Spherical Harmonics and Approximations on the Unit Sphere: An Introduction

1 Preliminaries.- 2 Spherical Harmonics.- 3 Differentiation and Integration over the Sphere.- 4 Approximation Theory.- 5 Numerical Quadrature.- 6 Applications: Spectral Methods.

From the reviews:

"The book concentrates on the theory of spherical harmonics on the unit sphere of a general d-dimensional Euclidian space. It summarizes the results related to Legendre and Gegenbauer polynomials as well as the theory of differentiation and integration over the d-dimensional unit sphere and the associated function spaces. ... The style of material presentation ... make the theory described in the book accessible to a wider audience of readers with only some basic knowledge in the functional analysis and measure theory." (Vladimir L. Makarov, Zentralblatt MATH, Vol. 1254, 2013)

"This is a very well-written, self-contained monograph on spherical harmonics. It is an excellent reference source for researchers and graduate students who are interested in polynomial approximation, numerical integration, differentiation and solution of partial differential and integral equations over the sphere." (Feng Dai, Mathematical Reviews, January, 2013)