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

Gröchenig, Karlheinz; Stöckler, Joachim

Duke Math. J. 162, No. 6, 1003-1031 (2013).

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.

Reviewer: Patricia Mariela Morillas (San Luis)

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

Keywords:

frames; Gabor frames; windows functions; totally positive functions; sampling theory; shift-invariant spaces; Nyquist rate

References:

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