Duke Mathematical Journal

Gabor frames and totally positive functions

Karlheinz Gröchenig and Joachim Stöckler

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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.

Article information

Source
Duke Math. J., Volume 162, Number 6 (2013), 1003-1031.

Dates
First available in Project Euclid: 22 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1366639398

Digital Object Identifier
doi:10.1215/00127094-2141944

Mathematical Reviews number (MathSciNet)
MR3053565

Zentralblatt MATH identifier
1277.42037

Subjects
Primary: 42C15: General harmonic expansions, frames
Secondary: 42C40: Wavelets and other special systems 94A20: Sampling theory

Citation

Gröchenig, Karlheinz; Stöckler, Joachim. Gabor frames and totally positive functions. Duke Math. J. 162 (2013), no. 6, 1003--1031. doi:10.1215/00127094-2141944. https://projecteuclid.org/euclid.dmj/1366639398


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