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International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 287218, 13 pages
http://dx.doi.org/10.1155/2008/287218
Research Article

Image of under the Hermite Semigroup

Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India

Received 9 June 2008; Revised 8 October 2008; Accepted 9 December 2008

Academic Editor: Misha Rudnev

Copyright © 2008 R. Radha and D. Venku Naidu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

It is shown that the Hermite (polynomial) semigroup maps into the space of holomorphic functions in for each , where is the Gaussian measure, is a scaled version of Gaussian measure with if and if with . Conversely if is a holomorphic function which is in a “slightly” smaller space, namely , then it is shown that there is a function such that . However, a single necessary and sufficient condition is obtained for the image of under , . Further it is shown that if is a holomorphic function such that or , then there exists a function such that , where and .