An analogue of the Paley–Wiener theorem and its applications to optimal recovery of entire functions

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 $1<p<\infty$. Full analogues of the Paley–Wiener theorem for the class $W^p$ and, in a multidimensional case, for the Plancherel–Pólya theorem on structure of the Fourier transform for any entire function $f\in W^2$, are obtained in a fundamentally new form in terms of the language of distributions. The results are applied to the problem of the best analytic continuation from a finite set of functions of the Wiener class. Of special interest is the description of the existence conditions for constructive algebraic formulae of characteristics for the optimal recovery of linear functionals.