On Partition of Unities Generated by Entire Functions and Gabor Frames in L2(Rd) and 2(Zd)
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- On Partition of Unities Generated by Entire Functions and Gabor Frames in L2(Rd) and 2(Zd)
- Christensen, Ole; Kim, Hongoh; Kim, Rae Young
- Issue Date
- JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS, v.22, no.5, pp.1121 - 1140
- 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).
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