Most of the Banach spaces of functions or distributions over LCA (locally compact Abelian) groups G are (isometrically) under both time-shifts and frequency shifts (also called modulations or multiplications with characters of the group, the pure frequencies in engineering terminology). There is a smallest space in this family, namely the Segal algebra S_0(G), introduced by the speaker in 1979. The dual space, also known as the space of mild distributions, contains all these spaces. Both spaces are Fourier invariant, and together with the Hilbert space they form the Banach Gelfand Triple (S_0,L_2,S_0'). The above-mentioned spaces are called Fourier Standard Spaces and are handy for the treatment of a variety of questions in Fourier Analysis (summability of Fourier series, multipliers, Shannon Sampling Theorem, Gabor expansions, etc.).

The setting appears to be suitable for the discussion of problems in Abstract Harmonic Analysis, but also for a new approach to Fourier Analysis which is much closer to the engineering view on the subject, for a discussion of the relation between the continuous Fourier Transform and the discrete FFT, realizable on a computer. Following the talk, the author is available for discussions. Corresponding publications can be found on the author's homepage:

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