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Zbl 0579.43008
Meaney, Christopher
Spherical functions and spectral synthesis. (English)
[J] Compos. Math. 54, 311-329 (1985). ISSN 0010-437X; ISSN 1570-5846/e

The article contains constructions of sets of nonsynthesis for the bi- invarant subalgebra ${}\sp KA(G)\sp K$ of the Fourier algebra on a Gel'fand pair (G,K); where G is a noncompact connected semisimple Lie group (with some additional hypotheses). In particular, suppose G is real and has finite center, K is a maximal compact subgroup, and G/K has rank one and dimension greater than two. By use of results of {\it M. Flensted-Jensen} and {\it T. H. Koornwinder} [Ark. Mat. 17, 139-151 (1979; Zbl 0409.33009)] about Jacobi functions and associated convolution algebras, the author demonstrates the existence of nontrivial point derivations on ${}\sp KA(G)\sp K$. This implies that each coset KaK, (a$\ne 1)$ is a set of nonsynthesis. \par Results about derivations on ${}\sp KA(G)\sp K$ are also presented for complex semisimple Lie groups G.
[Ch.F.Dunkl]

MSC 2000:: *43A45 Spectral synthesis on groups, etc.
43A90 Spherical functions (abstract harmonic analysis)
33C80 Connections of theory of special functions with groups and algebras
22E30 Analysis on real and complex Lie groups

Keywords: sets of nonsynthesis; Fourier algebra; Gel'fand pair; connected semisimple Lie group

Citations: Zbl 0409.33009

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