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

Enrico De Micheli
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Enrico De Micheli: IBF–Consiglio Nazionale delle Ricerche, Via De Marini, 6-16149 Genova, Italy

Mathematics, 2020, vol. 8, issue 2, 1-30

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 ) .

Keywords: holomorphic extension; spherical Laplace transform; non-Euclidean Fourier transform; Fourier–Legendre expansion (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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