Harmonicity of a function via harmonicity of its spherical means

TL;DR

This paper characterizes harmonic functions through the harmonicity of their spherical means and extends this characterization to solutions of the modified Helmholtz equation, offering new insights into their properties.

Contribution

It introduces a novel characterization of harmonic and panharmonic functions using iterated spherical means and pointwise mean equality, expanding understanding of these functions.

Findings

Harmonic functions are characterized by harmonicity of their spherical means.

Solutions to the modified Helmholtz equation are similarly characterized.

A new pointwise mean equality condition for harmonic functions is established.

Abstract

It is proved that harmonic functions are characterized by harmonicity of their spherical means, for which purpose the iterated spherical means are used. The similar characterization of solutions to the modified Helmholtz equation (panharmonic functions) is given. Another description of harmonic functions is the pointwise equality of a function and its iterated mean over an admissible pair of spheres.