Gabor Frames and Time-Frequency Analysis of Distributions

This paper lays the foundation for a quantitative theory of Gabor expansionsf(x)=∑k, nck, n e2πinαxg(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 coefficientsck, nis 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 onL2and the atomgis 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.