Pairs of dual Gabor frames generated by functions of Hilbert-Schmidt type

Lasse Hjuler Christiansen

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Abstract

We show that any two functions which are real-valued, bounded, compactly supported and whose integer translates each form a partition of unity lead to a pair of windows generating dual Gabor frames for (Formula presented.). In particular we show that any such functions have families of dual windows where each member may be written as a linear combination of integer translates of any B-spline. We introduce functions of Hilbert-Schmidt type along with a new method which allows us to associate to certain such functions finite families of recursively defined dual windows of arbitrary smoothness. As a special case we show that any exponential B-spline has finite families of dual windows, where each member may be conveniently written as a linear combination of another exponential B-spline. Unlike results known from the literature we avoid the usual need for the partition of unity constraint in this case.
Original languageEnglish
JournalAdvances in Computational Mathematics
Volume41
Issue number6
Pages (from-to)1101-1118
ISSN1019-7168
DOIs
Publication statusPublished - 2015

Keywords

  • Gabor frames
  • Dual frame pairs
  • Dual windows
  • Exponential B-splines

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