On Partition of Unities Generated by Entire Functions and Gabor Frames in L2(Rd) and ℓ2(Zd)

Ole Christensen, Hong Oh Kim, Rae Young Kim

Research output: Contribution to journalJournal articleResearchpeer-review


We characterize the entire functions P of d variables, d≥2, for which the Zd-translates of Pχ[0,N]d satisfy the partition of unity for some N∈N. In contrast to the one-dimensional case, these entire functions are not necessarily periodic. In the case where P is a trigonometric polynomial, we characterize the maximal smoothness of Pχ[0,N]d, as well as the function that achieves it. A number of especially attractive constructions are achieved, e.g., of trigonometric polynomials leading to any desired (finite) regularity for a fixed support size. As an application we obtain easy constructions of matrix-generated Gabor frames in L2(Rd), with small support and high smoothness. By sampling this yields dual pairs of finite Gabor frames in ℓ2(Zd).
Original languageEnglish
JournalJournal of Fourier Analysis and Applications
Issue number5
Pages (from-to)1121-1140
Publication statusPublished - 2016


  • Entire functions
  • Trigonometric polynomials
  • Partition of unity
  • Dual frame pairs
  • Gabor systems
  • Tight frames

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