Equiconvergence theorem for integral operator with involution

In the paper, the integral operator with kernel having discontinuities of the first kind at the lines $t=x$ and $t=1-x$ is studied. The equiconvergence of Fourier expansions for arbitrary integrable function $f(x)$ in eigenfunctions and associated functions of the considered operator and expansions of linear combination of functions $f(x)$ and $f(1-x)$ in trigonometric system is proved. The equiconvergence is studied using the method based on integration of the resolvent using spectral value. Methods, developed by A. P. Khromov in the study of spectral theory of integral operators are widely used. Recently, these methods are of use in studies of boundary value problems of mathematical physics using Fourier method with minimal smoothness conditions for the initial data.