, Volume 13, Issue 2, pp 239-254

The Balian-Low theorem and regularity of Gabor systems

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Abstract

For any positive real numbers A, B, and d satisfying the conditions \(\frac{1}{A} + \frac{1}{B} = 1\) , d>2, we construct a Gabor orthonormal basis for L2(ℝ), such that the generating function g∈L2(ℝ) satisfies the condition:∫|g(x)|2(1+|x| A )/log d (2+|x|)dx < ∞ and \(\int_{\hat {\mathbb{R}}} {\left| {\hat g(\xi )} \right|^2 (1 + \left| \xi \right|^B )/\log ^d (2 + \left| \xi \right|)d\xi< \infty } \) .

Communicated by Guido Weiss