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

TL;DR

The paper develops an operator-based framework to study the spherical mean and its generalizations via operational calculus, identifying the mean with a hypergeometric function of the Laplacian: $\\bar f_S(r,x) = {}_0F_1(; n/2; (r^2 \\Delta_x)/4) f(x)$. This representation yields explicit power-series expansions in $r$, enables inversion by a corresponding operator series, and leads to the Euler–Poisson–Darboux partial differential equation $\\partial^2_r \\bar f_S + \\frac{n-1}{r}\\partial_r \\bar f_S = \\Delta_x \\bar f_S$, while allowing iterates and fractional spherical means. Generalizing to radially symmetric kernels $K(u)$ produces a family of means with hypergeometric operator representations, plus addition formulas and special cases such as the $n$-ball mean and kernels with $\\beta=2$ or $\\beta=1$. Using the operator framework, the paper derives a rigorous inversion for the X-ray transform by relating X\\{f\\} to spherical means and expressing the inversion via the Riesz potential $1/\\sqrt{-\\Delta}$, yielding a Lai-type integral representation.

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.

Figures (1)

• Figure 1: Demonstration of different spherical means in 1D.(a) Comparison of the spherical mean obtained from the integral in Eq. \ref{['eq:sphericalMean']} (solid red line) with the spherical mean obtained from the operator in Eq. \ref{['eq:sphericalMeanOp']} (dashed red line) for the same radius applied to an arbitrary function $f(x)$. (b) Illustration of various fractional spherical means. The solid blue line represents the $\frac{1}{2}$-th spherical mean, and the solid green line represents the $\frac{1}{3}$-rd spherical mean. For reference, the full spherical mean is shown in red.