Flexible Gabor-wavelet atomic decompositions for L 2-Sobolev spaces

Hans G. Feichtinger, Massimo Fornasier

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17 Scopus citations


In this paper we present a general construction of frames, which allows one to ensure that certain families of functions (atoms) obtained by a suitable combination of translation, modulation, and dilation will form Banach frames for the family of L 2-Sobolev spaces on ℝ of any order. In this construction a parameter α [0,1) governs the dependence of the dilation factor on the frequency parameter. The well-known Gabor and wavelet frames (also valid for the same scale of Hilbert spaces) using suitable Schwartz functions as building blocks arise as special cases (α=0) and a limiting case (α→1), respectively. In contrast to those limiting cases, it is no longer possible to use group-theoretical arguments. Nevertheless, we will show how to constructively ensure that for Schwartz analyzing atoms and any sufficiently dense but discrete and well-structured family of parameters one can guarantee the frame property. As a consequence of this novel constructive technique, one can generate quasicoherent dual frames by an iterative algorithm. As will be shown in a subsequent paper, the new frames introduced here generate Banach frames for corresponding families of α-modulation spaces.

Original languageEnglish
Pages (from-to)105-131
Number of pages27
JournalAnnali di Matematica Pura ed Applicata
Issue number1
StatePublished - Feb 2006
Externally publishedYes


  • Continuous/discrete frames
  • Gabor and wavelet frames
  • Non-orthogonal expansions
  • Sobolev spaces
  • α-modulation spaces


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