Modulation spaces have been introduced in 1983 by the speaker, as part of an attempt to define smoothness spaces over LCA (locally compact Abelian) groups, but turned out meanwhile to be highly useful in the context of a modern form of Fourier analysis, called time-frequency analysis, or in its discretized form, the so-called Gabor analysis (which can be viewed as the mathematical backbone of the modern MP3 compression algorithm for audio signals even).

In the talk the idea of Wiener amalgam spaces will be discussed and
explained, which are function spaces characterized by BUPUs (bounded
uniform partitions of unity), using a local and a globald component of quite general type, denoted typically by W(B,Y), where B is some Banach space of tempered distributions and Y is Banach function spaces on a lattice. Modulation spaces can be thought of as spaces which, on the Fourier transform side, are space of the form W(FLp,lq_vs),where the globa component is a polynomially weighted lq-space. The analogy with Besov spaces (which make use od dyadic partitions of unity instead of uniform ones) will be explained.