Harmonic analysis on hyperboloids |
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Authors: | Robert S Strichartz |
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Institution: | Department of Mathematics, Cornell University, Ithaca, New York 14850 USA |
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Abstract: | The regular representation of O(n, N) acting on is decomposed into a direct integral of irreducible representations. The homogeneous space is realized as the Hyperboloid . The problem is essentially equivalent to finding the spectral resolution of a certain self-adjoint invariant differential operator □h on H, which is the tangential part of the operator □ = Δx ? Δt on Rn + N. The spectrum of □h contains a discrete part (except when N = 1) with eigenfunctions generated by restricting to H solutions of □u = 0 which vanish in the region , and a continuous part ?. As a representation of O(n, N), ? ⊕ ? is unitarily equivalent to the regular representation on L2 of the cone , and the intertwining operator is obtained by solving the equation □u = 0 with given boundary values on the cone. Explicit formulas are given for the spectral decomposition. The special case n = N = 2 gives the Plancherel formula for SL(2, R). |
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