This project deals with a variety of problems in modern harmonic analysis. It combines ideas and problems from abstract harmonic analysis and representation theory on Lie groups related to symmetric spaces with tools and questions well known from classical Euclidean harmonic analysis. Our list of problems includes a detailed study of series of representations occurring in the regular representation on Pseudo-Riemannian symmetric spaces and, in particular, geometric realizations of those representations using tools from complex analysis. Part of this study is the interplay between special functions and the spherical character of representations occurring discretely in the regular representation. Our study also includes compactification of symmetric spaces, application of representation theory to special functions, in particular, Laguerre functions and polynomials. On the other hand the proposal includes problems related to wavelet theory, function spaces on cones and other symmetric spaces, in particular, Besov spaces. The proposed work combines methods and ideas from several areas of mathematics: Complex analysis, group actions on manifolds and function spaces, in particular, Besov, Fock, and Hardy spaces, and classical harmonic analysis. It even borrows some ideas from applied mathematics. Parts of the proposed work will be done in collaboration with our students as well as specialists in USA and Europe.
Harmonic analysis has its origin in the work of Fourier on the heat equation, which led him to consider the expansion of a periodic functions into superposition of trigonometric functions. This can be interpreted either as the spectral decomposition of the differential operators with constant coefficients, or as decomposition of regular representation into irreducible representations. In short, the subject of harmonic analysis is to study functions or function spaces by decomposing the functions into simpler functions. In the theory of differential equations this decomposition means to write an arbitrary functions as a sum or integral of eigenfunctions. In several applications, as in image processing, the wavelets shows up as the basic atoms used to approximate or represent the signal. If we have a symmetry group acting on the system, then we would like to write an arbitrary function as a sum of functions that transforms in a simple and controllable way under the symmetry group, leading to representation theory of the symmetry group. Both aspects usually involve the study of integral transforms and the corresponding kernel function.
