(p, q)-Frame measures on LCA groups: Perturbations and Construction

TL;DR

This paper extends the concept of Fourier frames to Banach spaces on LCA groups by introducing (p, q)-frame measures, providing construction methods, analyzing their properties, and studying their stability under perturbations.

Contribution

It introduces (p, q)-frame measures on LCA groups, offers a general construction approach, and investigates their stability and uniformity properties.

Findings

Measures with (p, q)-frame measures exhibit uniform distribution on support.

Certain measure combinations do not admit (p, q)-frame measures.

(p, q)-frame measures are stable under small perturbations.

Abstract

Motivated to generalize the Fourier frame concept to Banach spaces we introduce (p, q)-Bessel/frame measures for a given finite measure on LCA groups. We also present a general way of constructing (p, q)-Bessel/frame measures for a given measure. Moreover, we prove that if a measure has an associated (p, q)-frame measure, then it must have a certain uniformity in the sense that the weight is distributed quite uniformly on its support. Next, we show that if the measures $\mu$ and $\lambda$ without atoms whose supports form a packing pair, then $\mu\ast\lambda+\delta_g\ast\mu$ does not admit any (p, q)-frame measure. Finally, we analyze the stability of (p, q)-frame measures under small perturbations. We prove new theorems concerning the stability of (p, q)-frame measures under perturbation in both Hilbert spaces and Banach spaces.