Zak transform characterization of S0

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We present a characterization of the modulation space So in terms of the Zak transform of its elements. We illustrate our result by considering Sh, where h is the standard Gaussian, S is the "frame" operator corresponding to the critical-density Gabor system (h, a = 1, b = 1), and λ ∈ [0,3/2). Both the proof of the main result and the example require basics from Gabor frame theory; these are developed in a separate section. We further use a result from recent work by Gröchenig and Leinert on Wiener-type theorems in a non-commutative setting. We also present an extension of our main result to more general modulation spaces.

Original languageEnglish
Pages (from-to)141-162
Number of pages22
JournalSampling Theory in Signal and Image Processing
Issue number2
Publication statusPublished - 1 May 2006
Externally publishedYes


  • Critical density
  • Feichtinger space S
  • Gabor system
  • Modulation space
  • Sampled short-time Fourier transform
  • Zak transform


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