# Mathematics

## The uniqueness of the solution of an initial boundary value problem for a hyperbolic equation with a mixed derivative and a formula for the solution

An initial boundary value problem for an inhomogeneous second-order hyperbolic equation on a finite segment with constant coefficients and a mixed derivative is investigated. The case of fixed ends is considered. It is assumed that the roots of the characteristic equation are simple and lie on the real axis on different sides of the origin. The classical solution of the initial boundary value problem is determined.

## On the approximation of bounded functions by trigonometric polynomials in Hausdorff metric

The article discusses a method for constructing a spline function to obtain estimates that are exact in order to approximate bounded functions by trigonometric polynomials in the Hausdorff metric. The introduction provides a brief history of approximation of continuous and bounded functions in the uniform metric and the Hausdorff metric. Section 1 contains the main definitions, necessary facts, and formulates the main result. An estimate for the indicated approximations is obtained from Jackson's inequality for uniform approximations.

## Rate of interpolation of analytic functions with regularly decreasing coefficients by simple partial fractions

We consider the problems of multiple interpolation of analytic functions $f(z)=f_0+f_1z+\dots$ in the unit disk with node $z=0$ by means of simple partial fractions (logarithmic derivatives of algebraic polynomials) with free poles and with all poles on the circle $|z|=1$. We obtain estimates of the interpolation errors under a condition of the form $|f_{m-1}|<C/\sqrt{m}$, $m=1,2,\dots$.

## On estimates of the order of the best M–term approximations of functions of several variables in the anisotropic Lorentz – Zygmund space

The article considers the anisotropic Lorentz – Karamata space of periodic functions of several variables and the Nikol'skii – Besov class in this space. The order-sharp estimates are established for the best $M$-term trigonometric approximations of functions from the Nikol'skii-Besov class in the norm of another Lorentz – Zygmund space.

## The Riemann problem on a ray for generalized analytic functions with a singular line

In this paper, we study an inhomogeneous Riemann boundary value problem with a finite index and a boundary condition on a ray for a generalized Cauchy – Riemann equation with a singular coefficient. For the solution of this problem, we derived a formula for the general solution of the generalized Cauchy – Riemann equation under constraints that led to an infinite index of logarithmic order of the accompanying problem for analytical functions.

## On the application of the qualitative theory of differential equations to a problem of heat and mass transfer

The possibilities of applying the qualitative theory of differential equations to one problem of heat and mass transfer in multilayer planar semiconducting structures are studied. The consideration is carried out on the example of a mathematical model of a stationary process of diffusion of nonequilibrium minority charge carriers generated by a wide excitation source.

## A new approach to the formation of systems of linear algebraic equations for solving ordinary differential equations by the collocation method

A new algorithm for the numerical solution of one-dimensional Cauchy problems and Poisson equations is implemented. The algorithm is based on the collocation method and representation of the solution as an expansion in Chebyshev polynomials. It is proposed instead of the usual approach, which consists in combining all known conditions — differential (the equation itself) and initial / boundary — into one system of approximate linear algebraic equations, to go to the method of solving the problem in several separate stages.

## Function correction and Lagrange – Jacobi type interpolation

It is well-known that the Lagrange interpolation based on the Chebyshev nodes may be divergent everywhere (for arbitrary nodes, almost everywhere), like the Fourier series of a summable function. On the other hand, any measurable almost everywhere finite function can be “adjusted” in a set of an arbitrarily small measure such that its Fourier series will be uniformly convergent. The question arises whether the class of continuous functions has a similar property with respect to any interpolation process.

## Application of the generalized degree method for constructing solutions of the quaternion variant of the Cauchy – Riemann system

This article indicates one of the ways to solve the generalized Cauchy – Riemann system for quaternionic functions in an eight-dimensional space. In previous works, some classes of solutions of this system were studied and it was stated that it is possible to use the method of generalized degrees to construct solutions of this system of differential equations. It is shown that the solution of the problem can be reduced to finding two arbitrary quaternionic harmonic functions in an eight-dimensional space.

## The Lezanski – Polyak – Lojasiewicz inequality and the convergence of the gradient projection algorithm

We consider the Lezanski – Polyak – Lojasiewicz inequality for a real-analytic function on a real-analytic compact manifold without boundary in finite-dimensional Euclidean space.