 
Summary: Asymptotic Expansions of Berezin Transforms
Jonathan Arazy Bent Orsted
June 22, 2001
Abstract
We study a new method of expanding Berezin transforms corresponding to various weighted volume
measures on symmetric domains. The main result is an explicit asymptotic expansion for such a
transform in terms of Pochhammer symbols associated with Cartan domains.
1 Introduction
In this paper we shall study asymptotic expansions of certain natural convolution operators and pseu
dodifferential operators in terms of one or several real parameters, the interpretation being analogous
to that of Planck's constant. A simple model example is that of the Taylor expansion of the translation
operator, viz.
e t(d=dx) f(x) =
1
X
n=0
t n
n! f (n) (x) = f(x + t)
which is an identity for analytic functions and otherwise the sum is an asymptotic expansion in the
usual sense when f is a smooth function. Note that to find the terms in the expansion we may work with
