Operational Methods Applied to the Spherical Mean and X-Ray Transform

Abstract

We employ the framework of operational calculus to derive the operators associated with the spherical mean and a class of related averaging means of a function in $n$-dimensional space. Beginning with the classical definition of the spherical mean, we obtain a compact operator representation in terms of confluent hypergeometric functions of the Laplacian. This operator-based formulation provides a straightforward approach to the analysis of spherical means, allowing us to determine their power series expansions, construct series solutions to the corresponding inversion problems, derive the partial differential equations they satisfy, and give meaning to iterated and fractional spherical means. Finally, we apply the spherical mean operator to derive the inversion formula for the X-ray transform in an operational manner.

v4: Complete rewrite