Gabor frames for rational functions

DOI10.1007/S00222-022-01151-8zbMath1512.42044arXiv2103.08959MaRDI QIDQ6362973

👤 Yurij I. Lyubarskij 👤 Aleksei Kulikov 👤 Yurii Belov

📅 16 March 2021

Abstract: We study the frame properties of the Gabor systems $$mathfrak{G}(g;alpha,�eta):={e^{2pi i �eta m x}g(x-alpha n)}_{m,ninmathbb{Z}}.$$ In particular, we prove that for Herglotz windows $g$ such systems always form a frame for $L^2(mathbb{R})$ if $alpha,�eta>0$, $alpha�etaleq1$. For general rational windows $gin L^2(mathbb{R})$ we prove that $mathfrak{G}(g;alpha,�eta)$ is a frame for $L^2(mathbb{R})$ if $0<alpha,�eta$, $alpha�eta<1$, $alpha�eta

otinmathbb{Q}$ and $hat{g}(xi)

eq0$, $xi>0$, thus confirming Daubechies conjecture for this class of functions. We also discuss some related questions, in particular sampling in shift-invariant subspaces of $L^2(mathbb{R})$.

Mathematics Subject Classification ID

Applications of dynamical systems (37N99) General harmonic expansions, frames (42C15) Entire and meromorphic functions of one complex variable, and related topics (30D99)

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