Article
Ufa Mathematical Journal
Volume 3, Number 1, pp. 16-29
An Analogue of the Paley-Wiener Theorem and its Applications to Optimal Recovery of Entire Functions.
Maergoiz L.S., Tarkhanov N.
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Let $W^p$ be the Wiener class of entire functions of exponential type in $\mathbb{C}^n$ belonging to $L^p(\mathbb{R}^n),$ where $1 < p <\infty$. For the class $W^p$ full analogues of the Paley-Wiener Theorem and (in multidimensional case) the Plancherel- P\'{o}lya Theorem on structure of Fourier transform for any entire function $f \in W^{2}$ are obtained on principle new form -- on distributions language. These results are applied for a solution of the problem of the best analytic continuation from a finite set for functions from the Wiener class. It is given description of existence conditions for constructive algebraic formulas for the optimal recovery characteristics of linear functionals exciting independent interest.