Frames and Oversampling Formulas for Band Limited Functions

TL;DR

This paper develops new frame constructions for band-limited functions using shift-invariant space theory, providing explicit formulas for duals, and demonstrating applications in sampling with numerical validation.

Contribution

It introduces novel frame and Riesz basis conditions for band-limited functions, including explicit dual generator formulas and practical sampling applications.

Findings

Established necessary and sufficient conditions for frames in B_ω.

Derived explicit dual generator formulas for sampling.

Validated the theory with numerical experiments.

Abstract

In this article we obtain families of frames for the space B_\omega of functions with band in [-\omega,\omega] by using the theory of shift-invariant spaces. Our results are based on the Gramian analysis of A. Ron and Z. Shen and a variant, due to Bownik, of their characterization of families of functions whose shifts form frames or Riesz bases. We give necessary and sufficient conditions for the translates of a finite number of functions (generators) to be a frame or a Riesz basis for B_\omega. We also give explicit formulas for the dual generators and we apply them to Hilbert transform sampling and derivative sampling. Finally, we provide numerical experiments which support the theory.