Gabor frames and time-frequency analysis of distributions

Abstract

This paper lays the foundation for a quantitative theory of Gabor expansions f(x) = ∑k, nck, ne2πinαx g(x-kβ). In analogy to wavelet expansions of Besov Triebel-Lizorkin spaces, we show that the correct class of spaces which can be characterized by the magnitude of the coefficients ck, n is the class of modulation spaces. To analyze the behavior of the coefficients, it is necessary to invert the Gabor frame operator on these spaces. We show that the frame operator is invertible on modulation spaces if and only if it is invertible on L2 and the atom g is in a suitable space of test functions. A similar statement for wavelet theory is false. The second part is devoted to Gabor analysis on general time-frequency lattices. © 1997 Academic Press.