The similarity between modulation spaces and Besov spaces, as discussed in the previous seminar consists in the fact that in both cases one described the corresponding spaces by a decompositions in the frequency domain. In the first case one uses smooth dyadic partitions of unity, obtained from a typical band-pass filter supported near $x = 1 = 2^0$ by dilation, in the modulation case one applies a frequency shift or {\it modulation operator} in order to obtain a uniform decomposition of unity. In both cases one has a way to charicterize the corresponding function spaces not only by the finiteness of some expression (defining and admissible norm on the space of tempered distributions), but also using atomic decompositions: One builds certain infinite series, where the atoms are obtained under the action of the ax+b group in the wavelet case, and under time-frequency shifts in the modulation space case.

Coorbit theory, developed jointly with K. Groechenig in Vienna,
now takes up the abstract structure: Given a Hilbert space with an irreducible, unitary representation which has the property of being INTEGRABLE allows to build such a theory. There is something like a general voice transform (unifying the idea of a continuous wavelet transform resp. the STFT, the Short-Time Fourier transform) which maps elements of certain distribution spaces into continuous functions on the acting group. Among the key results is the coincidence of coorbit spaces (distributions which show a certain behaviour of their voice transforms) and orbit spaces, which are space built as convergent series resp. infinite linear combinations of elements from the orbit of an atom (mother wavelet, Gabor atom, etc.) under the group action. For the modulation spaces this is the reduced Heisenberg group, closely related to the concept of phase-space, or the time-frequency plane in an engineering terminology.