)\pi/n$, $j=1,\ldots,2n$.

To calculate the exact $n$-widths of $\h$ we need the following theorem of
A.~Pinkus \cite[p.~180, 182]{Pi}.

\begin{theorem}\label{P}
Let $G$ satisfy Property~B. Set
$$\B:=\left\{\,a+G*h:\|h\|_\infty\le1,\ h\perp1,\ a\in\mathbb R\,\right\}.$$
Then:
\begin{enumerate}
\item
$$d_{2n-1}(\B,L_\infty)=\lambda_{2n-1}(\B,L_\infty)=d^{2n-1}(\B,L_
\infty)=\|G*h_n\|_\infty;$$
\item for each $1\le q\le\infty$
$$d_{2n}(\B,L_q)=\lambda_{2n}(\B,L_q)=d^{2n}(\B,L_q)=\|G*h_n\|_q.$$
\end{enumerate}
\end{theorem}

We say that a real, $2\pi$-periodic, continuous function $\mathcal K\in NCVD$
(nondegenerate cyclic variation diminishing) if $S_c(\mathcal K*f)\le S_c(f)$
for all real, $2\pi$-periodic, piecewise continuous functions $f$, and
$$\dim\spa\left\{\,\mathcal K(t_1-\cdot),\ldots,\mathcal K(t_n-\cdot)\,\right\}
=n$$
for every choice of $0\le t_1<\ldots<t_n<2\pi$ and all $n$. It is known (see
\cite[p.~128, 133]{Pi}) that $K_\beta\in NCVD$ and for each $r\ge2$, $D_r$
satisfies Property~B ($D_1(x)$ also satisfies all conditions of Property~B
except that it is not continuous at $x=0$). Therefore, $G_{r,\beta}$
satisfies Property~B for every $r\ge1$.

The Euler perfect splines are defined by
$$\pr(t):=\frac4{\pi n^r}\sk\frac{\sin((2k+1)nt-\pi r/2)}{(2k+1)^{r+1}},
\quad r=0,1,\ldots\,\,.$$
Several properties of this splines may be found, for example, in
\cite[p.~104]{Ko}. In particular,
$$\varphi_{n,0}=h_n,\quad\pr=D_r*h_n,\quad r=1,2,\ldots\,\,.$$
Put
$$\prr(t):=(K_\beta*\pr)(t)=\frac4{\pi n^r}\sk\frac{\sin((2k+1)nt-\pi r/2)}
{(2k+1)^{r+1}\cosh((2k+1)n\beta)}.$$
It was proved in \cite{Os1} that
\begin{equation}\label{phi}
\varphi_{n,0}^\beta(t)=(K_\beta*h_n)(t)=\frac4\pi\arctan\left(\sqrt\lambda
\sn\left(\frac{2n\Lambda}\pi t,\lambda\right)\right),
\end{equation}
where $\Lambda$ is the complete elliptic integral of the first kind with
modulus
\begin{equation}\label{l}
\lambda=4e^{-2\beta n}\left(\frac{\sk e^{-4\beta nk(k+1)}}{1+2\sum_{k=1}
^\infty e^{-4\beta nk^2}}\right)^2.
\end{equation}
By analogy to the Euler perfect splines it can be shown that
$$\kk:=\|\prr\|_\infty=\frac4{\pi n^r}\sk\frac{(-1)^{k(r+1)}}{(2k+1)^{r+1}
\cosh((2k+1)n\beta)}.$$

For $r\ge1$, $\prr=G_{r,\beta}*h_n$. Thus from Theorem \ref{P} ($r\ge1$) and
Theorems 4.8 and 4.9 of \cite[p.~179, 180]{Pi} ($r=0$) we obtain the
following result.

\begin{theorem}\label{h}
Let $r\ge0$. Then:
\begin{enumerate}
\item
$$d_{2n-1}(\h,L_\infty)=\lambda_{2n-1}(\h,L_\infty)=d^{2n-1}(\h,L_\infty)=
\kk;$$
\item for each $1\le q\le\infty$
$$d_{2n}(\h,L_q)=\lambda_{2n}(\h,L_q)=d^{2n}(\h,L_q)=\|\prr\|_q.$$
\end{enumerate}
\end{theorem}

For integer $r\ge1$ denote by $\W$ the class of real, $2\pi$-periodic
functions whose $(r-1)$st derivative is absolutely continuous and
whose $r$th derivative satisfies the condition $|f^{(r)}(t)|\le1$. Let $\A$
be the set of functions from $\W$ which are analytic on $[0,2\pi)$.

\begin{theorem}
Let $r\ge1$. Then:
\begin{enumerate}
\item
\begin{equation}\label{od}
d_{2n-1}(\A,L_\infty)=\lambda_{2n-1}(\A,L_\infty)=d^{2n-1}(\A,L_\infty)=
\frac{K_r}{n^r},
\end{equation}
where
$$K_r:=\frac4\pi\sk\frac{(-1)^{k(r+1)}}{(2k+1)^{r+1}};$$
\item for each $1\le q\le\infty$
\begin{equation}\label{ev}
d_{2n}(\A,L_q)=\lambda_{2n}(\A,L_q)=d^{2n}(\A,L_q)=\|\pr\|_q.
\end{equation}
\end{enumerate}
\end{theorem}

\begin{proof}
For the class $\W$ the equations analogous to \eqref{od} and \eqref{ev} are
a well-known fact (the details and references may be found in \cite{Pi} and
\cite{Ko}). Since $\A\subset\W$ it is sufficient to prove the lower bound.
For all $\beta>0$, \ $\h\subset\A$. Therefore, the lower bound follows from
Theorem~\ref{h} and the obvious equations
$$\lim_{\beta\to0}\kk=\frac{K_r}{n^r},\quad\lim_{\beta\to0}\|\prr\|_q=\|\pr\|_q.$$
The theorem is proved.
\end{proof}

\section{Exact $n$-Widths of $\HH$}

To calculate the exact $n$-widths of $\HH$ we shall need some preliminary
results.

\begin{proposition}
Let $\varphi$ be a continuous, odd, and strictly increasing function
defined on $[-1,1]$. Put
$$\LL:=\left\{\,\xi\in\Lambda_{2n}:\varphi(K_\beta*h_\xi)\perp1\,\right\}.$$
Then
$$\inf\left\{\,\|a+D_r*\varphi(K_\beta*h_\xi)\|_\infty:a\in\mathbb R,\ \xi\in
\LL\,\right\}=\|D_r*\varphi(K_\beta*h_n)\|_\infty.$$
\end{proposition}

\begin{proof}
Let $\xi\in\LL$. Set
$$f_\xi:=D_r*\varphi(K_\beta*h_\xi),\quad f_n:=D_r*\varphi(K_\beta*h_n).$$
Suppose that there exists an $a\in\mathbb R$ and a $\xi\in\LL$ for which
$\|a+f_\xi\|_\infty<\|f_n\|_\infty$. As $f_n(t+\pi/n)=-f_n(t)$ there exist
at least $2n$ points $0\le t_1<\ldots<t_{2n}<2\pi$ such that
$$f_n(t_j)=\varepsilon(-1)^j\|f_n\|_\infty,\quad j=1,\ldots,2n,$$
where $\varepsilon=1$ or $-1$. Denote by $Z(f)$ the number of distinct zeros
of a function $f$ on $[0,2\pi)$. We have
$$Z\bigl(f_n(\cdot+\alpha)\pm a\pm f_\xi(\cdot)\bigr)\ge2n$$
for every $\alpha\in\mathbb R$. By Rolle's Theorem
\begin{multline*}
S_c\bigl(f_n^{(r)}(\cdot+\alpha)\pm f_\xi^{(r)}(\cdot)\bigr)=S_c\bigl(
\varphi\left((K_\beta*h_n)(\cdot+\alpha)\right)\pm\varphi\left((K_\beta*h_
\xi)(\cdot)\right)\bigr)\\\ge2n.
\end{multline*}
Since $\varphi$ is an odd and strictly increasing function,
$$S_c\bigl((K_\beta*h_n)(\cdot+\alpha)\pm(K_\beta*h_\xi)(\cdot)\bigr)\ge2n.$$
From the fact that $K_\beta\in NCVD$ it follows that
$$S_c\bigl(h_n(\cdot+\alpha)\pm h_\xi(\cdot)\bigr)\ge2n.$$
Using Lemma 4.1 of \cite[p.~170]{Pi}, we obtain the existence of an
$\alpha\in\mathbb R$ and $\varepsilon=1$ or $-1$ for which
$$S_c\bigl(h_n(\cdot+\alpha)-\varepsilon h_\xi(\cdot)\bigr)\le2(n-1).$$
This contradiction proves the proposition.
\end{proof}

Set $\varphi_0(z):=\tan\frac\pi4z$,
$$\Phi_{n,0}^\beta:=\varphi_0(K_\beta*h_n),\quad\PP:=D_r*\varphi_0(K_\beta
*h_n),\quad r=1,2,\ldots\,\,.$$
In view of (\ref{phi}) and the representation (see \cite[p.~266]{Ak2})
$$\se=\frac\pi{\lambda\Lambda}\sk\frac{\sin((2k+1)nt)}{\sinh((2k+1)2n\beta)
}$$
we have
$$\PP(t)=\pp\frac{\sin((2k+1)nt-\pi r/2)}\tk,\quad r=0,1,\ldots\,\,.$$
It also follows from (\ref{phi}) that
$$\Phi_{n,0}^\beta(t)=\sqrt\lambda\se.$$
Using the same arguments as for $\prr$, we obtain
$$\|\PP\|_\infty=\pp\frac{(-1)^{k(r+1)}}\tk.$$

Put
$$\tn:=\begin{cases}\dfrac{\pi(j-1)}n,&r=2m,\\
\dfrac{\pi(j-1)}n+\dfrac\pi{2n},&r=2m+1,\end{cases}\quad j=1,\ldots,2n.$$
It is easily seen that $\PP(\tn)=0$, $j=1,\ldots,2n$.

\begin{proposition}\label{P2}
For all $t\in[0,2\pi)$ and $r\ge0$
$$\sup\left\{\,|f(t)|:f\in\HH,\ f(\tn)=0,\ j=1,\ldots,2n\,\right\}=|\PP(t)|.$$
\end{proposition}

\begin{proof}
Suppose there exists a $t^*\in[0,2\pi)$ and a function $f_0\in\HH$ for which
$f_0(\tn)=0$, $j=1,\ldots,2n$, and $|f_0(t^*)|>|\PP(t^*)|$. Set
$$\rho:=\PP(t^*)/f_0(t^*),\quad F:=\PP-\rho f_0.$$
The function $F$ has at least $2n+1$ distinct zeros at the points $\tn$,
$j=1,\ldots,2n$, and $t^*$. By Rolle's Theorem
$$F^{(r)}(t)=\sqrt\lambda\se-\rho f_0^{(r)}(t)$$
has at least $2n+1$ zeros on $[0,2\pi)$.

Denote by $H_\infty(\D)$ the set of functions which are analytic
on the annulus
$$\D:=\left\{\,z\in\mathbb C:e^{-\beta}<|z|<e^\beta\,\right\}$$
and which satisfy the condition $|f(z)|\le1$, $z\in\D$. If $f(z)\in\Ht$,
then $f\left(\dfrac1i\log z\right)\in H_\infty(\D)$. It is easy to check that the
function
$$G(z):=\sqrt\lambda\sn\left(\frac{2n\Lambda}{\pi i}\log z,\lambda\right)$$
has exactly $2n$ zeros on $\D$. Since on $\partial\D$
$$\left|G(z)-F^{(r)}\left(\dfrac1i\log z\right)\right|=\left|\rho f_0^{(r)}
\left(\dfrac1i\log z\right)\right|\le|\rho|<1\equiv|G(z)|,$$
Rouche's Theorem implies that $F^{(r)}(t)$ has $2n$ or fewer zeros on
$[0,2\pi)$. We thus reach a contradiction, which proves the proposition.
\end{proof}

\begin{theorem}
For all integers $r\ge0$
\begin{multline*}
d_{2n}(\HH,L_\infty)=\lambda_{2n}(\HH,L_\infty)=d^{2n}(\HH,L_\infty)\\
=\pp\frac{(-1)^{k(r+1)}}\tk,
\end{multline*}
where $\Lambda$ is the complete elliptic integral of the first kind with
modulus $\lambda$ defined by \eqref{l}.
\end{theorem}

\begin{proof}
The case $r=0$ follows from \cite{Os2} where the equalities
\begin{multline*}
d_{2n}(\Ht,L_q)=\lambda_{2n}(\Ht,L_q)=d^{2n}(\Ht,L_q)=\|\Phi_{n,0}^\beta
\|_q,\\
1\le q\le\infty,
\end{multline*}
were proved. So we shall assume that $r\ge1$.

We shall first prove the lower bound for the Kolmogorov widths. Set
\begin{gather*}
S^{2n}:=\left\{\,x=(x_1,\ldots,x_{2n+1})\in\