On polyharmonic polynomials

TL;DR

This paper investigates the structure and properties of polyharmonic polynomials, focusing on their orthogonal projections, decompositions, and convergence of series, advancing theoretical understanding in harmonic analysis.

Contribution

It introduces a new decomposition of homogeneous polynomials using Kelvin transforms and derivatives of fundamental solutions, and analyzes convergence issues of orthogonal series in polyharmonic spaces.

Findings

Derived a decomposition of homogeneous polynomials in terms of Kelvin transforms.

Analyzed the convergence of orthogonal series of polyharmonic polynomials.

Provided vector bases for the space of homogeneous polyharmonic polynomials.

Abstract

We study the orthogonal projection of homogeneous polynomials onto the space of homogeneous polyharmonic polynomials. To do this we derive the decomposition of homogeneous polynomials in terms of the Kelvin transform of derivatives of the fundamental solution $|x|^{2-n}$ or $\log |x|$. We consider also the vector bases of the space of homogeneous polyharmonic polynomials and study the problem of convergence of orthogonal series.