A new basis for eigenmodes on the sphere

DOI10.1088/0305-4470/37/1/014zbMATH Open1046.33009arXivmath/0304409OpenAlexW3121781732MaRDI QIDQ4470680

Publication date: 15 June 2004

Published in: Journal of Physics A: Mathematical and General (Search for Journal in Brave)

Abstract: The usual spherical harmonics

Y_{e l l m}

form a basis of the vector space

{c a l V}^{e l l}

(of dimension

2 e l l + 1

) of the eigenfunctions of the Laplacian on the sphere, with eigenvalue

l a m b d a_{e l l} = - e l l (e l l + 1)

. Here we show the existence of a different basis

P h i_{j}^{e l l}

for

{c a l V}^{e l l}

, where

P h i_{j}^{e l l} (X) e q u i v (X c d o t N_{j})^{e l l}

, the

e l l^{t h}

power of the scalar product of the current point with a specific null vector

N_{j}

. We give explicitly the transformation properties between the two bases. The simplicity of calculations in the new basis allows easy manipulations of the harmonic functions. In particular, we express the transformation rules for the new basis, under any isometry of the sphere. The development of the usual harmonics

Y_{e l l m}

into thee new basis (and back) allows to derive new properties for the

Y_{e l l m}

. In particular, this leads to a new relation for the

Y_{e l l m}

, which is a finite version of the well known integral representation formula. It provides also new development formulae for the Legendre polynomials and for the special Legendre functions.

Full work available at URL: https://arxiv.org/abs/math/0304409

zbMATH Keywords

spherical harmonics Legendre polynomials eigenmodes eigenfucntions Sobolev inner product, distribution of zeros

Mathematics Subject Classification ID

Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) (33C45) Connections of hypergeometric functions with groups and algebras, and related topics (33C80) Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable (33C50)