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 ) .
Highlights
It is well known that the classical Fourier transform refers to the decomposition of a function belonging to an appropriate space into exponentials, which can be viewed as the irreducible unitary representations of the additive group of the real numbers
One of our results consists of proving that the spherical Laplace transform reduces to the non-Euclidean Fourier transform at Re λ = − 21, which is precisely the value corresponding to the principal series of the unitary representations of the group
The harmonic analysis in causal symmetric spaces has been a subject of growing interest in the last three decades, and the research on these topics has flowed in various directions
Summary
It is worth recalling that the question of relating the harmonic analysis of different real forms of a complex symmetric space has been studied in the context of scattering theory and resonances [16,17,18,19,20]. We are led to develop (c) the harmonic analysis on the complex one-sheeted hyperboloid X2 , which contains as submanifolds either the Euclidean sphere (iR × R2 ) ∩ X2 on which the Fourier–Legendre expansion can be developed, and the real one-sheeted hyperboloid, which contains the support of the cut. Holomorphic Extension Associated with the Fourier–Legendre Expansion and the Spherical
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