Time-frequency analysis on modulation spaces Mmp,q, 0 < p,q ≤ ∞

The theory of modulation spaces M-m(p.q) is extended to the general case 0 < p, q less than or equal to infinity. It is shown that these spaces admit an atomic decomposition and are characterized by decay and summability properties of Gabor coefficients. A large space of admissible windows is constructed which contains the Schwartz class in the case of weights of polynomial growth and has the form of a union of modulation spaces. It is shown that different windows in this space yield equivalent modulation space quasi-norms. (C) 2003 Elsevier Inc. All rights reserved.