"The Universe of Mild Distributions"Feichtinger, HansTogether with the Segal algebra $S_0(R^d)$ and the Hilbert spaces $L^2(R^d)$ the dual space $S^*_0(R^d)$ of {\it mild distributions} forms the so-called Banach Gelfand Triple $(S_0,L^2,S^*_0)$. The talk will highlight the concept of ``mild convergence'' (usually known as weak-* convergence) and the relevance of this very natural concept of approximation which is a good way of turning heuristic concepts into solid mathematical statements. While the Fourier transform (even using Riemann integrals) leaves $S_0(R^d)$ invariant it extends in a unique way to $S^*_0(R^d)$. Classical variants such as the Fourier series expansion of periodic functions or the DFT (usually realized as FFT) for periodic discrete signals are special cases. Each type of signals (and the corresponding FT) can be considered as the natural extension of any of the other settings. As time permits it is planned to provide applications of this setting which allow to derive results about Gabor multipliers (viewed as Anti-Wick operators with mild distributions as upper symbols), but also questions of classical Fourier Analysis (such as the definition of the spectrum of a bounded function), or results relevant for engineering applications such as Shannon's Sampling Theorem. Finally insight relevant for Quantum Harmonic Analysis could be presented. |
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