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These notes provide an introduction to the theory of spherical harmonics in an arbitrary dimension as well as an overview of classical and recent results on some aspects of the approximation of functions by spherical polynomials and numerical integration over the unit sphere. The notes are intended for graduate students in the mathematical sciences and researchers who are interested in solving problems involving partial differential and integral equations on the unit sphere, especially on the unit sphere in three-dimensional Euclidean space. Some related work for approximation on the unit disk in the plane is also briefly discussed, with results being generalizable to the unit ball in more dimensions.
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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)
"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)