Relationships Involving Transforms and Convolutions Via the Translation Theorem

Abstract

In this article, we use the translation theorem to obtain several relationships involving integral transforms and convolution products. In particular, we obtain several useful formulas involving various functionals, which arise naturally in quantum mechanics.

Keywords:

• Fubini theorem
• Generalized Brownian motion process
• Generalized convolution product
• Generalized integral transform
• Translation theorem

Mathematics Subject Classification:

• 60J65
• 28C20

Acknowledgments

The authors thank the referees for their helpful suggestions which led to the present version of this paper.

Notes

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2012R1A1A1004774). This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0014552).