Semigroup asymptotics, the Funk-Hecke identity and the Gegenbauer coefficients associated with the spherical Laplacian

Abstract

A trace formulation of the Maclaurin spectral coefficients of the Schwartzian kernel of functions of the spherical Laplacian is given. A class of polynomials $\mathscr {P}^\nu _l(X)$ $(l \ge 0$, $\nu \gt -1/2)$ linking to the classical Gegenbauer polynomials through a differential-spectral identity is introduced, and its connection to the above spectral coefficients and their asymptotics analyzed. The paper discusses some applications of these ideas combined with the Funk-Hecke identity and semigroup techniques to geometric and variational-energy inequalities on the sphere and presents some examples.