Generalized Fock space and fractional derivatives with Applications to Uniqueness of Sampling and Interpolation Sets

TL;DR

This paper introduces a generalized Fock space associated with Gelfond-Leontiev derivatives, including fractional derivatives and Dunkl operators, and establishes density theorems for sampling and interpolation within this framework.

Contribution

It develops a new Fock space linked to a broad class of derivatives and proves density theorems for sampling and interpolation, extending classical results to this generalized setting.

Findings

Established a modified Bargmann transform for the new Fock space.

Proved density theorems for sampling and interpolation.

Derived lattice conditions for frame construction from integral transforms.

Abstract

In this paper we introduce a Fock space related to derivatives of Gelfond-Leontiev type, a class of derivatives which includes many classic examples like fractional derivatives or Dunkl operators. For this space we establish a modified Bargmann transform as well as density theorems for sampling and interpolation. These density theorems allow us to establish lattice conditions for the construction of frames arising from integral transforms which are linked by the modified Bargmann transform with the Fock space.