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

### **Enrico De Micheli †**

IBF –Consiglio Nazionale delle Ricerche, Via De Marini, 6-16149 Genova, Italy; enrico.demicheli@cnr.it † Dedicated to the memory of Professor Giovanni Alberto Viano.

Received: 17 January 2020; Accepted: 17 February 2020; Published: 20 February 2020

**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, <sup>1</sup>).

**Keywords:** holomorphic extension; spherical Laplace transform; non-Euclidean Fourier transform; Fourier–Legendre expansion

**MSC:** 42A38; 44A10; 44A12; 42C10
