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\begin{document}

\title[Optimal recovery of periodic functions]{Optimal recovery of periodic functions from Fourier coefficients
given with an error}
\author{K. Yu. Osipenko}
\thanks{The research was supported in part by Russian Foundation of
Fundamental Research (Grant 93--01--00237) and by the International
Scientific Fund (Grant MR 1000)}
\address{Department of Mathematics, Moscow State University of Aviation
Technology, Moscow, Russia, 103767}
\begin{abstract}
We construct optimal methods of recovery of $2\pi$-periodic functions analytic in a strip and its derivatives at a point $t\in[0,2\pi)$, using information about the Fourier coefficients given with an error in the uniform norm. The same problem is solved for the Sobolev space $\widetilde W_2^r$. \end{abstract}

\maketitle

\section{Introduction}

Let $X$ and $Y$ be linear spaces over the field $K=\mathbb R$ or $\mathbb C$, $W\subset X$ and $U\subset Y$ balanced convex sets and $I\:W\to Y$ a linear
operator. Denote by $X'$ the set of all linear functionals on $X$. We
consider the problem of optimal recovery of $<x',x>$ where $x'\in X'$ and $
x\in W$, using information about approximate values of the operator $I$. A
method of recovery is any function $\varphi\:Y\to K$. The value
\begin{equation}\label{1}
e(x',I,W,U):=\infp_\varphi\sup_{x\in W}\sup_{\substack{y\in Y\\Ix-y\in U}}|<x
',x>-\varphi(y)|
\end{equation}
is called the intrinsic error in the recovery problem. Any $\varphi_0$ for
which
$$e(x',I,W,U)=\sup_{x\in W}\sup_{\substack{y\in Y\\Ix-y\in U}}|<x',x>-\varphi_0
(y)|$$
is said to be an optimal method.

Many examples and other settings of optimal recovery problems can be found
in \cite{1}--\cite{6}. It follows from Magaril-Il'yaev and Osipenko \cite{6} that there is
a linear optimal method $\varphi_0(y)=<y',y>$, $y'\in Y'$, and the
following equality
\begin{equation}\label{2}
e(x',I,W,U)=\sup_{\substack{x\in W\\Ix\in U}}|<x',x>|
\end{equation}
holds. On the other hand, since $U$ and $W$ are balanced, we have
\begin{multline}\label{3}
e(x',I,W,U)=\infp_{y'\in Y'}\sup_{\substack{x\in W\\z\in U}}|<x',x>
-<y',Ix-z>|\\
=\infp_{y'\in Y'}\left(\sup_{x\in W}|<x',x>-<y',Ix>|+\sup_{z\in U}|<y',z>|
\right).
\end{multline}

In this paper, we consider the problem of optimal recovery of $2\pi
$-periodic functions analytic in a strip and its derivatives from the
Hardy--Sobolev and Bergman--Sobolev spaces based on the information about
Fourier coefficients given with an error in the uniform norm. We also
obtain an optimal method of recovery in the analogous problem for the
Sobolev space $\widetilde W_2^r$.

A similar problem for the estimation of functions in the $L_2$-norm was
considered in Melkman and Micchelli~\cite{4}. The case when the $l_2$-norm is
used to measure the error in the Fourier coefficients was analyzed by
Micchelli and Rivlin~\cite{1}. In Boyanov~\cite{7} the problem of optimal recovery of
periodic functions from the Sobolev space $\widetilde W_q^r$, $1\le q\le
\infty$, was solved for the case when the Fourier coefficients are known
exactly.

\section{Optimal recovery in Hilbert spaces from inaccurate Fourier
coefficients}

Let $X$ be a Hilbert space and $e_1,e_2,\ldots$ a complete orthonormal
system in $X$. For $x\in X$ denote by $x_j:=(x,e_j)$ the Fourier
coefficients of $x$. Consider the problem~\eqref1 for $W=BX:=\{\,x\in X:\|
x\|\le1\,\}$, $<x',x>=(x,f)$, $f\in X$, $|f_j|>0$, $j=1,2,\ldots$, $Ix=(x_1,
\dots,x_n)$ and
$$U=\{\,y=(y_1,\ldots,y_n):|y_j|\le\delta_j,\ j=1,\ldots,n\,\}.$$
Thus we consider the problem of optimal recovery of the linear functional $
(x,f)$ from approximate Fourier coefficients $(\tilde x_1,\ldots,\tilde x_n)
$ such that
$$|x_j-\tilde x_j|\le\delta_j,\quad j=1,\ldots,n.$$
In this case the intrinsic error will be denoted by $e(f,I,BX,\delta)$.

For $a\in\mathbb R$ put
$$a_+:=\begin{cases}a,&a>0,\\
0,&a\le0.\end{cases}$$

\begin{theorem}\label{T1} 
Let $\lambda\in(0,\|f\|]$ be a solution of the equation
\begin{equation}\label{4}
\|f\|^2-\sum_{j=1}^n\left(|f_j|^2-\lambda^2\delta_j^2\right)_+-\lambda^2=
0.
\end{equation}
Then
\begin{equation}\label{5}
(x,f)\approx\sum_{j=1}^n\left(1-\lambda\delta_j|f_j|^{-1}\right)_+
\overline f_j\tilde x_j
\end{equation}
is an optimal method of recovery and
$$e(f,I,BX,\delta)=\lambda+\sum_{j=1}^n\delta_j\left(|f_j|-\lambda\delta_j
\right)_+.$$
\end{theorem}

\begin{proof} 
First we show that the equation \eqref4 has a solution $\lambda\in(0,\|f\|]$. Denote by $\varphi(\lambda)$ the function on the left hand side of \eqref4. This function is continuous for all $\lambda\ge0$.
Moreover,
$$\varphi(0)=\|f\|^2-\sum_{j=1}^n|f_j|^2>0.$$
Since $\varphi(\|f\|)<0$, there exists a $\lambda\in(0,\|f\|]$ which is a
solution of \eqref4.

For such $\lambda$ consider the method \eqref5. In view of \eqref3 we
have
\begin{multline*}
e(f,I,BX,\delta)\le\sup_{x\in BX}\left|(x,f)-\sum_{j=1}^
n\left(1-\lambda\delta_j|f_j|^{-1}\right)_+\overline f_jx_j\right|\\
+\sum_{j=1}^n\delta_j|f_j|\left(1-\lambda\delta_j|f_j|^{-1}\right)_+=\sup_{
x\in BX}(x,f_\lambda)+\sum_{j=1}^n\delta_j\left(|f_j|-\lambda\delta_j\right
)_+
\end{multline*}
where
$$(f_\lambda)_j=\begin{cases}f_j,&j\ge n+1,\\
f_j-f_j\left(1-\lambda\delta_j|f_j|^{-1}\right)_+,&1\le j\le n.\end{cases}$$
It can be easily shown that
$$\|f_\lambda\|^2=\|f\|^2-\sum_{j=1}^n\left(|f_j|^2-\lambda^2\delta_j^2
\right)_+=\lambda^2.$$
Consequently
$$e(f,I,BX,\delta)\le\lambda+\sum_{j=1}^n\delta_j\left(|f_j|-\lambda\delta_
j\right)_+.$$

Put
$$x_0:=\frac{f_\lambda}{\|f_\lambda\|}=\lambda^{-1}f_\lambda.$$
Let $1\le j\le n$. If $1-\lambda\delta_j|f_j|^{-1}>0$ then
$$|(x_0)_j|=\lambda^{-1}|(f_\lambda)_j|=\delta_j.$$
If $1-\lambda\delta_j|f_j|^{-1}\le0$ then
$$|(x_0)_j|=\lambda^{-1}|f_j|\le\delta_j.$$
Thus $Ix_0\in U$. Using \eqref2 we obtain
\begin{multline*}
e(f,I,BX,\delta)\ge|(x_0,f)|=\lambda^{-1}\left(\|f\|^2-\sum_{j=1}^n|f_j|^2\left(1-\lambda
\delta_j|f_j|^{-1}\right)_+\right)\\
=\lambda^{-1}\left(\|f\|^2-\sum_{j=1}^n\left(|f_j|+\lambda\delta_j
\right)\left(|f_j|-\lambda\delta_j\right)_++\lambda\sum_{j=1}^n\delta_j\left(|f_j|-
\lambda\delta_j\right)_+\right)\\
=\lambda+\sum_{j=1}^n\delta_j\left(|f_j|-\lambda\delta_j\right)_+.
\end{multline*}
This completes the proof of the theorem.
\end{proof}

Now let $\delta_j=\delta\lambda_j$, $\lambda_j>0$, $j=1,\ldots,n$, and $\delta\ge0$.

\begin{theorem}\label{T2}  
Suppose that
$$|f_1|\lambda_1^{-1}\ge\ldots\ge|f_n|\lambda_n^{-1}.$$
Set
$$\mu_k:=\left(\sum_{j=1}^k\lambda_j^2+|f_k|^{-2}\lambda_k^2
\sum_{j=k+1}^\infty|f_j|^2\right)^{-1/2},\quad k=1,\ldots,n,$$
$\mu_0:=+\infty$, $\mu_{n+1}:=0$, and $\Delta_k:=[\mu_{k+1},\mu_k)$, $k=0,
\ldots,n$. Then for $\delta\in\Delta_k$, $0\le k\le n$, the method
$$(x,f)\approx\sum_{j=1}^k\left(1-\delta\frac{\lambda_j}{|f_j|}\sqrt{\frac{
\sum_{j=k+1}^\infty|f_j|^2}{1-\delta^2\sum_{j=1}^k\lambda_j^2}}\,\right)
\overline f_j\tilde x_j$$
is optimal and
$$e(f,I,BX,\delta)=\sqrt{\sum_{j=k+1}^\infty|f_j|^2}\sqrt{1-\delta^
2\sum_{j=1}^k\lambda_j^2}+\delta\sum_{j=1}^k\lambda_j|f_j|.$$
\end{theorem}

\begin{proof} 
The equation \eqref4 now takes the following form
\begin{equation}\label{6}
\|f\|^2-\sum_{j=1}^n\left(|f_j|^2-\lambda^2\delta^2\lambda_j^2\right)_+-
\lambda^2=0.
\end{equation}
If $\delta=0$, then the solution of \eqref6 is evident and the theorem
follows from Theorem~\ref{T1} immediately. If $\delta>0$, then \eqref6 is
equivalent to the equation
\begin{equation}\label{7}
\frac {c^2}{\|f\|^2-\sum_{j=1}^n\left(|f_j|^2-c^2\lambda_j^2\right)_+}=
\delta^2
\end{equation}
where $c=\lambda\delta$. Denote by $\varphi(c)$ the function on the left
hand side of \eqref7. It is easy to show that $\varphi(c)$ is
monotonically increasing for $c\ge0$. Furthermore,
$$\varphi\left(|f_k|\lambda_k^{-1}\right)=\mu_k^2,\quad k=1,\dots,n.$$
Hence for $\delta\in\Delta_k$, $0\le k\le n$,
$$c=\delta\sqrt{\frac{\sum_{j=k+1}^\infty|f_j|^2}{1-\delta^2\sum_{j=1}^k
\lambda_j^2}}$$
is the solution of \eqref7. Now the theorem follows from Theorem~\ref{T1}.
\end{proof}

For $\lambda_1=\ldots=\lambda_n=1$, Theorem~\ref{T2} was proved in \cite8 using more
complicated arguments.

Denote by $L$ the linear space of vectors $x=(x_1,x_2,\ldots)$, $x_j\in\mathbb
C$, which satisfy the condition
$$\sum_{j=1}^\infty\gamma_j|x_j|^2<\infty$$
where $\gamma_1\ge0$ and $\gamma_j>0$, $j>1$. Let $x'$ be the linear
functional on $L$ defined by the following equality
$$\langle x',x\rangle:=\sum_{j=1}^\infty x_j\overline f_j$$
where $|f_j|>0$, $j>1$, and
$$\sum_{j=2}^\infty\gamma_j^{-1}|f_j|^2<\infty.$$

Consider the problem of optimal recovery of the functional $x'$ on the set
$BL:=\{\,x\in L:\sum_{j=1}^\infty\gamma_j|x_j|^2\le1\,\}$ from information
$(\tilde x_1,\ldots,\tilde x_n)$ such that
$$|x_j-\tilde x_j|\le\delta\lambda_j,\quad\lambda_j>0,\quad j=1,\ldots,n.$$

Put
$$m:=\begin{cases}1,&\gamma_1f_1\ne0,\\
2,&\gamma_1f_1=0.\end{cases}$$

\begin{theorem}\label{T3} 
Suppose that
\begin{equation}\label{8}
\frac{|f_m|}{\lambda_m\gamma_m}\ge\ldots\ge\frac{|f_n|}{\lambda_n\gamma_n}.
\end{equation}
Set
$$\mu_{km}:=\left(\sum_{j=m}^k\gamma_j\lambda_j^2+\gamma_k^2|f_k|^{
-2}\lambda_k^2\sum_{j=k+1}^\infty\gamma_j^{-1}|f_j|^2\right)^{-1/2}
,\quad k=m,\ldots,n,$$
$\mu_{m-1,m}:=+\infty$, $\mu_{n+1,m}:=0$, and $\Delta_{km}:=[\mu_{k+1,m},
\mu_{km})$, $k=m-1,\ldots,n$. Then for $\delta\in\Delta_{km}$, $m-1\le k\le
n$, the method
\begin{equation}\label{9}
\langle x',x\rangle\approx(m-1)\overline f_1\tilde x_1+\sum_{j=m}^k\nu_{j
m}\overline f_j\tilde x_j,
\end{equation}
where
$$\nu_{jm}=1-\delta\frac{\gamma_j\lambda_j}{|f_j|}\sqrt{\frac{\sum_{j=k+1}^
\infty\gamma_j^{-1}|f_j|^2}{1-\delta^2\sum_{j=m}^k\gamma_j\lambda_j^2}},$$
is optimal and
$$e(x',I,BL,\delta)=\sqrt{\sum_{j=k+1}^\infty\gamma_j^{-1}|f_j|^2}\sqrt{1-
\delta^2\sum_{j=m}^k\gamma_j\lambda_j^2}+\delta\sum_{j=m}^k\lambda_j|f_j|.$$
\end{theorem}

\begin{proof} 
Consider the case $m=1$. Then $L$ is a Hilbert space with the inner product
$$(x,y)_L:=\sum_{j=1}^\infty\gamma_jx_j\overline y_j.$$
The vectors $e_1,e_2,\ldots$, 
$$(e_j)_s:=\begin{cases}0,&s\ne j,\\
\gamma_j^{-1/2},&s=j,\end{cases}$$
form a complete orthonormal basis in $L$. The Fourier coefficients of $x$
are equal to $(x,e_j)=\sqrt{\gamma_j}x_j$. Now we can use Theorem~\ref{T2} in
which we have to replace $\lambda_j$ and $f_j$ by $\gamma_j^{1/2}\lambda_j$
and $\gamma_j^{-1/2}f_j$, respectively.

Suppose that $\gamma_1=0$. Denote by $L_0$ the space of all vectors $x\in L$ for which $x_1=0$. The space $L_0$ is a Hilbert space with the inner product
$$(x,y)_{L_0}:=\sum_{j=2}^\infty\gamma_jx_j\overline y_j.$$
From Theorem~\ref{T2} it follows that the method
$$\langle x',x\rangle\approx\sum_{j=2}^k\nu_{jm}\overline f_j\tilde x_j$$
is optimal for the set $BL_0$, and
$$e(x',I,BL_0,\delta)=\sqrt{\sum_{j=k+1}^\infty\gamma_j^{-1}|f_j|^2}\sqrt{1
-\delta^2\sum_{j=2}^k\gamma_j\lambda_j^2}+\delta\sum_{j=2}^k\lambda_j|f_j|.$$
From \eqref2 we have
\begin{equation}\label{10}
e(x',I,BL,\delta)=\delta\lambda_1|f_1|+e(x',I,BL_0,\delta).
\end{equation}
On the other hand, from \eqref3 it follows that for the method \eqref9
\begin{multline*}
e(x',I,BL,\delta)\le\sup_{x\in BL_0}\left|\langle x',x\rangle-
\sum_{j=2}^k\nu_{jm}\overline f_jx_j\right|+\delta\lambda_1|f_1|\\
+\sup_{|z_j|\le\delta\lambda_j}\left|\sum_{j=2}^k\nu_{jm}\overline f
_jz_j\right|=\delta\lambda_1|f_1|+e(x',I,BL_0,\delta).
\end{multline*}
In view of \eqref{10} the method \eqref9 is optimal for the set $BL$.

Now assume that $f_1=0$. Since from \eqref2
$$e(x',I,BL,\delta)=e(x',I,BL_0,\delta),$$
it suffices to construct an optimal method for the set $BL_0$. It can be
immediately obtained from Theorem~\ref2. The theorem is proved.
\end{proof} 

\section{Optimal recovery in Hardy--Sobolev and Bergman--Sobolev spaces}

Let $W$ be a shift invariant class of sufficiently smooth and $2\pi$-periodic functions. Consider the problem of optimal recovery of $f^{(s)}(t)$, $t\in[0,2\pi)$, $f\in W$, using information about the Fourier coefficients
$$c_k=\frac1{2\pi}\int_0^{2\pi}f(t)e^{-ikt}\,dt,\quad|k|\le n,$$
given with error at most $\delta$ in the uniform norm, i.e., by $\tilde c_k$ such that
$$|c_k-\tilde c_k|\le\delta,\quad|k|\le n.$$
Denote by $e_{ns}(W,\delta)$ the intrinsic error for this problem (from \eqref2 it follows that it does not depend on $t$).

Let $\h$ be the space of all $2\pi$-periodic functions analytic in the strip $S_\beta:=\{\,z\in{\mathbb C}:|\IM z|<\beta\,\}$ which satisfy the condition
$$\|f\|_{\h}:=\sup_{0\le\eta<\beta}\left(\frac1{4\pi}\int_0^{2\pi}\left(|f(
t+i\eta)|^2+|f(t-i\eta)|^2\right)\,dt\right)^{1/2}<\infty.$$
The Hardy--Sobolev space $\hh$ is the set of all $2\pi$-periodic functions analytic in the strip $S_\beta$ for which $f^{(r)}\in\h$. Set
$$B\hh:=\left\{\,f\in\hh:\|f^{(r)}\|_{\h}\le1\,\right\},\quad r=0,1,\ldots\,\,.$$

Functions from $\h$ have finite boundary values almost everywhere and the space $\h$ can be considered as a Hilbert space with the inner product
$$(f,g)_{\h}:=\frac1{4\pi}\int_0^{2\pi}\left(f(t+i\beta)\overline{g(t+i
\beta)}+f(t-i\beta)\overline{g(t-i\beta)}\right)\,dt.$$
It is easy to verify that the functions $e_j(z):=e^{ijz}$, $j=0,\pm1,\ldots$
form a complete orthogonal basis in $\h$ and $\|e_j\|_{\h}^2=\cosh2j\beta$.
Thus $f\in B\hh$ iff
$$f(z)=\sum_{j=-\infty}^{+\infty}c_je^{ijz}$$
and
$$\sum_{j=-\infty}^{+\infty}|c_j|^2j^{2r}\cosh2j\beta\le1.$$

For $p=\{p_j\}_{-\infty}^{+\infty}$, $p_j>0$, we introduce the following notation
\begin{gather*}
\mu_{kr}(p,s):=\left(\sum_{|j|\le k}j^{2r}p_j+k^{4r-2s}p_k^2\sum_{
|j|>k}j^{2(s-r)}p_j^{-1}\right)^{-1/2},\quad1\le k\le n,\\
\mu_{n+1,r}(p,s):=0,\quad\mu_{00}(p,s):=\left(\sum_{|j|\ge0}p_j^{-1}\right)
^{-1/2},\\
\mu_{0r}(p,s):=+\infty,\ r\ge1,\\
\Delta_{kr}(p,s):=\big[\mu_{k+1,r}(p,s),\mu_{kr}(p,s)\big),\quad0\le k\le n
,\ r\ge0,\\
\Delta_{-1,0}(p,s):=\big[\mu_{00}(p,s),+\infty\big).
\end{gather*}

Using Theorem~\ref3 with $\overline f_j=(ij)^se^{ijt}$, $\lambda_j=1$ and $\gamma_j=j^{2r}\cosh2j\beta$, we obtain the following result.

\begin{theorem}\label{T4} 
Let $r$ and $s$ be nonnegative integers such that $0\le s\le2r$. Put $p_j=\cosh2j\beta$, $j=0,\pm1,\ldots\,\,$. For $\delta\in\Delta_{kr}(p,s)$ the method
\begin{equation}\label{11}
f^{(s)}(t)\approx\sum_{|j|\le k}\nu_{jk}(p,s,\delta)\tilde c_j(ij)^se^{ijt},
\end{equation}
where
$$\nu_{jk}(p,s,\delta)=1-\delta|j|^{2r-s}p_j\sqrt{\frac{\sum_{|j|>k}j^{2(s-
r)}p_j^{-1}}{1-\delta^2\sum_{|j|\le k}j^{2r}p_j}},$$
is optimal for the class $B\hh$, and
\begin{multline*}
e_{ns}(B\hh,\delta)=E_{kr}(p,s,\delta):=\sqrt{\sum_{|j|>k}j^{2(s-
r)}p_j^{-1}}\sqrt{1-\delta^2\sum_{|j|\le k}j^{2r}p_j}\\
+\delta\sum_{|j|\le k}|j|^s.
\end{multline*}
\end{theorem}

Denote by $\aA$ the space of all $2\pi$-periodic functions analytic in the strip $S_\beta$ which satisfy the condition
$$\|f\|_{\aA}:=\left(\frac1{4\pi\beta}\int_0^{2\pi}\!\int_{-\beta}^\beta|f(t
+i\eta)|^2\,dtd\eta\right)^{1/2}<\infty.$$
The Bergman--Sobolev space $\aaA$ is the set of all $2\pi$-periodic functions analytic in the strip $S_\beta$ for which $f^{(r)}\in\aA$. Set
$$B\aaA:=\left\{\,f\in\aaA:\|f^{(r)}\|_{\aA}\le1\,\right\},\quad r=0,1,\ldots\,\,.$$
Consider the problem of optimal recovery of $f^{(s)}(t)$ for the class $B\aaA$.

$\aA$ is a Hilbert space with the inner product
$$(f,g)_{\aA}:=\frac1{4\pi\beta}\int_0^{2\pi}\!\int_{-\beta}^\beta f(t+i\eta
)\overline{g(t+i\eta)}\,dtd\eta.$$
It can be easily shown that the functions $e_j(z)$, $j=0,\pm1,\ldots$ form a
complete orthogonal basis in $\aA$ and
$$\|e_0\|_{\aA}=1,\quad\|e_j\|_{\aA}^2=\frac{\sinh2j\beta}{2j\beta},\quad j=
\pm1,\pm2,\ldots\,\,.$$
Therefore $f\in B\aaA$ iff
$$f(z)=\sum_{j=-\infty}^{+\infty}c_je^{ijz}$$
and
$$\sum_{j=-\infty}^{+\infty}|c_j|^2j^{2r}\|e_j\|_{\aA}^2\le1.$$
Analogously to Theorem~\ref{T4} we have

\begin{theorem}\label{T5} 
Let $0\le s\le2r$. Put
$$p_0=1,\quad p_j=\frac{\sinh2j\beta}{2j\beta},\quad j=\pm1,\pm2,\ldots\,\,.$$
For $\delta\in\Delta_{kr}(p,s)$ the method \eqref{11} is an optimal method
for the class $B\aaA$ and 
$$e_{ns}(B\aaA,\delta)=E_{kr}(p,s,\delta).$$
\end{theorem}

{\it Remark.} We need the condition $0\le s\le2r$ to satisfy \eqref8.
For $s>2r$ optimal methods of recovery for the classes $B\hh$ and $B\aaA$
can be constructed by Theorem~\ref{T1}.

Almost the same arguments as in Theorem~\ref{T4} and Theorem~\ref{T5} enable us to obtain
an optimal method of recovery of $f^{(s)}(t)$, $0\le s\le r-1$, for the Sobolev class $B\widetilde W_2^r$ which is the set of all real-valued $2\pi$-periodic functions such that $f^{(r-1)}$ is absolutely continuous and
$$\frac1{2\pi}\int_0^{2\pi}|f^{(r)}(t)|^2\,dt\le1.$$

\begin{theorem}\label{T6} 
Let $0\le s\le r-1$. Put $p_j=1$, $j=0,\pm1,\ldots\,\,$. For $\delta\in\Delta_{kr}(p,s)$ the method \eqref{11} is an optimal method for the class $B\widetilde W_2^r$ and
$$e_{ns}(B\widetilde W_2^r,\delta)=E_{kr}(p,s,\delta).$$
\end{theorem}


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\end{document}