On the Fourier transform of regularized Bessel functions on complex numbers and Beyond Endoscopy over number fields

Abstract

In this article, we prove certain Weber-Schafheitlin type integral formulae for Bessel functions over complex numbers. A special case is a formula for the Fourier transform of regularized Bessel functions on complex numbers. This is applied to extend the work of A. Venkatesh on Beyond Endoscopy for $\mathrm{Sym}^2$ on $\mathrm{GL}_2$ from totally real to arbitrary number fields.