Summability properties of Gabor expansions

Anton Baranov (1) , Yurii Belov (2) , Alexander Borichev (3)

1 Department of Mathematics and Mechanics
2 Chebyshev Laboratory
3 I2M - Institut de Mathématiques de Marseille

We show that there exist complete and minimal systems of time-frequency shifts of Gaussians in $L^2(\mathbb{R})$ which are not strong Markushevich basis (do not admit the spectral synthesis). In particular, it implies that there is no linear summation method for general Gaussian Gabor expansions. On the other hand we prove that the spectral synthesis for such Gabor systems holds up to one dimensional defect.

Domaines: Variables complexes [math.CV]

Anton Baranov, Yurii Belov, Alexander Borichev. Summability properties of Gabor expansions. Journal of Functional Analysis, 2018, 274 (9), pp.2532-2552. ⟨10.1016/j.jfa.2017.12.009⟩. ⟨hal-02065968⟩