ABSTRACT:
Summability methods are an important principle that arise from the classical theory of Fourier series, but are equally important for the inversion of Fourier transforms or the proof of Plancherel's Theorem. The usual ``good summability kernels'' are also very suitable for signal processing applications, where they are used to localize an ongoing function or distribution in order to define its short-time Fourier transform or sliding window Fourier transform. This opens the way to modern time-frequency analysis and Gabor analysis, its discretized and more practical version. Function spaces defined by the behaviour of their STFT, usually called modulation spaces, are well suited in order to handle e.g. questions about pseudo-differential operators or symbolic calculi for operators. Elements from the Segal algebra $S_0(R^d)$ (also called $M^1(R^d)$) are very suitable both in the context of Gabor analysis and as kernels for classical problems. Pertinent results in this direction are obtained in a series of joint papers with Ference Weisz.