## Spherical Harmonics and Approximations on the Unit Sphere: An IntroductionThese 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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### Contents

An Introduction Chapter 1 Preliminaries | 1 |

An Introduction Chapter 2 Spherical Harmonics | 10 |

An Introduction Chapter 3 Differentiation and Integration over the Sphere | 87 |

An Introduction Chapter 4 Approximation Theory | 131 |

An Introduction Chapter 5 Numerical Quadrature | 165 |

An Introduction Chapter 6 Applications Spectral Methods | 211 |

An Introduction References | 237 |

242 | |

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addition theorem apply associated Legendre functions Assume f basis functions best approximation bound change of variables coefficients compute consider constant Corollary deﬁne degree of precision denote derivatives dimension discuss error evaluating example expansion ﬁnd ﬁrst ﬁxed following result function f G N0 G Sdil Galerkin method Gegenbauer polynomials given Hence homogeneous harmonic inner product integral equation integral identity integral operator integrand interpolation introduce invariant Jacobi polynomials Laplace Laplace series Legendre polynomials Lemma linear mapping matrix nodes norm notation numerical integration obtain oo Nn,d orthogonal polynomials orthogonal projection orthonormal basis polynomial of degree product Gauss formula Proof Proposition quadrature formula rate of convergence Recall recursion relation satisﬁes SdI1 Sect smooth Sobolev spaces solution spectral method spherical harmonics spherical polynomials subspace triangulation uniform convergence unit disk unit sphere vector write