Gabor meets Littlewood–Paley: Gabor expansions in $L^p({{\mathbb R}}^d)$

Volume 146 / 2001

Karlheinz Gröchenig, Christopher Heil Studia Mathematica 146 (2001), 15-33 MSC: Primary 42B25; Secondary 42C15, 42C40, 46B15. DOI: 10.4064/sm146-1-2


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.


  • Karlheinz GröchenigDepartment of Mathematics
    The University of Connecticut
    Storrs, Connecticut 06269-3009, U.S.A.
  • Christopher HeilSchool of Mathematics
    Georgia Institute of Technology
    Atlanta, Georgia 30332-0160, U.S.A.

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