Recovering functions from the Paley-Wiener amalgam space

TL;DR

This paper demonstrates that functions in the Paley-Wiener amalgam space can be reconstructed using a family of interpolators and sequences, extending classical recovery methods to a broader function space.

Contribution

It introduces a novel recovery framework for Paley-Wiener amalgam space functions, paralleling classical Paley-Wiener space techniques.

Findings

Functions in PW,l^1 have similar recovery properties as classical Paley-Wiener functions.

A family of interpolators and sequences can be used for function reconstruction.

The method extends classical sampling theory to the amalgam space.

Abstract

In this paper we show that functions from the Paley-Wiener amalgam space $(PW,l^1)=\{f\in L^2(\mathbb{R}): \sum\|\hat{f}(\xi+2\pi m) \|_{L^2([-\pi,\pi])} < \infty\}$ enjoy similar recovery properties as the classical Paley-Wiener space. Specifically, if $\{\phi_\alpha(x): \alpha\in A\}$ is a regular family of interpolators and $\{x_n: n\in \mathbb{Z}\}$ is a complete interpolating sequence for $L^2([-\pi,\pi])$, then the family $\{ e^{2\pi i m x}\phi_{\alpha}(x-x_n): m,n\in \mathbb{Z}, \alpha\in A \} $ may be used to recover $f\in(PW,l^1)$.