Banach Gabor frames with Hermite functions: polyanalytic spaces from the Heisenberg group

TL;DR

This paper explores Banach Gabor frames with Hermite functions, linking them to sampling in polyanalytic Fock spaces via the polyanalytic Bargmann transform, and extends sampling theorems using coorbit space theory.

Contribution

It introduces Banach spaces of polyanalytic functions and analyzes the polyanalytic Bargmann transform's properties on modulation spaces, deriving explicit sampling theorems.

Findings

Establishes the unitarity of the polyanalytic Bargmann transform in the L^2 case.

Derives explicit polyanalytic sampling theorems for Banach spaces.

Connects Gabor frames with polyanalytic Fock spaces through the Heisenberg group representation.

Abstract

Gabor frames with Hermite functions are equivalent to sampling sequences in true Fock spaces of polyanalytic functions. In the L^2-case, such an equivalence follows from the unitarity of the polyanalytic Bargmann transform. We will introduce Banach spaces of polyanalytic functions and investigate the mapping properties of the polyanalytic Bargmann transform on modulation spaces. By applying the theory of coorbit spaces and localized frames to the Fock representation of the Heisenberg group, we derive explicit polyanalytic sampling theorems which can be seen as a polyanalytic version of the lattice sampling theorem discussed by J. M. Whittaker in Chapter 5 of his book "Interpolatory Function Theory".