Algebraic properties of Mehler-Fock convolution and applications

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,\infty ) $ L1(1,infinity) 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.