On an equality for the iterated weighted spherical mean and its applications

TL;DR

This paper generalizes iterated spherical means to weighted versions using generalized translation operators and explores their applications in solving hyperbolic equations and transmission tomography problems.

Contribution

It introduces a new class of weighted iterated spherical means and demonstrates their utility in partial differential equations and integral geometry.

Findings

Weighted spherical means are effective in solving hyperbolic equations

Applications to transmission tomography show improved reconstruction methods

Generalized translation operators extend the applicability of spherical means

Abstract

Spherical means are well-known useful tool in the theory of partial differential equations with applications to solving hyperbolic and ultrahyperbolic equations and problems of integral geometry, tomography and Radon transforms. We generalize iterated spherical means to weighted ones based on generalized translation operators and consider applications to B-hyperbolic equations and transmission tomography problems.