# Mathematics

## About Asymptotic Polynomials, Orthogonal on Any Grids

Asymptotic properties of polynomials orthogonal ln(x), with weight e −xj ∆tj on any infinite set points from semi-axis [0, ∞) are investigated. Namely an asymptotic formula is proved in which asymptotic behaviour of these polynomials as n tends to infinity together with N is closely related to asymptotic behaviour of the polynomials by Lagerra.

## Nonorthogonal Multiresolution Analysis on Zero-Dimensional Locally Compact Groups

We given necessary and sufficient condition under which the solution of refinement equation with compactly supported Fourier transform generate the multiresolution analysis.

## On Characterization Determining Entire Functions and Consistent with Riman’s Type Equation Dirichlet’s Series with Finetly-Valued Coefficients

In the investigation were founded specifications for Dirichlet’s series coefficients, wherein this series determine entire function and measure up functional Riman’s type equation. Were shown that exist infinit multitude of such series that are different from Dirichlet’s L-functions.

## Approximation of Functions by Borel Means of Fourier Series with Respect to Multiplicative Systems

In the present paper we consider Borel means of Fourier series with respect to Vilenkin systems with bounded generating sequence and obtain some estimates of approximation by this means in L p , uniform and Holder type norm in classes of functions with given majorant of best approximation or modulus of continuity. In the trigonometric case similar results were established by P.Chandra, L.Rempulska and K.Tomczak.

## On Iterative Method of Constructing Optimal Control for Singularly Perturbed Systems with Delay with Quadratic Constraints

The control problem for the singularly perturbed system with delay with indeterminate initial conditions and integral quadratic constraints on the control resources according to the minimax criterion is considered. Iterative procedure of constructing control response that approximates the optimal solution with given accuracy with respect to a small positive parameter is proposed.

## Problem with Conditions on all Boundary for One 6-th Order Pseudoparabolic Equation

Here consider characteristic problem with conditions, setting on all boundary, in two order space for 6th order equation with 3-times taken old particular derivative. The existence and uniqueness of the solution are proved.

## Optimality Solutions in Games with Preference Relations

For n person games with preference relations some types of optimality solutions are introduced. Elementary properties of their solutions are considered. One sufficient condition for nonempty Cα-core is found.

## Parametrization of Bivariate Nonseparable Haar Wavelets

A parametrization of all orthogonal wavelet bases for Haar multiresolution analysis is derived. The bases generated by three piecewise constant wavelets {ηi(x, y)}, i = 1, 2, 3, supported on [0, 1] × [0, 1], with values aij ∈ R, i = 1, 2, 3, j = 1, 2, 3, 4 are considered.

## The Characteristic of Stability of the Solution in the Problem of Convex Compact Set Asphericity

We consider the problem of stability of the solution in the problem of asphericity of a convex set with respect to the error of defining the compact set. It is shown that the optimal value of the criterion function (an asphericity indicator) is stable. Properties of the setvalued mapping, that puts to a convex compact compact set the centers of its asphericity are also investigated. It is proved that this mapping is semicontinious from above everywhere in the space of convex compact sets.

## Λ-Summability and Multiplicators of Holder Classes of Fourierseries with Respect ̈ to Character Systems

Let G be a Vilenkin group of bounded type. We obtain nessesary and sufficient conditions of uniform Λ-summability for all Fourier series of f ∈ C(G) and one of Λ-summability in L 1 (G) for all Fourier series of f ∈ L 1 (G). Also we extend some T. Quek and L. Yap results to the case of general modulus of continuity.