Explicit constructions and properties of generalized shift-invariant systems in $L^2(\mathbb{R})$

TL;DR

This paper investigates the properties of generalized shift-invariant systems in $L^2(R)$, revealing new insights into frame bounds, constructing explicit frames and duals, and analyzing the local integrability condition.

Contribution

It provides the first analysis showing the Calderón sum may not be bounded by the lower frame bound in GSI systems, offers explicit constructions of frames and dual pairs, and deepens understanding of the LIC.

Findings

Calderón sum not necessarily bounded by the lower frame bound in GSI frames

Explicit constructions of GSI frames and dual pairs without characteristic functions

Enhanced understanding of the local integrability condition (LIC)

Abstract

Generalized shift-invariant (GSI) systems, originally introduced by Hern\'andez, Labate & Weiss and Ron & Shen, provide a common frame work for analysis of Gabor systems, wavelet systems, wave packet systems, and other types of structured function systems. In this paper we analyze three important aspects of such systems. First, in contrast to the known cases of Gabor frames and wavelet frames, we show that for a GSI system forming a frame, the Calder\'on sum is not necessarily bounded by the lower frame bound. We identify a technical condition implying that the Calder\'on sum is bounded by the lower frame bound and show that under a weak assumption the condition is equivalent with the local integrability condition introduced by Hern\'andez et al. Second, we provide explicit and general constructions of frames and dual pairs of frames having the GSI-structure. In particular, the setup applies to wave packet systems and in contrast to the constructions in the literature, these constructions are not based on characteristic functions in the Fourier domain. Third, our results provide insight into the local integrability condition (LIC).