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\title{Optimal recovery of analytic functions from their Fourier
coefficients given with an error}
\author{K. Yu. Osipenko (Moscow)}

\maketitle

Let $H$ be a Hilbert space and $\{e_j\}$ a complete orthonormal system in $H$. We
consider the problem of optimal recovery of the linear functional $(x,f)$, $f\in H$,
from approximate values of Fourier coefficients $x_j=(x,e_j)$.

Put
$$e_n(f,\delta):=\infp_{\varphi\colon\mathbb C^n\to\mathbb C}\sup_{\substack{x\in H\\\|x\|\le1}}\,\sup_{\substack{\widetilde x_j,\ j=1,\ldots,n\\|\widetilde x_j-x_j|\le\delta_j}}|(x,f)-\varphi(\widetilde x_1,\ldots,\widetilde x_n)|.$$
Set
$$a_+:=\begin{cases}a,&a\ge0,\\
0,&a<0.\end{cases}$$

\begin{theorem}
The method
$$(x,f)\approx\sum_{j=1}^n(1-\lambda\delta_j|f_j|^{-1})_+\overline f_j\widetilde x_j$$
is an optimal method of recovery and
$$e_n(f,\delta)=\lambda+\sum_{j=1}^n\delta_j(|f_j|-\lambda\delta_j)_+,$$
where $\lambda\in(0,\|f\|]$ is a solution of the equation
$$\|f\|^2-\sum_{j=1}^n(|f_j|^2-\lambda\delta_j^2)_+-\lambda^2=0.$$
\end{theorem}

Using this Theorem we construct optimal methods of recovery of $2\pi$-periodic and
analytic in a strip functions and its derivatives from approximate values of their
Fourier coefficients.

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