Abstract
AbstractIn this chapter we develop polynomial function systems for scalar-valued functions on spheres in the three-dimensional Euclidean space, namely the scalar spherical harmonics. Scalar spherical harmonics are essential for any analysis of spherical functions. The main features are the addition theorem, the Funk-Hecke formula, and the orthogonal invariance leading to expressions in the terms of Legendre polynomials. The scalar spherical harmonics also provide the foundation for vector spherical harmonics (see Chapter 6) as well as tensor spherical harmonics (see Chapter 7).
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