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On the $a b c$ -problem in Weyl-Heisenberg frames

Xinggang He; Haixiong Li — 2014

Czechoslovak Mathematical Journal

Let $a, b, c > 0$ . We investigate the characterization problem which asks for a classification of all the triples $(a, b, c)$ such that the Weyl-Heisenberg system ${e^{2 π i m b x} χ_{[n a, n a + c)} : m, n \in ℤ}$ is a frame for $L^{2} (ℝ)$ . It turns out that the answer to the problem is quite complicated, see Gu and Han (2008) and Janssen (2003). Using a dilation technique, one can reduce the problem to the case where $b = 1$ and only let $a$ and $c$ vary. In this paper, we extend the Zak transform technique and use the Fourier analysis technique to study the problem for the case of...

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