Mathematics

Discrete Transform with Stick Property Based on {sinx sinkx} and Second Kind Chebyshev Polynomials Systems

Authors: Sharapudinov I. I., Akniev G. G.

In this paper we introduce the discrete series with the «sticking»-property of the periodic ({sinx sinkx} system) and non-periodic (using the system of the second kind of Chebyshev polynomials Uk(x)) cases. It is shown that series of the system {sinx sinkx}
have an advantage over cosine Fourier series because they have better approximation properties near the bounds of the [0, π] segment. Similarly discrete series of the system Uk(x) near the bound of the [−1, 1] approximates given function significantly

Some Special Two-dimensional Series of {sinx sinkx} System and Their Approximation Properties

Authors: Sharapudinov I. I.

In present paper there were introduced two-dimensional special series of the system {sinx sinkx}. It’s shown that these series have the advantage over two-dimensional cosine Fourier series, because they have better approximation properties near the bounds of the square [0, 1]2. It’s given convergence speed estimate of special series partial sums to functions f(x, y) from the space of even 2π-periodic continuous functions.

The Intermediate Case of Regularity in the Problem of Differentiation of Multiple Integrals

Authors: Fufaev D. V.

The paper deals with generalization of Lebesgue and Jessen –Marcinkiewicz – Zygmund theorems of the differentiation of multiple integrals for the intermediate case of regularity of the system of sets. The application of the result to the Fourier-Haar series and to orthorecursive expansions with respect to system of indicators of multi-dimensional intervals is considered.

Orthogonal Basis of Shifts in Space of Trigonometric Polynomials

Authors: Lukashenko T. P.

The orthonormal basis of a system of shifts of one trigonometric polynomial exist in the space of complex trigonometric polynomials with components from m to n and in the space of real trigonometric polynomials with components from 0 to n. Under condition 0 < m < n there is no orthogonal basis of shifts of one trigonometric polynomial in this space real trigonometric polynomials with components from m to n. The system of shifts of two trigonometric polynomials are orthogonal basis in this space.

New Properties of Almost Nilpotent Variety of Exponent 2

Authors: Shulezhko O. V.

In the presented work we consider numerical characteristics of almost nilpotent variety of exponent 2, which was first constructing in article [1]. The main result of this paper is introduce the exact values of the multiplicities of the irreducible modules appearing in the expansion of the multilinear part of the variety. Meanwhile, we obtain as a consequence the formulas of codimension and colength of the variety of exponent 2.

The Gradient Methods for Solving the Cauchy Problem for a Nonlinear ODE System

Authors: Fominykh A. V.

The article considers the Cauchy problem for a nonlinear system of ODE. This problem is reduced to the variational problem of minimizing some functional on the whole space. For this functional necessary minimum conditions are presented. On the basis of these conditions the steepest descent method and the method of conjugate directions for the considered problem are described. Numerical examples of the implementation of these methods are presented. The Cauchy problem with the system which is not solved with respect to derivatives is additionally investigated.

Approximation of the Riemann–Liouville Integrals by Algebraic Polynomials on the Segment

Authors: Tyuleneva A. A.

The direct approximation theorem by algebraic polynomials is proved for Riemann–Liouville integrals of order r>0. As a corollary, we obtain asymptotic equalities for ε-entropy of the image of a Hölder type class under Riemann–Liouville integration operator.

Approximation of Functions by Fourier–Haar Sums in Weighted Variable Lebesgue and Sobolev Spaces

Authors: Magomed-Kasumov M. G.

It is considered weighted variable Lebesgue Lp(x)w and Sobolev Wp(⋅),w spaces with conditions on exponent p(x)≥1 and weight w(x) that provide Haar system to be a basis in Lp(x)w. In such spaces there were obtained estimates of Fourier–Haar sums convergence speed. Estimates are given in terms of modulus of continuity Ω(f,δ)p(⋅),w, based on mean shift (Steklov's function).

Approximation of Functions in Symmetrical and Connected Holder Spaces by Linear Means of Fourier–Vilenkin Series

Authors: Likhacheva T. V.

In this paper some summation methods are applied to Fourier-Vilenkin series in so called symmetric spaces. These methods use triangular matrix with sums in rows tending to zero and with some conditions on difference of coefficients. The triginometric counterpart of our results are due to M. L. Mittal, B. E. Rhoades, A. Guven, etc.

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