Asymptotic Expansions of Finite Hankel Transforms and the Surjectivity of Convolution Operators

Abstract

A compactly supported distribution is called invertible in the sense of Ehrenpreis-Hörmander if the convolution with it induces a surjection from 𝒞∞(ℝⁿ) to itself. We give sufficient conditions for radial functions to be invertible. Our analysis is based on the asymptotic expansions of finite Hankel transforms. The dominant term may come from the origin or the boundary of the support of the function. To provide proof, we propose a new method for calculating the asymptotic expansions of finite Hankel transforms of functions with singularities at points other than the origin.