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On the Connection between Spherical Laplace Transform and Non-Euclidean Fourier Analysis

Author

Listed:
  • Enrico De Micheli

    (IBF–Consiglio Nazionale delle Ricerche, Via De Marini, 6-16149 Genova, Italy
    Dedicated to the memory of Professor Giovanni Alberto Viano.)

Abstract

We prove that, if the coefficients of a Fourier–Legendre expansion satisfy a suitable Hausdorff-type condition, then the series converges to a function which admits a holomorphic extension to a cut-plane. Next, we introduce a Laplace-type transform (the so-called Spherical Laplace Transform ) of the jump function across the cut. The main result of this paper is to establish the connection between the Spherical Laplace Transform and the Non-Euclidean Fourier Transform in the sense of Helgason. In this way, we find a connection between the unitary representation of SO ( 3 ) and the principal series of the unitary representation of SU ( 1 , 1 ) .

Suggested Citation

  • Enrico De Micheli, 2020. "On the Connection between Spherical Laplace Transform and Non-Euclidean Fourier Analysis," Mathematics, MDPI, vol. 8(2), pages 1-30, February.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:2:p:287-:d:322936
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