A Generalization of the Kravchenko–Kotelnikov Theorem by Spectra of Compactly Supported Infinitely Differentiable Functions \(h_{a}^{{(m)}}(x)\)

Abstract

A new generalization of the Kravchenko–Kotelnikov theorem by spectra of compactly supported infinitely differentiable functions \(h_{{\mathbf{a}}}^{{(m)}}(x)\) is considered. These functions are solutions of linear integral equations of a special form. The spectrum of \(h_{{\mathbf{a}}}^{{(m)}}(x)\) is a multiple infinite product of the spectra of the atomic functions \({{h}_{a}}(x)\) dilated with respect to the argument. The resulting generalized series is characterized by fast convergence, which is confirmed by the truncation error bound presented in the study and by the results of a numerical experiment.