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Gabor frames and totally positive functions. (English) Zbl 1277.42037

A fundamental problem of Gabor analysis, that has its origins in the foundations of quantum mechanics and of information theory, is to determine triples \((g,\alpha,\beta)\) consisting of a window \(L^{2}\)-function \(g\) and lattice parameters \(\alpha,\beta > 0\) such that the set \(G(g,\alpha,\beta)=\{e^{2\pi i \beta l t}g(t-\alpha k):k,l \in \mathbb{Z}\}\) is a frame for \(L^{2}(\mathbb{R})\).
In this paper it is proved that if \(g\) is a totally positive function of finite type, that is, \(\widehat{g}(\xi)=\prod_{\nu=1}^{M}(1+2\pi i\delta_{\nu}\xi)^{-1}\) for \(\delta_{\nu} \in \mathbb{R}\), \(\delta_{\nu} \neq 0\), and \(M \geq 2\), then \(G(g,\alpha,\beta)\) is a frame if and only if \(\alpha\beta < 1\). This result is a first positive contribution to a conjecture of Daubechies from 1990. It also increases the number of known window functions with a complete characterization of lattice parameters from six to uncountable many. Exploiting the connection with Gabor frames, new and sharp sampling theorems for shift-invariant spaces are proved.

MSC:

42C15 General harmonic expansions, frames
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
94A20 Sampling theory in information and communication theory
41A30 Approximation by other special function classes
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