Harmonic analysis and Lie groups

AbstractOlafssonThis project deals with a variety of problems in modern harmonic analysis.It combines ideas and problems from abstract harmonic analysisand representation theory on Lie groups related tosymmetric spaces with tools and questions wellknown from classical Euclidean harmonic analysis. Our list ofproblems includes a detailed study of series of representations occurringin the regular representation on Pseudo-Riemannian symmetric spaces and, inparticular, geometric realizations of those representations usingtools from complex analysis. Part of this study is the interplay betweenspecial functions and the spherical character of representationsoccurring discretely in the regular representation.Our study also includes compactificationof 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 areasof mathematics: Complex analysis, group actions on manifolds and functionspaces, 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 bedone 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 superpositionof trigonometric functions. This can be interpreted either as the spectraldecomposition of the differential operators with constant coefficients, oras decomposition of regular representation into irreducible representations.In short, the subject of harmonic analysis is to study functions orfunction spaces by decomposing the functions into simpler functions. Inthe theory of differential equations this decomposition means to writean arbitrary functions as a sum or integral of eigenfunctions. In severalapplications, as in image processing, the wavelets shows up as the basicatoms used to approximate or represent the signal. If we have a symmetrygroup acting on the system, then we would like to write an arbitraryfunction as a sum of functions that transforms in a simple and controllableway under the symmetry group, leading to representation theory ofthe symmetry group. Both aspects usually involve the study of integraltransforms and the corresponding kernel function.

AbstractOlafssonThis project deals with a variety of problems in modern harmonic analysis.It combines ideas and problems from abstract harmonic analysisand representation theory on Lie groups related tosymmetric spaces with tools and questions wellknown from classical Euclidean harmonic analysis. Our list ofproblems includes a detailed study of series of representations occurringin the regular representation on Pseudo-Riemannian symmetric spaces and, inparticular, geometric realizations of those representations usingtools from complex analysis. Part of this study is the interplay betweenspecial functions and the spherical character of representationsoccurring discretely in the regular representation.Our study also includes compactificationof 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 areasof mathematics: Complex analysis, group actions on manifolds and functionspaces, 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 bedone 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 superpositionof trigonometric functions. This can be interpreted either as the spectraldecomposition of the differential operators with constant coefficients, oras decomposition of regular representation into irreducible representations.In short, the subject of harmonic analysis is to study functions orfunction spaces by decomposing the functions into simpler functions. Inthe theory of differential equations this decomposition means to writean arbitrary functions as a sum or integral of eigenfunctions. In severalapplications, as in image processing, the wavelets shows up as the basicatoms used to approximate or represent the signal. If we have a symmetrygroup acting on the system, then we would like to write an arbitraryfunction as a sum of functions that transforms in a simple and controllableway under the symmetry group, leading to representation theory ofthe symmetry group. Both aspects usually involve the study of integraltransforms and the corresponding kernel function.