Gabor windows supported on [−1, 1] and construction of compactly supported dual windows with optimal smoothness

Jakob Lemvig, Kamilla Haahr Nielsen

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Abstract

We consider Gabor frames {e2πibm·g(· − ak)}m,k∈Z with translation parameter a = L/2, modulation parameter b ∈ (0, 2/L) and a window function g ∈ Cn(R) supported on [x0, x0+L] and non-zero on (x0, x0+L) for L > 0 and x0 ∈ R. The set of all dual windows h ∈ L2 (R) with sufficiently small support is parametrized by 1-periodic measurable functions z. Each dual window h is given explicitly in terms of the function z in such a way that desirable properties (e.g., symmetry, boundedness and smoothness) of h are directly linked to z. We derive easily verifiable conditions on the function z that guarantee, in fact, characterize, compactly supported dual windows h with the same smoothness, i.e., h ∈ Cn (R). The construction of dual windows is valid for all values of the smoothness index n ∈ Z≥0 ∪ {∞} and for all values of the modulation parameter b < 2/L; since a = L/2, this allows for arbitrarily small redundancy (ab)−1 > 1. We show that the smoothness of h is optimal, i.e., if g /∈ Cn+1(R) then, in general, a dual window h in Cn+1(R) does not exist.
Original languageEnglish
Article number105304
JournalJournal of Approximation Theory
Volume249
Number of pages27
ISSN0021-9045
DOIs
Publication statusPublished - 2020

Keywords

  • Dual frame
  • Dual window
  • Gabor frame
  • Optimal smoothness
  • Redundancy 2010 MSC
  • Primary 42C15
  • Secondary 42A60

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