Representations of Almost-Periodic Functions Using Generalized Shift-Invariant Systems in ℝd

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Abstract

The problem of estimating the AP-norm of univariate almost-periodic functions using a Gabor system or a wavelet system was studied by several authors, and culminated in the characterization given in a recent paper of the present author. The present article unifies and generalizes the various efforts via the study of this problem in the context of (multivariate) generalized shift-invariant (GSI) systems. The main result shows that the sought-for norm estimation of the AP functions is valid if and only if the given GSI system is an L2(ℝd)-frame. Moreover, the frame bounds of the system are also the sharpest bounds in our estimation.

Original languageEnglish
Pages (from-to)857-876
Number of pages20
JournalJournal of Fourier Analysis and Applications
Volume19
Issue number4
DOIs
StatePublished - Aug 2013

Keywords

  • Almost-periodic functions
  • Dual Gramian
  • Frames
  • Generalized shift-invariant (GSI) systems
  • Wavelet systems

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