Mathematics

About the Norms of Interpolation Processes with Fixed Nodes

Authors: Smotritski K. A., Dirvuk Y. V.

The object of study is interpolating rational Lagrange functions. The aim of the research — the study of approximation properties of these functions in the space of square integrated functions. In the introduction the relevance of the research is indicated, references to some works related to this article are given. We also describe the construction of the apparatus of approximation — interpolating rational Lagrange functions. In the main part the norm of the interpolating rational function in the space of the square integrated functions is calculated.

On Spectrum of Schrödinger Operator on Manifold of a Special Type

Authors: Svetlov A. V.

The main subject of the paper is spectrum of the Schrödinger operator on weighted quasimodel manifold with an end, which is warped product of a special type. We prove the criterion of discreteness for the spectrum of the operator in terms of metric coefficients and potential of the operator. As the conclusion we made some remarks on the corollaries of the proved theorem and on its extension to more complex quasimodel manifolds.

On Multiple Completeness of the Root Functions of a Certain Class of Pencils of Differential Operators with Constant Coefficients

Authors: Rykhlov V. S., Blinkova O. V.

A polinomial pencil of ordinary differential operators of n-th order generated by a homogeneous differential expression with constant coefficients and by two-point boundary conditions of a special structure with l conditions in zero only (1 ≤ l ≤ n−1) is considered in the space L2[0,1]. The case is studied, when the roots of the characteristic equation lie on a ray coming from the origin.

Martingales and Theorems of Cantor–Young–Bernstein and de la Vallée Poussin

Authors: Plotnikov M. G., Plotnikova J. A.

Uniqueness problems for one-dimensional Haar series and for multiple ones have understood in numerous works. It is well-known that the subsequence of the partial sums S2k of an arbitrary Haar series can be represented as a discrete-time martingale on some filtered probability space (­Ω, F, (Fk), P). In paper the concept of a U -set for martingales is presented and some uniqueness theorems for martingales on arbitrary compact filtered probability spaces are established.

Riescz Basis Property of Eigen and Associated Functions of Integral Operators with Discontinuous Kernels, Containing Involution

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

For invertible integral operator which kernel is discontinuous on the diagonals of the unit square Riescz basis property of its eigen and associated functions in L2[0, 1] is proved.

Green Function of the Dirichlet Boundary Value Problem for Polyharmonic Equation in a Ball Under Polynomial Data

Authors: Karachik V. V.

The classical Dirichlet boundary value problem for the polyharmonic equation in the unit ball is considered. For this problem with polynomial right-hand side and zero boundary data a polynomial solution is constructed. Our approach is based on the Almansi representation of polyharmonic functions and on the previously obtained an explicit representation of the harmonic components, expressed through the given polyharmonic function. In the case of the harmonic equation the known representation of the solution through the Green function is obtained.

Uniqueness of Solution of the Inverse Scattering Problem for Various Order Differential Equation on the Simplest Noncompact Graph with Cycle

Authors: Ignatyev M. Y.

An inverse scattering problem is studied for variable orders differential operators on simplest noncompact graph with cycle. A uniqueness theorem of recovering coefficients of operators from the scattering data is provided.

On Equivalence of the Method of Steepest Descent and the Method of Hypodifferential Descent in Some Constrained Optimization Problems

Authors: Dolgopolik M. V., Tamasyan G. S.

The method of exact penalty functions is widely used for the study of constrained optimization problems. The approach based on exact penalization was successfully applied to the study of optimal control problems and various problems of the calculus of variations, computational geometry and mathematical diagnostics. It is worth mentioning that even if the constrained optimization problem under consideration is smooth, the equivalent unconstrained optimization problems constructed via exact penalization technique is

About the Retrofit of the Valle’e-Poussin’s Algorithm for Approximations of Multivalued Mappings by Algebraic Polynomial with Type Constraint Equality

Authors: Vygodchikova I. Y.

The discrete approximation of noisy data by algebraic polynomial with restriction of type equality is studied. The aimof the investigation is to obtain the fundamental properties of solution of the problem and development by them the new algorithm, more effective, in comparison with existing methods of the solution. The tasks of the research — gets the properties of the solution of the problem, presentation of the algorithm and the demonstration of its implementation. Research methodology continues P. L. Chebyshjov’s and Valle-Pussen’s method. Results.

Embedding Theorems for P-nary Hardy and VMO Spaces

Authors: Volosivets S. S.

In the present paper several embedding theorems of P. L. Ul’yanov type for H¨older spaces connected with P-nary Hardy, VMO, L1 and uniform metric on Vilenkin groups are proved. Its sharpness is also established. The sufficient conditions for the convergence of Fourier series with respect to multiplicative systems in Hardy space and uniform metric are also given.

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