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\newcommand{\h}{\tilde h^r_{\infty,\beta}}
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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\mathbb R^{2n+1}:\sum
_{k=1}^{2n+1}|x_k|=2\pi\,\right\},\\
\tau_0(x):=0,\quad\tau_j(x):=\sum_{k=1}^j|x_k|,\quad j=1,\ldots,2n+1.
\end{gather*}
For each $x\in S^{2n}$ put
\begin{align*}
g_x(t)&:=\sign x_j,\quad\tau_{j-1}(x)\le t<\tau_j(x),\quad j=1,\ldots,2n+1,\\
f_x&:=D_r*\varphi_0(K_\beta*g_x).
\end{align*}
Let $X_{2n}$ be any $2n$-dimensional subspace of $L_q$, $1<q<\infty$, such
that $1\in X_{2n}$. Suppose that $X_{2n}=\spa\{f_1,\ldots,f_{2n}\}$ and
$f_1(t)\equiv1$. Let $a_1(x),\ldots,a_{2n}(x)$ be the coefficients of
$f_1,\ldots,f_{2n}$, respectively, in the best approximation to $f_x$ from
$X_{2n}$. The mapping
$$A(x):=(b(x),a_2(x),\ldots,a_{2n}(x)),$$
where
$$b(x):=\int_0^{2\pi}\varphi_0\bigl((K_\beta*g_x)(t)\bigr)\, dt,$$
is a continuous map of $S^{2n}$ into $\mathbb R^{2n}$. By Borsuk's Theorem
there exists an $x^*\in S^{2n}$ for which $A(x^*)=0$. As the function
$\varphi_0(z)=\tan\dfrac\pi4z$ maps the strip $|\RE z|<1$ conformally onto
the open unit disk, for all $x\in S^{2n}$, $f_x\in\HH$. Thus,
\begin{multline}\label{lb}
 \sup_{f\in\HH}\,\infp_{g\in X_{2n}\vphantom{\HH}}\|f-g\|_q\ge\sup_{x\in S^
{2n}}\infp_{g\in X_{2n}}\|f_x-g\|_q\ge\|f_{x^*}-a_1(x^*)\|_q\\
\ge\inf\left\{\,\|a+D_r*\varphi_0(K_\beta*h_\xi)\|_q:a\in\mathbb R,\ \xi\in
\Lambda_{2n}^{\varphi_0}\,\right\}.
\end{multline}
If $1\notin X_{2n}$, then the left-hand side of \eqref{lb} is equal to
$+\infty$.  Hence,
$$d_{2n}(\HH,L_q)\ge\inf\left\{\,\|a+D_r*\varphi_0(K_\beta*h_\xi)\|_q:a\in
\mathbb R,\ \xi\in\Lambda_{2n}^{\varphi_0}\,\right\}.$$
By passing to the limit $q\to\infty$ we obtain
$$d_{2n}(\HH,L_\infty)\ge\|\PP\|_\infty.$$

Let us now prove the lower bound for the Gel'fand widths. Suppose that
$$X^{2n}:=\left\{\,f\in L_\infty:\la l_j,f\ra=0,\ j=1,\ldots,2n,\ l_j\in L_
\infty'\,\right\}.$$
If $\la l_j,1\ra=0$, $j=1,\ldots,2n$, then
$$\sup_{f\in\HH\cap X^{2n}}\|f\|_\infty=\infty.$$
Assume that $\la l_1,1\ra\ne0$. Set
$$L_j:=l_j-\frac{\la l_j,1\ra}{\la l_1,1\ra}l_1,\quad j=2,\ldots,2n.$$
For each $x\in S^{2n}$ denote by $A_1$ the mapping
$$A_1(x):=(b(x),\la L_2,f_x\ra,\ldots,\la L_{2n},f_x\ra).$$
Since $A_1\colon S^{2n}\to\mathbb R^{2n}$ is an odd and continuous map, by
Borsuk's Theorem there exists an $x^*$ for which $A_1(x^*)=0$. Then
$$f^*:=f_{x^*}-\frac{\la l_1,f_{x^*}\ra}{\la l_1,1\ra}\in X^{2n}.$$
Consequently,
\begin{multline*}
\sup_{f\in\HH\cap X^{2n}}\|f\|_\infty\ge\|f^*\|_\infty\\
\ge\inf\left\{\,\|a+D_r*\varphi_0(K_\beta*h_\xi)\|_\infty:a\in\mathbb R,\
\xi\in\Lambda_{2n}^{\varphi_0}\,\right\}
\ge\|\PP\|_\infty.
\end{multline*}
Thus,
$$d^{2n}(\HH,L_\infty)\ge\|\PP\|_\infty.$$

Now let us prove the upper bound for the linear widths. We shall use a
modification of the proof for the nonperiodic case from \cite{Fi}. Denote
by $E_p$ the set of functions $2\pi$-periodic and analytic on $S_\beta$
which satisfy
\begin{align*}
\|f\|_{E_p}&:=\sup_{0<\rho<1}\left(\f|f(\rho z)|^p|dz|\right)^{1/p}<
\infty,\quad1\le p<\infty,\\
\|f\|_{E_\infty}&:=\sup_{z\in S_\beta}|f(z)|<\infty,\quad p=\infty,
\end{align*}
where $\Gamma:=[2\pi+i\beta,i\beta]\cup[-i\beta,2\pi-i\beta]$. If the
$2\pi$-periodic functions $\omega(z),\omega_1(z)\in C(\Gamma)$ then, using
the mapping $z=\dfrac1i\log w$, it can be proved (see \cite{Kh}) that for
all $1<p\le\infty$
\begin{multline}\label{du}
\sup\left\{\,\Bigl|\f f(z)\omega(z)\,dz\Bigr|:\|f\|_{E_p}\le1,\ \int_
\Gamma f(z)\omega_1(z)\,dz=0\,\right\}\\
=\inf\left\{\,\left(\f|\omega(z)-c\omega_1(z)-\chi(z)|^q|dz|\right)^{1/q}
:c\in\mathbb C,\ \chi\in E_q\,\right\}\\
=\|\omega\|_{L_q(\Gamma)/E_{q,1}},
\end{multline}
where $1/p+1/q=1$, $E_{q,1}:={E_q}_{|_\Gamma}+\spa\{\omega_1\}$, and
${E_q}_{|_\Gamma}$ is the space of boundary values on $\Gamma$ of functions
from $E_q$.

By the same mapping $z=\dfrac1i\log w$ and the Residue Theorem we obtain
\begin{equation}\label{re}
f(t)=\frac1{2\pi}\int_\Gamma\frac{f(z)e^{iz}}{e^{iz}-e^{it}}\,dz,\quad t
\in S_\beta,
\end{equation}
for all $f\in E_\infty$. For $x,t_1\in[0,2\pi)$ we define
$$K(z,x):=\frac1\pi\int_0^{2\pi}\frac{D_r(x-t)e^{iz}}{e^{iz}-e^{it}}\,dt,
\quad V(z,x):= K(z,x)-K(z,t_1).$$
From \eqref{re} it follows that all functions $2\pi$-periodic and analytic
on $S_\beta$ whose $r$th derivative lies in $E_\infty$ satisfy the
following equation
\begin{equation}\label{rep}
f(x)-f(t_1)=\f V(z,x)f^{(r)}(z)\,dz.
\end{equation}

Set
$$\omega_1(z):=\begin{cases}1,&z\in[2\pi+i\beta,i\beta],\\
0,&z\in[-i\beta,2\pi-i\beta].\end{cases}$$
Then the condition $f\perp1$ can be written in the form
$$\int_\Gamma f(z)\omega_1(z)\,dz=0.$$
For distinct points $t_1,t_2,\ldots,t_{2n}\in[0,2\pi)$ put
$$D_q(x):=\inf_{c_2,\ldots,c_{2n}}\Bigl\|V(\cdot,x)-\st V(\cdot,t_j)\Bigr\|
_{L_q(\Gamma)/E_{q,1}}.$$
Since $L_q(\Gamma)/E_{q,1}$ is uniformly convex for $1<q<\infty$ there are
continuous functions on $[0,2\pi]$, $c_2(x),\ldots,c_{2n}(x)$, such that
$$D_q(x)=\Bigl\|V(\cdot,x)-\st(x)V(\cdot,t_j)\Bigr\|_{L_q(\Gamma)/E_{q,1}}
.$$
It follows from \eqref{du} and \eqref{rep} that
\begin{multline*}
\sup_{x\in[0,2\pi]}D_q(x)\\
=\sup_{\|f\|_{E_p}\le1,\ f\perp1}\Bigl\|\f\Bigl(V(z,\cdot)-\st(\cdot)V(z,t_j)\Bigr)f(z)\,dz
\Bigr\|_\infty\\
\ge\sup_{\|f^{(r)}\|_{E_\infty}\le1}\Bigl\|\f\Bigl(V(z,\cdot)-\st(\cdot)
V(z,t_j)\Bigr)f^{(r)}(z)\,dz\Bigr\|_\infty\\
\ge\sup_{f\in\HH}\Bigl\|f(\cdot)-f(t_1)-\st(\cdot)(f(t_j)-f(t_1))\Bigr\|
_\infty\ge\lambda_{2n}(\HH,L_\infty).
\end{multline*}
The function $D_q$ is continuous on $[0,2\pi]$, $1\le q<\infty$, and
$D_q\searrow D_1$ uniformly as $q\searrow1$. Letting $q$ decrease to 1, we
obtain
\begin{multline*}
\lambda_{2n}(\HH,L_\infty)\le\sup_{x\in[0,2\pi]}D_1(x)\\
=\Bigl\|\infp_{c_2,\ldots,c_{2n}\vphantom{f^(}}\sup_{\|f^{(r)}\|_{E_
\infty}\le1\vphantom{f^(}}\Bigl|\f\Bigl(V(z,\cdot)-\st V(z,t_j)\Bigr)f^{(r)
}(z)\,dz\Bigr|\Bigr\|_\infty\\
=\sup_{x\in[0,2\pi]}\sigma(x),
\end{multline*}
where
\begin{multline}\label{dua}
\sigma(x):=\sup\left\{\,\Bigl|\f V(z,x)f^{(r)}(z)\,dz\Bigr|:\|f^{(r)}\|_{
E_\infty}\le1,\right.\\
\left.\int_\Gamma V(z,t_j)f^{(r)}(z)\,dz=0,\ j=2,\ldots,2n\,\right\}.
\end{multline}

Using the same methods as in \cite{Kh}, it can be shown that the solution
of \eqref{dua} is unique up to a factor $e^{i\alpha}$, $\alpha\in\mathbb R$. By
\eqref{rep} we have
\begin{multline*}
\sigma(x)=\sup\left\{\,|f(x)-f(t_1)|:\|f^{(r)}\|_{E_\infty}\le1,\
f(t_j)-f(t_1)=0,\right.\\
\left. j=2,\ldots,2n\,\right\}.
\end{multline*}
Therefore, if $f^*(z)$ is a solution of \eqref{dua}, then $\overline{f^*(
\bar z)}$ is also a solution of \eqref{dua}. Consequently, there exists an
extremal function which is real on $\mathbb R$. Thus,
$$\sigma(x)=\infp_{c_2,\ldots,c_{2n}\vphantom{\HH}}\sup_{f\in\HH}\Bigl|f(x)-
f(t_1)-\st(f(t_j)-f(t_1))\Bigr|.$$

Put
$$\sigma_1(x):=\infp_{c_1,\ldots,c_{2n}\vphantom{\HH}}\sup_{f\in\HH}\Bigl|f(
x)-\so f(t_j)\Bigr|.$$
Obviously, $\sigma_1(x)\le\sigma(x)$. On the other hand, if $\so\ne1$, then
$$\sup_{f\in\HH}\Bigl|f(x)-\so f(t_j)\Bigr|\ge\sup_{c\in\mathbb R}\Bigl|c\Bigl(
1-\so\Bigr)\Bigr|=\infty.$$
Hence, $\sigma_1(x)=\sigma(x)$. Now we have
\begin{multline*}
\lambda_{2n}(\HH,L_\infty)\le\sup_{x\in[0,2\pi]}\sigma_1(x)\\
=\sup_{x\in[0,2\pi]}\sup\left\{\,|f(x)|:f\in\HH,\ f(t_j)=0,\ j=1,\ldots,2n
\,\right\}.
\end{multline*}
For $t_j=\tn$, $j=1,\ldots,2n$, it follows from Proposition \ref{P2} that
$$\lambda_{2n}(\HH,L_\infty)\le\|\PP\|_\infty.$$
Since $\lambda_{2n}\ge d_{2n}$ and $\lambda_{2n}\ge d^{2n}$ we obtain
$$d_{2n}(\HH,L_\infty)=\lambda_{2n}(\HH,L_\infty)=d^{2n}(\HH,L_\infty)=\|
\PP\|_\infty.$$
The theorem is proved.
\end{proof}

\noindent{\bf Acknowledgement.} The research was supported in part by Grant 93--01--00237 from Russian Foundation of Fundamental Research and by Grant MP~1300 from the ISF\&RG.



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\end{document}