On the Gabor frame set of polynomial-Gaussian windows
Abstract
We prove two sufficient conditions for the frame property of Gabor systems 𝒢(g,a,b)={e2πi mb ·g(·−ak):k,m∈ℤ } generated by polynomial--Gaussian windows g(t)=P(t)e−ct2, where c>0 and P is a nonzero complex polynomial of degree n. The conditions depend only on the product ab of the lattice parameters and the degree of P. First, 𝒢(g,a,b) is a frame whenever ab<1 is irrational. Second, if ab=p/q<1, with p,q∈ℕ coprime, then 𝒢(g,a,b) is a frame whenever q−p>n. Consequently, in the region ab<1, failure of the frame property can occur only on rational hyperbolas whose reduced denominator and numerator differ by at most n. The proof extends the translated-Wronskian method of Faulhuber and Petersen [\emph{The frame set of the first Hermite function}, arXiv:2609.02610] from the first Hermite function to arbitrary polynomial multiples of a Gaussian, combining a multiplicity estimate for Wronskians with Jensen's formula.
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BibTeX
@misc{lemvig2026gabor,
title = {{On the Gabor frame set of polynomial-Gaussian windows}},
author = {Jakob Lemvig},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05337},
eprint = {2611.05337},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05337},
}
RIS
TY - UNPB
TI - On the Gabor frame set of polynomial-Gaussian windows
AU - Jakob Lemvig
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05337
UR - https://arxiv.org/abs/2611.05337
AB - We prove two sufficient conditions for the frame property of Gabor systems 𝒢(g,a,b)={e^2πi mb ·g(·−ak):k,m∈ℤ } generated by polynomial--Gaussian windows g(t)=P(t)e^−ct^2, where c>0 and P is a nonzero complex polynomial of degree n. The conditions depend only on the product ab of the lattice parameters and the degree of P. First, 𝒢(g,a,b) is a frame whenever ab<1 is irrational. Second, if ab=p/q<1, with p,q∈ℕ coprime, then 𝒢(g,a,b) is a frame whenever q−p>n. Consequently, in the region ab<1, failure of the frame property can occur only on rational hyperbolas whose reduced denominator and numerator differ by at most n. The proof extends the translated-Wronskian method of Faulhuber and Petersen [\emph{The frame set of the first Hermite function}, arXiv:2609.02610] from the first Hermite function to arbitrary polynomial multiples of a Gaussian, combining a multiplicity estimate for Wronskians with Jensen's formula.
ER -