A restriction of a homogeneous harmonic polynomial of degree in variables to the unit sphere of the Euclidean space , . In particular, when , the spherical harmonics are the classical spherical functions.
Let , , , . The basic property of spherical harmonics is the property of orthogonality: If and are spherical harmonics of degree and , respectively, with , then
The simplest spherical harmonics are the zonal spherical harmonics. For any and any , a zonal spherical harmonic exists which is constant on any parallel of the sphere that is orthogonal to the vector . The zonal spherical harmonics differ from the Legendre polynomials , when , or from the ultraspherical polynomials , when , only by a constant factor:
where the polynomials are defined, when , by the generating function
, , . The polynomials , are orthogonal with weight and form an orthogonal basis of the space . If is a function in with , then there is a unique set of spherical harmonics such that
where the series converges in the norm of .
Expansions in spherical harmonics are largely analogous to expansions in Fourier series, of which they are essentially a generalization. The homogeneous harmonic polynomials are sometimes called spatial spherical harmonics. By virtue of the homogeneity
spherical harmonics are sometimes also called surface spherical harmonics.
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Spherical harmonics. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Spherical_harmonics&oldid=12872