On a polyconvolution with a weight function for Fourier cosine and Laplace transforms
Convolutions and generalized convolutions have remained common nomenclatures with several investigations related with the integral transforms and their applications in science and mathematics. A general definition of polyconvolution was proposed by \textit{V. A. Kakichev} [in: Mathematical modelling and complex analysis. Proceedings of the 2nd international conference, Vilnius, Lithuania, June 3--4, 1997. Vilnius: Technika. 79--83 (1997; Zbl 0963.44500)] for \((n+1)\) arbitrary integral transforms with certain weight function of functions for which the factorization property holds true. This notion is found to have been applied to integral transforms, viz. Fourier, Fourier cosine, Fourier sine, or Hartley and types of Fourier transforms (citations of such applications can be seen in the current paper, under review). The present paper claims to have studied a new polyconvolutions with a weight function for the Fourier cosine and Laplace transform and proves the existence of the same in certain function spaces, and also includes the proofs of the same for the factorization equality and algebraic properties (one may refer to Theorems 3.2-- 3.4 in Section 3). Boundedness of polyconvolution under certain prescribed conditions is obtained in Section 4, even one can notice Equation (13) in this context. Integral transforms related to this polyconvolution are studied in Section 5. A class of Toeplitz-Hankel integral equations and certain systems of integral equations are studied by virtue of the polyconvolution (proposed in the present paper) and it is proved that these equations can be solved in a closed form in Section 6.
scientific article; zbMATH DE number 6678128