Interpolation Formulas With Derivatives in De Branges Spaces

TL;DR

This paper extends an interpolation formula involving derivatives for entire functions of exponential type to general L^p de Branges spaces, utilizing de Branges' Hilbert space theory and Hilbert-type inequalities, with applications to Bessel functions and band-limited approximations.

Contribution

It generalizes Vaaler's interpolation formula to L^p de Branges spaces and applies it to problems involving Bessel functions and extremal approximations.

Findings

Extended interpolation formula to L^p de Branges spaces.

Derived applications involving Bessel functions.

Proved a uniqueness result for band-limited approximations.

Abstract

The purpose of this paper is to prove an interpolation formula involving derivatives for entire functions of exponential type. We extend the interpolation formula derived by J. Vaaler in [37, Theorem 9] to general $L^p$ de Branges spaces. We extensively use techniques from de Branges' theory of Hilbert spaces of entire functions as developed in [6], but a crucial passage involves the Hilbert-type inequalities as derived in [15]. We give applications to homogeneous spaces of entire functions that involve Bessel functions and we prove a uniqueness result for extremal one-sided band-limited approximations of radial functions in Euclidean spaces.