Series expansions of the transform with respect to the Gaussian process

Abstract

In this paper, we consider the Fourier-type functionals introduced in [Chung HS, Tuan VK. Fourier-type functionals on Wiener space. Bull Korean Math Soc. 2012;49:609–619] and used in [Lee IY, Chung HS, Chang SJ. Series expansions of the analytic Feynman integral for the Fourier-type functional. J Korean Soc Math Educ Ser B. 2012;19:87–102]. We first establish the existence of the transform with respect to the Gaussian process for the Fourier-type functionals. We then obtain the various relationships for the transform with respect to the Gaussian process of the Fourier-type functionals which involved the ⋄-product and in addition, establish various integration formulas.

Keywords:

• Fourier transform
• Fourier-type functional
• Gaussian process
• ⋄-product
• confluent hypergeometric series

AMS Subject Classification:

• Primary: 60J25
• 28C20

Acknowledgments

The authors thank the referees for their helpful suggestions which led to the present version of this paper. The present research was conducted by the research fund of Dankook University in 2015.