Stability of shifts, interpolation, and crystalline measures

TL;DR

This paper investigates the stability and basis properties of shift-invariant function spaces generated by shifts of a function, establishing conditions on the shift set that relate to interpolation and crystalline measures.

Contribution

It provides necessary and sufficient conditions on the shift set for the shifts of certain generators to form an unconditional basis in the space.

Findings

Characterization of when shifts form an unconditional basis

Conditions linking basis properties to interpolation and crystalline measures

Analysis of stability for various generator functions

Abstract

Let $V^p_\Gamma(\mathcal{G}),1\leq p\leq\infty,$ be the quasi shift-invariant space generated by $\Gamma$-shifts of a function $\mathcal{G}$, where $\Gamma\subset\mathbb{R}$ is a separated set. For several large families of generators $\mathcal{G}$, we present necessary and sufficient conditions on $\Gamma$ that imply that the $\Gamma$-shifts of $\mathcal{G}$ form an unconditional basis for $V^p_\Gamma(\mathcal{G})$. The connection between this property, interpolation, universal interpolation, and crystalline measures is discussed.