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The Banach Gelfand Triple and Fourier Standard Spaces

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The Banach Gelfand Triple and Fourier Standard Spaces


Hans G. Feichtinger


Central objects of classical Fourier Analysis are the Fourier transform, convolution operators, periodic and non-periodic functions and so on. Distribution theory widens the scope by allowing larger families of Banach spaces of functions or generalized functions and extending many of the concepts to this more general setting. Although, according to A. Weil the natural setting for Fourier Analysis (leading to the spirit of Abstract Harmonic Analysis: AHA) most of the time one works in the setting of the Schwartz space of rapidly decreasing functions and its dual space, the tempered distributions. In this setting weighted Lp-spaces and Sobolev spaces correspond to each other in a very natural way. In this talk we will summarize the advantages with respect to the level of technical sophistication and theoretical background which is possible when one uses instead of the Schwartz-Bruhat space the Segal algebra S0(G) and the resulting Banach Gelfand Triple, which appears to be suitable for the description of most problems in AHA as well as for many engineering applications (this part is beyond the scope of the current talk). Among others the use of Wiener amalgam spaces and modulation spaces (introduced by the author in the 1980s) belong to a comprehensive family of Banach spaces, which we call Fourier Standard Spaces. These spaces have a double module structure, with respect to convolution by integrable functions and pointwise multiplication with functions from the Fourier algebra.



2020-02-24 at 3:00 pm (subject to variability)