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Asymptotic Expansions of Berezin Transforms Jonathan Arazy Bent Orsted

Summary: Asymptotic Expansions of Berezin Transforms
Jonathan Arazy Bent Orsted
June 22, 2001
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) =
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


Source: Arazy, Jonathan - Department of Mathematics, University of Haifa


Collections: Mathematics