In his seminal paper from 1946 Denes Gabor was suggesting to expand every function as a double series constituted by Gauss-functions, shifted along the integer lattice and multiplied with harmonic frequencies, as they are used in the classical theory of Fourier series. His suggestions for parameters indicates that he was hoping to have a Riesz basis for the Hilbert space L2(R) in this way, allowing to represent every element f in L2 as a double series, using suitable (and hopefully uniquely determined) coefficients, which can then interpreted as the local energy at a given frequency.

The corresponding theory has been developed only after 1980 and is flourishing until now. Concepts like frames and Riesz basic sequences, the Balian-Low Theorem, Gabor multipliers, and modulation spaces, or Banach Gelfand Triples are concepts that have developed during the last decades and have become meanwhile part of the standard repertoire for mathematical analysis. In this talk, some of these concepts will be illustrated which might be useful for the development of mathematical analysis at large.

Standard reference: K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, 2001.