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Frames and oversampling formulas for band limited functions

Abstract:
In this article, we obtain families of frames for the space Bω of functions with band in [−ω,ω] by using the theory of shift-invariant spaces. Our results are based on the Gramian analysis of Ron and Shen and a variant, due to Bownik, of their characterization of families of functions whose shifts form frames or Riesz bases. We give necessary and sufficient conditions for the translates of a finite number of functions (generators) to be a frame or a Riesz basis for Bω. We also give explicit formulas for the dual generators, and we apply them to Hilbert transform sampling and derivative sampling. Finally we provide numerical experiments that support the theory.

Keywords:
Frame · Riesz basis · Shift-invariant space · Sampling formula · Band-limited function

MSC:
42, 46