Mathematics

On Convergence of Bernstein – Kantorovich Operators sequence in Variable Exponent Lebesgue Spaces

Authors: Shakh-Emirov T. N.

Let E = [0, 1] and let a function p(x) > 1 be measurable and essentially bounded on E. We denote by L p(x) (E) the set of measurable function f on E for which R E |f(x)| p(x) dx < ∞. The convergence of a sequence of operators of Bernstein – Kantorovich {Kn(f, x)} ∞n=1 to the function f in Lebesgue spaces with variable exponent L p(x) (E) is studied.

Sobolev Orthogonal Polynomials Generated by Meixner Polynomials

Authors: Sharapudinov I. I., Gadzhieva Z. D.

The problem of constructing Sobolev orthogonal polynomials mα r,n(x, q) (n = 0, 1, . . .), generated by classical Meixner’s polynomials is considered.

On a Limit Value of a Remainder of the Lagrange Constant Corresponding to the Lagrange Trigonometrical Polynomial

Authors: Shakirov I. A.

The behavior of Lebesgue constant of a trigonometrical Lagrange polynomial interpolating the periodic function in an odd number of clusters is studied. The limit value of the remainder in the known asymptotic formula for this constant is found. A special representation of a remainder allowed us to establish its strict decreasing. On this basis, for a Lebesgue constant, a non-improvable uniform bilateral logarithmic function estimate is received.

On Operators with Discontinuous Range

Authors: Khromova G. V.

With the use of operators from approximation function theory we construct integral operators with discontinuous range of values, which make it possible to obtain uniform approximations of continuous functions on the whole interval of their definition. 

Necessary and Sufficient Conditions for the Uniform on a Segment Sinc-approximations Functions of Bounded Variation

Authors: Trynin A. Y.

The necessary and sufficient conditions for the uniform convergence of sinc-approximations of functions of bounded variation is obtained. Separately we consider the conditions for the uniform convergence in the interval (0, π) and on the interval [0, π]. The impossibility of uniform approximation of arbitrary continuous function of bounded variation on the interval [0, π] is settled.

On the Solvability of the Discrete Analogue of the Minkowski – Alexandrov Problem

Authors: Klyachin V. A.

The article deals with the multidimensional discrete analogue of the Minkowski problem in the production of A. D. Aleksandrov on the existence of a convex polyhedron with given curvatures at the vertices. We find the conditions for the solvability of this problem in a general setting, when the curvature measure at the polyhedron vertices is defined by an arbitrary continuous function defined on a field F : S n−1 → (0, +∞).

Iterative Procedure of Constructing Optimal Solving in the Minimax Problem of Control for Singularly Perturbed System with Delay with Geometric Constraints

Authors: Grebennikova I. V., Kremlev A. G.

The control problem for the singularly perturbed system with delay with indeterminate initial conditions and geometric 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.

Admissible Hypercomplex Structures on Distributions of Sasakian Manifolds

Authors: Galaev S. V.

The notions of admissible (almost) hypercomplex structure and almost contact hyper-Kahlerian structure are introduced. On a ¨ manifold M with an almost contact metric structure (M, ~ξ, η, ϕ, D) an interior symmetric connection ∇ is defined. In the case of a contact manifold of dimension bigger than or equal to five, it is proved that the curvature tensor of the connection ∇ is zero if and only if there exist adapted coordinate charts with respect to that the coefficients of the interior connection are zero.

Orthogonal Shift Systems in the Field of p-adic Numbers

Authors: Vodolasov A. M., Lukomskii S. F.

In 2010 S. Albeverio, S. Evdokimov and M. Skopina proved that if the shift system (ϕ(x−˙ h)) of a step function ϕ is orthonormal and ϕ generates p-adic MRA then its Fourier transform lies in the unit ball. We prove then in some cases the condition "ϕ generates MRA" is possible to be omitted. In general, we indicate the number of linearly independent step-functions, which shifts form an orthonormal system.

Affine System of Walsh Type. Completeness and Minimality

Authors: Al-Jourany K. H., Mironov V. A., Terekhin P. A.

The question on completeness and minimality of Walsh affine systems is considered. On the basis of functional-analytical structure of multishift in Hilbert space, which being the generalized analogue of the operator of simple one-side shift and closely connected with Cuntz algebra representations, we give definition of Walsh affine system. Various criteria and tests of completeness of affine systems of functions are established. A biorthogonal conjugate system is found and its completeness is proved.