On the \(abc\)-problem in Weyl-Heisenberg frames. (English) Zbl 1340.42073

He, Xinggang; Li, Haixiong

Czech. Math. J. 64, No. 2, 447-458 (2014).

Let \(a,b,c > 0\). Authors investigate the characterization problem which asks for a classification of all the triples \((a, b, c)\) such that the Weyl-Heisenberg system \[ \{e^{2\pi imbx} \times \chi_{[na,na+c)} : m, n \in \mathbb Z\} \] is a frame for \(L^2(\mathbb R^+)\). Without loss of generality it is possible to assume that \(b = 1\) and \(0 < a < 1\). It is slightly surprising that the classification of all \(a, c \in \mathbb R^+\) such that the \( (a, 1,c)\)-Weyl-Heisenberg system is a frame is a very difficult problem (called the \(abc\)-problem). In this paper, by virtue of the technique of Fourier analysis, the authors make some progress on this problem.

Reviewer: S. F. Lukomskii (Saratov)

MSC:

42C15	General harmonic expansions, frames
42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems

Keywords:

\(abc\)-problem; Weyl-Heisenberg frame; Zak transform

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