Chapter 10 - Compendium on spherical harmonics and polyharmonic functions

Abstract

This chapter presents a compendium on spherical harmonics and polyharmonic functions. The main purpose of this compendium is to present basic results about the representation of polyharmonic functions in the spherically symmetric domains—ball and annulus. As a necessary prelude, an exposition of the basics of the theory of spherical harmonics is provided. The theory of spherical harmonics is very important for the theory of polysplines (when the break-surfaces are concentric spheres). The compendium contains many well-known results but also some which are far from popular, such as the Almansi representation of polyharmonic functions in the annulus. The notion of spherical coordinates and the Laplace operator is discussed in the chapter. Fourier series and basic properties are elaborated. The chapter attempts to find the point of view from which the spherical harmonics are a natural generalization of the Fourier series. This chapter also discusses homogeneous polynomials in Rn, examples of homogeneous polynomials, and Gauss representation of homogeneous polynomials.