Time-frequency analysis on modulation spaces \(M_{m}^{p,q}\), \(0 < p,q \leqslant \infty\). (Q1421493)

Let \(g\) be a non-zero Schwartz function. The short-time Fourier transform of a function \(f\) on \(\mathbb R^{d}\) w.r.t. \(g\) is defined by \(V_g(f)(x,w)= \int_{\mathbb R^d} f(t)\overline{g(t-x)}e^{-2\pi i w\cdot t}dt\). Given a weight function \(m\) on \(\mathbb R^{d}\) and \(0<p,q\leq \infty\), Feichtinger defined the modulation space \(M_m^{p,q}(\mathbb R^d)\) as the set of tempered distributions \(f\) for which the quasi-norm \[ \left( \int \left( \int | V_g(f)(x,w)| ^pm(x,w)^p dx\right)^{q/p}dw \right)^{1/q} \] is finite. The space is independent of \(g\). Modulation spaces are known to play an important role in time-frequency analysis. Most of the analysis on these spaces has been done in the case \(1\leq p,q\leq \infty\); the present paper extends the analysis to the general case. It is proved that the spaces admit an atomic decomposition and are characterized by the decay and summability properties of the Gabor coefficients.