# Sukhorolskyi M. A.

### Bessel functions of two complex mutually conjugated variables and their application in boundary-value problems of mathematical physics

Ukr. Mat. Zh. - 2017. - 69, № 3. - pp. 381-396

We formulate boundary-value problems for the eigenvalues and eigenfunctions of the Helmholtz equation in simply connected domains by using two complex mutually conjugated variables. The systems of eigenfunctions of these problems are orthogonal in the domain. They are formed by Bessel functions of complex variables and the powers of conformal mappings of the analyzed domains onto a circle. The boundary-value problems for the main equations of mathematical physics are formulated in an infinite cylinder with the use of complex and time variables. The solutions of the boundaryvalue problems are obtained in the form of series in the systems of eigenfunctions. The Cauchy problem for the main equations of mathematical physics with three independent variables is also considered.

### Boundary-value problems for the Helmholts equation in domains of the complex plane

Ukr. Mat. Zh. - 2016. - 68, № 3. - pp. 363-376

By using the conformal mappings of plane with elliptic hole and a plane with cross-shaped hole into the outside of the circle, we construct systems of functions playing the role of bases in the spaces of the functions analytic in these domains. The Faber polynomials are biorthogonal with the basis functions. We construct the solutions of the Helmholtz equation in the plane with holes whose boundary values coincide with the boundary values of analytic functions represented in the form of series in these bases.

### Expansion of functions in a system of polynomials biorthogonal on a closed contour with a system of functions regular at infinitely remote point

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 238–254

We study properties of the systems of polynomials constructed according to the schemes similar to the schemes used for the Bernoulli and Euler polynomials, formulate conditions for the existence of functions associated with polynomials and conditions of representation of polynomials by contour integrals, and present the classes of analytic functions expandable in series in the systems of polynomials. The expansions of functions are illustrated by examples.

### Domain of convergence of the Euler transform for the power series of an analytic function

Ukr. Mat. Zh. - 2008. - 60, № 8. - pp. 1144–1152

We consider the Euler transform of the power series of an analytic function playing the role of its expansion in a series in a system of polynomials and study the domain of convergence of the transform depending on the parameter of transformation and the character of singular points of the function. It is shown that the transform extends the function beyond the boundaries of the disk of convergence of its series on the interval of the boundary located between two singular points of the function. In particular, it is established that the power series of the function whose singular points are located on a single ray is summed by the transformation in the half plane.

### On the order of local approximation of functions by trigonometric polynomials that are partial sums of averaging operators

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 706–714

We study the order of polynomial approximations of periodic functions on intervals which are internal with respect to the main interval of periodicity and on which these functions are sufficiently smooth. The estimates obtained contain parameters which characterize the smoothness and alternation of signs of nuclear functions and parameters that determine classes of approximated functions.