On the Fourier Transform of Bessel Functions over Complex Numbers---I: the Spherical Case

TL;DR

This paper derives a Fourier transform formula for spherical Bessel functions over complex numbers, motivated by representation theory and the Waldspurger correspondence, extending classical results to the complex setting.

Contribution

It introduces a new formula for the Fourier transform of spherical Bessel functions over complex numbers, linking harmonic analysis with representation theory.

Findings

Derived a Fourier transform formula for spherical Bessel functions over complex numbers

Connected the formula to the Waldspurger correspondence in representation theory

Extended classical Fourier analysis results to the complex domain

Abstract

In this note, we shall prove a formula for the Fourier transform of spherical Bessel functions over complex numbers, viewed as the complex analogue of the classical formulae of Hardy and Weber. The formula has strong representation theoretic motivations in the Waldspurger correspondence over the complex field.