Abstract
Abstract The restriction to the unit sphere Ω of any homogeneous harmonic polynomial of degree nis called a spherical harmonic of order n.It turns out that any spherical harmonic is an eigenfunction of the Beltrami operator. On the sphere, spherical harmonics are the analogues of the exponential functions for Fourier analysis. They were introduced in the 1780s to study gravitational theory (cf. P.S. DE LAPLACE (1785), A.M. LEGENDRE (1785)). Early books on the theory of potentials and spherical harmonics are due to E. HEINE (1878) and F. NEUMANN (1887). Indeed, spherical harmonics are essential for the analysis of any phenomena with spherical symmetry, e.g. celestial mechanics, terrestrial magnetism, earthquakes, solar corona etc. The use of spherical harmonics in all geosciences for the purpose of representing fields is a well established technique. A great incentive came from the fact that global geomagnetic data became available in the first half of the last century (cf. C.F. GAUSS (1839)).
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