The Zero-Removing Property in Hilbert Spaces of Entire Functions Arising in Sampling Theory

Abstract

In the topic of sampling in reproducing kernel Hilbert spaces, sampling in Paley–Wiener spaces is the paradigmatic example. A natural generalization of Paley–Wiener spaces is obtained by substituting the Fourier kernel with an analytic Hilbert-space-valued kernel K. Thus we obtain a reproducing kernel Hilbert space \({\mathcal{H}_{K}}\) of entire functions in which the Kramer property allows to prove a sampling theorem. A necessary and sufficient condition ensuring that this sampling formula can be written as a Lagrange-type interpolation series concerns the stability under removal of a finite number of zeros of the functions belonging to the space \({\mathcal{H}_{K}}\); this is the so-called zero-removing property. This work is devoted to the study of the zero-removing property in \({\mathcal{H}_{K}}\) spaces, regardless of the Kramer property, revealing its connections with other mathematical fields.