Spectral analysis on upper light cone in \(R^ 3\) and the Radon transform. (English) Zbl 0684.43005

Wawrzyñczyk, Antoni

Can. J. Math. 40, No. 6, 1458-1481 (1988).

Let G be the Lorentz group SO(2,1). In the usual notation of the subject [see S. Helgason, Groups and geometric analysis (1984; Zbl 0543.58001)] the author considers the upper hyperboloid \(H=G/K\) and the upper light cone \(L=G/N\). The Radon transform on H and the dual Radon transform on L as well as the Fourier transform on H are introduced. The study of spectral analysis in L is pursued and it is found that not only continuous series but also finite dimensional and discrete series representations of G appear. A theorem of Pompeiu type is formulated for L. That is a necessary and sufficient condition for a compactly supported distribution on L to span a dense subset of \({\mathcal E}(L)\) by translations and linear combinations.

Reviewer: A.Terras

MSC:

43A85	Harmonic analysis on homogeneous spaces
43A80	Analysis on other specific Lie groups
22E43	Structure and representation of the Lorentz group
43A30	Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.

Keywords:

Lorentz group; upper light cone; Radon transform; Fourier transform; spectral analysis; continuous series; discrete series representations; theorem of Pompeiu type; compactly supported distribution