Banach Gelfand Triples and their use in Fourier Analysis

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Date(s) - 15/09/2014
10 h 00 min - 11 h 00 min

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Banach Gelfand triples are triples of Banach spaces included in each other in a very special way: Formally one assumes that there is an embedding of a Banach space (of test functions) into its dual (a Banach space of generalized functions), with a Hilbert space in the middle. There is a specific Banach Gelfand triple, based on the Segal Algebra S0(ℝd) (in fact it can be defined for general LCA groups), which has its roots in time-frequency analysis. In fact, this space (and its dual as well as the intermediate Hilbert space L2(ℝd)) are so-called modulation spaces and can be characterized via their Gabor coefficients. This BGT is suitable in describing the Fourier transform in a general way, but also the transition from operators to their spreading representation. One of the cornerstones is the analog of matrix representations for linear mappings on ℝn: the so-called kernel theorem, where one usually has to resort to the Schwartz space of rapidly descreasing function, a prototype of a nuclear Frechet space (hence not a Banach space!).

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Olivier CHABROL
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