# IJPAM: Volume 61, No. 1 (2010)

**THE CHARACTERIZATION FOR**

THE ROTATION-INVARIANT SEGAL-BARGMANN SPACE

THE ROTATION-INVARIANT SEGAL-BARGMANN SPACE

Department of Mathematics

Faculty of Science

Burapha University

Chonburi, 20131, THAILAND

e-mail: [email protected]

Department of Mathematics

Faculty of Science

Chulalongkorn University

Bangkok, 10330, THAILAND

e-mail: [email protected]

**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