Modulation Spaces

Abstract

The main tenet of time-frequency analysis is the interpretation of the short— time Fourier transform, the Wigner distribution, and other representations as a measure of simultaneous time— frequency information. The pointwise values V _g f (x, w) or W f (x, w) measure the content of the frequency band at w in a neighborhood of time x, and V _g f or W f itself encodes the global time— frequency distribution of f However, so far we have made little use of this global time— frequency information. In fact, the only global information that we can handle so far with the STFT or with Gabor frames is the membership of a function in L ²(R^d). But such a characterization can be accomplished with any orthonormal basis or any frame with less effort and more simplicity.