Coherent states on the unit ball of C n and asymptotic expansion of their associated covariant symbol

Abstract

Starting from a complete family for the unit sphere S n in the complex n-space C n (whose elements are coherent states attached to the Barut–Girardello space), we obtain an asymptotic expansion for the associated Berezin transform. The proof involves the computation of the asymptotic behaviour of functions in the complete family. Furthermore, in an analogous manner (slightly weaker) we obtain the asymptotic expansion of the covariant symbol of a pseudo-differential operator on L 2 ( S n ).