Uniform Aproximations of Solutions to a Strongly Singular Integral Equation of the First Kind

Abstract

The article is devoted to uniform approximations of the solution of a periodic strongly singular integral equation of the first kind. We note that an integral equation of the first kind is an ill-posed problem. However, the presence of certain features in the kernels makes it possible to establish a correct formulation of the problem on specially selected function spaces. After this selection, various approximation methods can be applied to solve the equation under consideration. In the present article, the authors use this approach, previously applied in the periodic and non-periodic cases to integral equations of the first kind with a logarithmic singularity in the kernel, to integral equations with the Cauchy kernel on a segment of the real axis, and to some singular integro-differential equations. The spaces of the required elements and right-hand sides are taken to be certain narrowings of the space of $2\pi$-periodic continuous functions with a certain norm. The boundedness of the operator, inverse to the characteristic one on a pair of selected spaces, is established. Next, projection methods for solving the characteristic and complete equations are considered. Constructive estimates of the approximation of a function by partial sums of its Fourier series and Lagrange interpolation polynomials in the proposed spaces make it possible to establish uniform approximations of solutions to the equation using the Galerkin and collocation methods.