IJPAM: Volume 61, No. 1 (2010)
THE ROTATION-INVARIANT SEGAL-BARGMANN SPACE



Faculty of Science
Burapha University
Chonburi, 20131, THAILAND
e-mail: areerak@buu.ac.th

Faculty of Science
Chulalongkorn University
Bangkok, 10330, THAILAND
e-mail: Wicharn.L@chula.ac.th
Abstract.The Segal-Bargmann space is the set of holomorphic functions on that are square-integrable with
respect to the complex Gaussian measure
, and is denoted by
.
In this work, we consider the rotation-invariant subspace of the Segal-Bargmann space. The complex rotation-invariant function
is determined by its values on
and it is a complex even function. Conversely, any even holomorphic
function on
has an extension to a complex rotation-invariant holomorphic function on
. Thus the space of complex
rotation-invariant functions in
can be expressed as an
-space of holomorphic functions on
with
respect to some non-Gaussian measure. This non-Gaussian measure is absolutely continuous with respect to Lebesgue measure on
.
We give a characterization for a complex function to be in the rotation-invariant subspace of the Segal-Bargmann space.
Received: March 22, 2010
AMS Subject Classification: 32Axx, 46E50
Key Words and Phrases: Segal-Bargmann space, rotation-invariance, Segal-Bargmann transform
Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 61
Issue: 1