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plenche
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There are 7 letters in PLENCHE ( C3E1H4L1N1P3 )
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In mathematics, the Plancherel theorem for spherical functions is an important result in the representation theory of semisimple Lie groups, due in its final form to Harish-Chandra. It is a natural generalisation in non-commutative harmonic analysis of the Plancherel formula and Fourier inversion formula in the representation theory of the group of real numbers in classical harmonic analysis and has a similarly close interconnection with the theory of differential equations. * It is the special case for zonal spherical functions of the general Plancherel theorem for semisimple Lie groups, also proved by Harish-Chandra. The Plancherel theorem gives the eigenfunction expansion of radial functions for the Laplacian operator on the associated symmetric space X; it also gives the direct integral decomposition into irreducible representations of the regular representation on L2(X). In the case of * hyperbolic space, these expansions were known from prior results of Mehler, Weyl and Fock. * The main reference for almost all this material is the encyclopedic text of Helgason (1984). |