Math Seminars
O. Musin
(Stanford and Moscow)
Multivariate positive definite functions on spheres
In 1942 I.J. Schoenberg proved that a function is positive definite in the
unit sphere if and
only if this function is a positive linear combination of the Gegenbauer
polynomials. In this
paper we extend Schoenberg's theorem for multivariate Gegenbauer polynomials.
This extension
derives new positive semidefinite constraints for the distance distribution
which can be applied for
spherical codes. In particular, using this method can be obtained new upper
bounds for the kissing
numbers in several dimensions.