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

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