# integral operator

## Structure of the inverse for the integral operator of special kind

Algebra (with identity) generated by integral operators on the spaces of continuous periodic functions is considered. This algebra is proved to be an inverse-closed subalgebra in the algebra of all bounded linear operators.

## The problem of Leont'ev on entire functions of completely regular growth

We consider an entire function of exponential type with all its zeros are simple and form a sequence with the index condensation zero. On the set of zeros a function of its derivative is growing quickly. Required to determine whether original function have complete regularity of growth. This problem, which arose in the theory of representation of analytic functions by exponential series was posed by A. F. Leontiev more than forty years ago and has not yet been solved.

## Equiconvergence Theorem for Expansions in Eigenfunctions of Integral Operators with Discontinuous Involution

In the paper we consider the equiconvergence of expansions in trigonometric Fourier series and in eigen- and associated functions of integral operators with involution having discontinuities of the ﬁrst type.

## Operator Integration with an Involution Having a Power Singularity

Spectral properties of the integral operator with an involution of special type in the upper limit are studied and an equiconvergence theorem for its generalized eigenfunction expansions is obtained.

## The Theorem on Equiconvergence for the Integral Operator on Simplest Graph with Cycle

The paper deals with integral operators on the simplest geometric two-edge graph containing the cycle. The class of integral operators with range of values satisfying continuity condition into internal node of graph is described. The equiconvergence of expansions in eigen and adjoint functions and trigonometric Fourier series is established.

## On Riesz Basises of Eigenfunctions of Integral Operators with Kernels Discontinuous on Broken Lines

For the integral operator, which kernel has jump discontinuities on the sides and diagonals of the four equal subsquares of the unit square 0 ≤ x, t ≤ 1, Riesz basisness of its eigen and associated functions is proved.

## 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.