nLab spherical function

Idea

Classical spherical functions

Classical spherical functions/harmonics are certain functions on $n$-sphere which are solutions of the Laplace equation restricted to the sphere.

The most classical case is the case when $n=2$ where spherical functions $Y_{l m}(\theta,\phi)$ (see at Legendre polynomial) appear in Fourier method of separation of variables for solving Laplace equation in space in spherical coordinates.

Spherical functions from representation theory

In representation theory, one considers matrix elements of certain representations with distinguished covariance properties with respect to a subgroup; this can be interpreted in terms of the homogeneous space. Various generalities are considered with main contributions from Israel Gelfand‘s school and Roger Godement and later by Sigurdur Helgason.

Literature

• eom: spherical functions
• wikipedia covers the classical case only: spherical harmonics
• Roger Godement, A theory of spherical functions. I, Trans. Amer. Math. Soc. 73: 3 (1952) 496–556 jstor
• I. M. Gel'fand, Dokl. Akad. Nauk SSSR , 70 (1950) pp. 5–8; transl. as Spherical functions on symmetric spaces Transl. Amer. Math. Soc. 37 (1964) 39–44
• I. M. Gel’fand, “Spherical functions on symmetric spaces” Transl. Amer. Math. Soc. , 37 (1964) pp. 39–44 Dokl. Akad. Nauk SSSR , 70 (1950) pp. 5–8
• F. A. Berezin, I. M. Gelfand, Some remarks on the theory of spherical functions on symmetric Riemannian manifolds, Amer. Math. Soc. Transl.(2), 1962 gBooks
• N. Ja. Vilenkin, Special functions and the theory of group representations, Transl. Math. Monog.
• Sigurdur Helgason, Groups and geometric analysis, Acad. Press (1984)