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

THE CHARACTERIZATION FOR
THE ROTATION-INVARIANT SEGAL-BARGMANN SPACE

Areerak Chaiworn, Wicharn Lewkeeratiyutkul
Department of Mathematics
Faculty of Science
Burapha University
Chonburi, 20131, THAILAND
e-mail: areerak@buu.ac.th
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 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.