Mathematics

Uniform Convergence of the Series with Respect to Multiplicative Systems

Authors: Fadeev R. N.

Two theorems on uniform convergence and boundedness of partial sums for the series with generalized monotone coefficients with respect to multiplicative systems are proved.

Affine Quantum Frames and Their Spectrum

Authors: Terekhin P. A.

The problem of coefficients quantization for polynomials is solved for affine frames. The problem about coefficients quantization for frame decomposition is considered also. The notion of a spectrum of the quantum frame is introduced. The spectrum of family of affine frames is estimated.

Limit Discrete Meixner Series and Their Approximative Properties

Authors: Sultanov E. S.

In this article the problemof function approximation by discrete series by Meixner polynomials orthogonal on uniform net {0, 1, . . .} is investigated. We constructed new series by these polynomials for which partial sums coincidewith input function f(x) in x = 0. These new series were constructed by the passage to the limit of Fourier series Σk=0∞fαkmαk(x) by Meixner polynomials when α → −1.

A Countably Connected Domain is not Homeomorphic to an Uncountably Connected Domain

Authors: Starkov V. V.

In 1923 Керекьярто proved, that a countably connected domain is not homeomorphic to an uncountaby connected domain. We give another proof of this statement.

Expansion in Eigenfunctions of Quadratic Strongly Irregular Pencils of Differential Operators of the Second Order

Authors: Rykhlov V. S.

We consider a quadratic strongly irregular pencil of 2-d order ordinary differential operators with constant coefficients and positive roots of the characteristic equation. Both the amounts of double expansions in a series in the derivative chains of such pencils and necessary and sufficient conditions for convergence of these expansions to the decomposed vector-valued function are found.

On Relationship between Derivative of Multifunction and Its Support Function

Authors: Polovinkin E. S.

We obtain sufficient conditions under which the support function of the derivative of a set-valued mapping coincides with the derivative of the support function of a set-valued mapping in some sence. The example showing the difference between these concepts and the example of a Lipschitz set-valued mapping whose support function at any point does not have the mixed derivatives are obtained.

Qualitative Properties of Mild Solutions of the Cauchy Problem

Authors: Kaluzhina N. S.

In this paper we study the qualitative properties of a mild solution of the problem Cauchy problem for the heat equation. We prove that every mild Cauchy problem is a slowly varying at infinity function.
The result is applied to study solutions of the Neumann problem for the heat equation.

About the Classical Solution of the Mixed Problem for the Wave Equation

Authors: Khromov A. P.

The classic solution of the mixed problem for a wave equation with a complex potential and minimal smoothness of initial data is established by the Fourier method. The resolvent approach consists of constructing formal solution with the help of the Cauchy – Poincaré method of integrating the resolvent of the corresponding spectral problem over spectral parameter. The method requires no information about eigen and associated functions and uses only the main part of eigenvalues asymptotics. Krylov’s idea of accelerating the convergence of Fourier series is essentially employed.

The Solution of the Homogeneous Riemann Boundary Value Problem with a Countable Set of Points of Discontinuity of the First Kind its Coefficient

Authors: Salimov R. B.

We consider the Riemann homogeneous boundary value problem with a countable set of points of discontinuity of the first kind in the case, when it is required to find two functions, analytic, respectively, in the upper and lower half-plane, for a given linear boundary condition on the real axis, connecting the boundary values of the unknown functions.

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