Algebraic properties of Mehler–Fock convolution and applications

Abstract

In this paper, we study some properties of the Mehler–Fock convolution operator. We also analyse the Banach algebraic structure on the space of integrable functions $L_1(1,\mathrm{\infty })$ with the multiplication being the Mehler–Fock convolution. The Titchmarsh-type theorem for this convolution operator is also obtained. As applications, we apply these properties of the convolution operator to solve some classes of Fredholm integral and integro-differential equations and prove some priori estimations under the given conditions.

Keywords:

• Banach algebra
• convolution
• Mehler–Fock transform
• Fredholm integral equation
• integro-differential equation

2020 Mathematics Subject Classifications:

• 46J10
• 44A35
• 45E10
• 45J05
• 47A30

Acknowledgments

The authors would like to express his sincere thanks to Professor Vu Kim Tuan for comments and suggestions that lead to a vast improvement of the manuscript, and the referees for the careful reading of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research received no specific grant from any funding agency.