Gabor meets Littlewood–Paley: Gabor expansions in $L^p({{\mathbb R}}^d)$
Volume 146 / 2001
Studia Mathematica 146 (2001), 15-33
MSC: Primary 42B25; Secondary 42C15, 42C40, 46B15.
DOI: 10.4064/sm146-1-2
Abstract
It is known that Gabor expansions do not converge unconditionally in $L^p$ and that $L^p$ cannot be characterized in terms of the magnitudes of Gabor coefficients. By using a combination of Littlewood–Paley and Gabor theory, we show that $L^p$ can nevertheless be characterized in terms of Gabor expansions, and that the partial sums of Gabor expansions converge in $L^p$-norm.
