Translation partitions of unity, symmetry properties, and Gabor frames

Ole Christensen*, Say Song Goh, Hong Oh Kim, Rae Young Kim

*Corresponding author for this work

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We consider the general question of constructing a partition of unity formed by translates of a compactly supported function g : RdC. In particular, we prove that such functions have a special structure that simplifies the construction of partitions of unity with specific properties. We also prove that it is possible to modify the function g in such a way that it becomes symmetric with respect to a given symmetry group on Zd. The results are illustrated with constructions of dual pairs of Gabor frames for L2(Rd). In addition, we obtain general approaches to construct dual Gabor frames whose window functions are symmetric with respect to an arbitrary symmetry group. Through sampling and periodization, these dual Gabor frames for L2(Rd) lead.
Original languageEnglish
Article number49
JournalAdvances in Computational Mathematics
Issue number4
Number of pages26
Publication statusPublished - 2021


  • Translation partitions of unity
  • Symmetry
  • Gabor systems
  • Dual frame pairs


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