ABSTRACT
This chapter gives a brief survey on the theory of spherical harmonics of dimension including the addition theorem, the Funk–Hecke formula, the closure and completeness properties and the characterization of spherical harmonics as eigensolutions of the Beltrami operator. It also gives a short approach to the theory of Bessel functions. The chapter provides necessary tools of the theory of special functions in a notation especially suitable for the development and formulation of lattice point and Shannon-type sampling identities in Euclidean spaces. It begins with the theory on homogeneous harmonic polynomials in Euclidean space and Legendre polynomials. The chapter explains that the Neumann function, which together with the Bessel function implies the Hankel functions.
