Wawrzyñczyk, Antoni Spectral analysis and mean-periodic functions on rank one symmetric spaces. (English) Zbl 0652.22012 Bol. Soc. Mat. Mex., II. Ser. 30, No. 2, 15-29 (1985). Let X be a symmetric space of rank one and of noncompact type. The manifold X can be represented as a homogeneous space \(X=G/K\), where G is a semisimple noncompact Lie group of finite center and K is a maximal compact subgroup of G. The action of G on the space E(X) of smooth functions with the usual Fréchet topology is defined by the formula: \(L_ gf(x):=f(g^{-1}x)\). The subject of the present paper is the spectral analysis and synthesis in E(X) with respect to the family \(e_{\lambda,b}\) of plane waves on X. MSC: 22E46 Semisimple Lie groups and their representations 43A85 Harmonic analysis on homogeneous spaces 33C80 Connections of hypergeometric functions with groups and algebras, and related topics 53C35 Differential geometry of symmetric spaces 43A90 Harmonic analysis and spherical functions Keywords:symmetric space; homogeneous space; semisimple noncompact Lie group; smooth functions; spectral analysis and synthesis; plane waves PDFBibTeX XMLCite \textit{A. Wawrzyñczyk}, Bol. Soc. Mat. Mex., II. Ser. 30, No. 2, 15--29 (1985; Zbl 0652.22012)