IJPAM: Volume 61, No. 1 (2010)

THE CHARACTERIZATION FOR
THE ROTATION-INVARIANT SEGAL-BARGMANN SPACE

Areerak Chaiworn$^1$, Wicharn Lewkeeratiyutkul$^2$
$^1$Department of Mathematics
Faculty of Science
Burapha University
Chonburi, 20131, THAILAND
e-mail: areerak@buu.ac.th
$^2$Department of Mathematics
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 $\C^d$ that are square-integrable with respect to the complex Gaussian measure $\mu_t(z)dz=(\pi t)^{-d}e^{-z^2/t}dz$, and is denoted by $\h L^2(\C^d, \mu_t)$.

In this work, we consider the rotation-invariant subspace of the Segal-Bargmann space. The complex rotation-invariant function $F$ is determined by its values on $(z,0,\dots, 0)\cong \C^1$ and it is a complex even function. Conversely, any even holomorphic function on $\C^1$ has an extension to a complex rotation-invariant holomorphic function on $\C^d$. Thus the space of complex rotation-invariant functions in $\h L^2(\C^d, \mu_t)$ can be expressed as an $L^2$-space of holomorphic functions on $\C^1$ with respect to some non-Gaussian measure. This non-Gaussian measure is absolutely continuous with respect to Lebesgue measure on $\C$. 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