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15 April 2013 Gabor frames and totally positive functions
Karlheinz Gröchenig, Joachim Stöckler
Duke Math. J. 162(6): 1003-1031 (15 April 2013). DOI: 10.1215/00127094-2141944

Abstract

Let g be a totally positive function of finite type, that is, ĝ(ξ)=ν=1M(1+2πiδνξ)1 for δνR, δν0, and M2, and let α,β>0. Then the set {e2πiβltg(tαk):k,lZ} is a frame for L2(R) if and only if αβ<1. This result is a first positive contribution to a conjecture of Daubechies from 1990. Until now, the complete characterization of lattice parameters α, β that generate a frame has been known for only six window functions g. Our main result now yields an uncountable class of window functions. As a by-product of the proof method, we also derive new sampling theorems in shift-invariant spaces and obtain the correct Nyquist rate.

Citation

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Karlheinz Gröchenig. Joachim Stöckler. "Gabor frames and totally positive functions." Duke Math. J. 162 (6) 1003 - 1031, 15 April 2013. https://doi.org/10.1215/00127094-2141944

Information

Published: 15 April 2013
First available in Project Euclid: 22 April 2013

zbMATH: 1277.42037
MathSciNet: MR3053565
Digital Object Identifier: 10.1215/00127094-2141944

Subjects:
Primary: 42C15
Secondary: ‎42C40, 94A20

Rights: Copyright © 2013 Duke University Press

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Vol.162 • No. 6 • 15 April 2013
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