Bangalore Harmonic Analysis Seminar

This is an extended seminar talk, explaining how one can go from linear algebra through MATLAB experiments to the foundations of Gabor or time-frequency analysis, emphasizing the role of the so-called Banach Gelfand triple.
It consists of the Segal algebra S_0(Rd) (which is also known as Feichtinger's algebra, and which can also be characterized as the smallest Fourier invariant Segal algebra in the sense of Hans Reiter), the Hilbert space L2(Rd) and the dual space, meanwhile called the space of mild distributions.

Since the Schwartz space S(Rd) of rapidely decreasing functions is continuously embedded into S_0(Rd), the dual space is a subspace of tempered distributions, namely the space of all tempered distributions which have a bounded spectrogram (STFT, sliding-window Fourier transform).

It is demonstrated by a few applications where and why SO and the (Fourier invariant) dual space SO* (Rd) are useful for classical Fourier analysis, for Gabor or time-frequency analysis (e.g. the discussion of Gabor multipliers), but also for the mathematically correct derivation of claims made in engineering books (using the mystic sifting property of the Dirac delta, or inversion formulas for the Fourier transform obtained by using appearently divergent integrals, and so on). For example, we show how to derive (with MATLAB illustrations) a practical version of the Shannon sampling theorem or how to obtain an approximate realization of the continuous FT as defined on S_0(Rd) just using the Riemann integral, making use of the FFT (DFT) in a suitable way.

Questions can be sent to the speaker at