The equiconvergence theorem for an integral operator with piecewise constant kernel

The paper is devoted to the equiconvergence of the trigonometric Fourier series and the expansions in the eigen and associated functions of the integral operator $ A $, the kernel of which has jumps on the sides of the square inscribed in the unit square. An equivalent integral operator in the space of 4-dimension vector-functions is introduced. This operator is remarkable for the fact that the components of its kernel have discontinuities only on the line diagonal. Necessary and sufficient conditions of the invertibility of the operator $ A $ are obtained in the form that a certain 4$^{\text{th}}$ order determinant is not zero. The Fredholm resolvent of the operator $ A $ is studied and its formula is found. The constructing of this formula is reduced to the solving of the boundary value problem for the first order differential system in the 4-dimension vector-functions space. To overcome the difficulties of this solving the transformation of the boundary value problem is carried out. Conditions analogous to Birkhoff regularity conditions are also obtained. These conditions mean that some 4$^{\text{th}}$ order determinants are not zero and can be easily verified. Under these conditions the determinant, which zeros are the eigenvalue of the boundary value problem, can be estimated. The equiconvergence theorem for the operator $A$ is formulated. The basic method used in the proof of this theorem is Cauchy–Poincare method of integrating the resolvent of the operator $A$ over expanding contours in the complex plane of the spectral parameter. An example is also given of the integral operator with piecewise constant kernel, which satisfies all the requirements obtained in the paper.