Convolutions generated by the dirichlet problem of the sturm-liouville operator

Abstract

This paper is devoted to approximations of the product of two continuous functions on a finite segment by some special convolutions. The accuracy of the approximation depends on the length of the segment on which the functions are defined. These convolutions are generated by the Sturm-Liouville boundary value problems. The paper indicates that each boundary value problem for a second order differential equation generates its own individual convolution and its own individual Fourier transform. At that the Fourier transform of the convolution is equal to the product of the Fourier transforms. The latter property makes it possible to approximately solve nonlinear Burgers-type equations by first replacing the nonlinear term with a convolution of two functions. Similar methods of studying nonlinear partial differential equations can be found in the works of A. Y. Kolesov, N. H. Rozov, V. A. Sadovnichy. In this paper, we construct a concrete convolution generated by the Dirichlet boundary value problem for twofold differentiation. The properties of the constructed convolution and their connection with the corresponding Fourier transform are derived. In the final part of the paper, the convergence of convolution is proved (g(x) sin(x)) ∗ (f(x) sin(x)) defined on a segment C[0, b] to the product g(x)f(x) with b tending to zero for any two continuous functions f(x) and g(x).