Group theoretical methods and wavelet theory: coorbit theory and applications (2013)

Before the invention of orthogonal wavelet systems by Yves Meyer<sup>1</sup> in 1986 Gabor expansions (viewed as discretized inversion of the Short-Time Fourier Transform<sup>2</sup> using the overlap and add OLA) and (what is now perceived as) wavelet expansions have been treated more or less at an equal footing. The famous paper on <i>painless expansions </i>by Daubechies, Grossman and Meyer<sup>3</sup> is a good example for this situation. The description of atomic decompositions for functions in modulation spaces<sup>4</sup> (including the classical Sobolev spaces) given by the author<sup>5</sup> was directly modeled according to the corresponding atomic characterizations by Frazier and Jawerth,<sup>6, 7</sup> more or less with the idea of replacing the dyadic partitions of unity of the Fourier transform side by uniform partitions of unity (so-called BUPU's, first named as such in the early work on Wiener-type spaces by the author in 1980<sup>8</sup>). Watching the literature in the subsequent two decades one can observe that the interest in wavelets "took over", because it became possible to construct <i>orthonormal wavelet systems </i>with compact support and of any given degree of smoothness,<sup>9</sup> while in contrast the Balian-Low theorem is prohibiting the existence of corresponding Gabor orthonormal bases, even in the multi-dimensional case and for general symplectic lattices.<sup>10</sup> It is an interesting historical fact that<sup>*</sup> his construction of band-limited orthonormal wavelets (the Meyer wavelet, see<sup>11</sup>) grew out of an attempt to prove the impossibility of the existence of such systems, and the final insight was that it <i>was not impossible to have such systems</i>, and in fact quite a variety of orthonormal wavelet system can be constructed as we know by now. Meanwhile it is established wisdom that wavelet theory and time-frequency analysis are two different ways of decomposing signals in orthogonal resp. non-orthogonal ways. The unifying theory, covering both cases, distilling from these two situations the common group theoretical background lead to the theory of coorbit spaces,<sup>12, 13 </sup>established by the author jointly with K. Gröchenig. Starting from an integrable and irreducible representation of some locally compact group (such as the "ax+b"-group or the Heisenberg group) one can derive families of Banach spaces having natural atomic characterizations, or alternatively a continuous transform associated to it. So at the end function spaces of locally compact groups come into play, and their generic properties help to explain why and how it is possible to obtain (nonorthogonal) decompositions. While unification of these two groups was one important aspect of the approach given in the late 80th, it was also clear that this approach allows to formulate and exploit the analogy to Banach spaces of analytic functions invariant under the Moebius group have been at the heart in this context. Recent years have seen further new instances and generalizations. Among them shearlets or the Blaschke product should be mentioned here, and the increased interest in the connections between wavelet theory and complex analysis. The talk will try to summarize a few of the general principles which can be derived from the general theory, but also highlight the difference between the different groups and signal expansions arising from corresponding group representations. There is still a lot more to be done, also from the point of view of applications and the numerical realization of such non-orthogonal expansions.