Colloquium: "Dirichlet spaces, wavelet theory, regularity spaces, and heat kernel estimates"

Abstract: Wavelet theory was developed by Meyer, Daubechies, Lemarie, Mallat, Cohen and scores of other mathematicians more than thirty years ago, after the work of Frazier, Jawerth, and Weiss ([5]). Lots of applications have been made thereafter. For example, the Littlewood-Paley analysis and wavelet theory have proved to be a very useful tool in nonparametric statistic analysis. This is essentially due to the fact that most of the regularity (Sobolev and Besov) spaces can be characterized by wavelet sparse coefficients. In turn, in the nineties the wavelet theory allowed to develop ([4]) an adaptive estimator of the density of a probability law with no apriori knowledge of the regularity. Then it appeared that the Euclidian analysis is not always appropriate because many statistical problems have their own geometry. For instance, this is the situation in Tomography, where one uses Harmonic analysis of the ball, and in the study of the Cosmological Microwave Background, which requires Harmonic analysis on the sphere ([2]).

At the same time the wavelet theory was extended in various geometric and nonclassical frameworks. Extensions of this kind have already been implemented in the cases of the interval ([10]), the ball ([11]), the sphere ([8, 9]), and have been extensively used in statistical applications (see for instance ([6]).

In recent years the Littlewood-Paley analysis and wavelet theory were developed in the general framework of Riemannian manifolds and furthermore in the general setting of a positive operator associated to a suitable Dirichlet space with a good behavior of the associated heat kernel ([1, 7]) on a homogeneous space ([3]).

In this talk we will review the topics mentioned above and present some new results.

Host: Todd Kuffner

Tea @ 3:45 in room 200