Gabor frames and time-frequency analysis of distributions
The authors show that the correct class of spaces which can be characterized by the magnitude of the coefficients \(C_{k,n}\) in the Gabor expansion \(\sum_{k,n} C_{k,n}e^{2\pi in\alpha x}g(x- k\beta)\) is the class of modulation spaces. They prove that the Gabor frame operator is invertible on the modulation space if and only if it is invertible on \(L^2\) and the atom \(g\) is in a suitable space of test functions. They show that an analogous statement for wavelet theory is false.
scientific article; zbMATH DE number 1077106