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

Abstract

Generalized shift-invariant (GSI) systems, originally introduced by Hernandez et al. and Ron and 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 Calderon sum is not necessarily bounded by the lower frame bound. We identify a technical condition implying that the Calderon 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 Hernandez 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).