Mathematics

Factorization of Entire Symmetrical Functions of Exponential Type

Authors: Shishkin A. B.

Let π be an entire function of minimal type of order 1. The entire function F is called π-symmetric if it is represented in the form of a composition f ◦ π, where the f is an entire function. The article deals with the following question. Can we present every π-symmetric function of exponential type as a product of two functions with a close growth, each of which is itself an entire π-symmetric function?

Special Wavelets Based on Chebyshev Polynomials of the Second Kind and their Approximative Properties

Authors: Sultanakhmedov M. S.

The system of wavelets and scalar functions based on Chebyshev polynomials of the second kind and their zeros is considered. With the help of them we construct a complete orthonormal system of functions.

About New Approach to Solution of Riemann’s Boundary Value Problem with Condition on the Half-line in Case of Infinite Index

Authors: Salimov R. B.

To solve a homogeneous Riemann boundary value problem with infinite index and condition on the half-line we propose a new approach based on the reduction of the considered problem to the corresponding task with the condition on the real axis and finite index.

Justification of Fourier Method in a Mixed Problem for Wave Equation with Non-zero Velocity

Authors: Gurevich A. P., Kurdyumov V. P., Khromov A. P.

In the paper, using contour integration of the resolvent of the corresponding spectral problem operator, justification of Fourier method in two mixed problems for wave equation with trivial initial function and non-zero velocity is given. The boundary conditions of these problems, together with fixed endpoint conditions, embrace all cases of mixed problems with the same initial conditions for which the corresponding spectral operators in Fourier method have regular boundary conditions. The problems are considered under minimal requirements on initial data. A. N.

Stochastic Simulation of Diffusion Filtering

Authors: Arutyunyan R. V.

Formulated and investigated is the system of kinetic equations describing the process of diffusion filtering based on a stochastic approach. The theorem of existence and uniqueness of the solution for the case of a continuous density is prove.

On the Least Type of Entire Functions of Order ½ ∈ (0, 1) with Positive Zeros

Authors: Sherstyukova O. V.

The paper is devoted to the theory of extremal problems in classes of entire functions with constraints on the growth and distribution of zeros and is associated with problems of completeness of exponential systems in the complex domain. The question of finding the exact lower bound for types of all entire functions of order p ∈ (0, 1) whose zeros lie on the ray and have prescribed upper p-density and p-step is discussed. It is shown that the infimum is attained in this problem, and a detailed construction of the extremal function is given.

Dominant Integrands Growth Estimates and Smoothness of Variational Functionals in Sobolev Spaces

Authors: Orlov I. V., Romanenko I. A.

For variational functionals in Sobolev spaces {W1,p} (1 ≤ p < ∞) we introduce a sequence of so-called dominant "growth estimates" for the gradient of appropriate order of the integrand, each of which guarantees the appropriate level of smoothness of variational functional in the C1-smooth points of the Sobolev space. Earlier studied K-pseudopolynomial representations of the integrand are particular cases of dominant growth estimates.

Interpolation of Continuous in Ordered H-variation Functions

Authors: Novikov V. V.

In 1972 D. Vaterman introduced a class of functions of Λ-bounded variation (in particular, a harmonic variation or an H-variation). Later he introduced also the class of functions of ordered ¤-bounded variation and the class of continuous in Λ-variation functions. These classes have been used by many authors in studies on the convergence and summability of the Fourier series.

Estimates of Speed of Convergence and Equiconvergence of Spectral Decomposition of Ordinary Differential Operators

Authors: Lomov I. S.

The present review contains results of V. A. Il’in and his pupils concerning an assessment of speed of convergence and equiconvergence with a trigonometrical series of Fourier of spectral decomposition of functions on root functions of linear ordinary differential
operators both self-conjugate, and not self-conjugate, set on a final piece of a numerical straight line. The first theorem of V. A. Ilyin of equiconvergence of spectral decomposition for the differential operator of any order is provided. Theorems of the speed of

On Riescz Bases of Eigenfunction of 2-nd Order Differential Operator with Involution and Integral Boundary Conditions

Authors: Kurdyumov V. P.

Riesz basisness with brackets of the eigen and associated function is proved for a 2-nd order differential operator with involution in the derivatives and with integral boundary conditions. To demonstrate this the spectral problem of the initial operator is reduced to the spectral problem of a 1-st order operator without involution in the 4-dimensional vector-function space.

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