Mathematics

The Il’in Spectral Method for Determination of the Properties of the Basis Property and the Uniform Convergence of Biorthogonal Expansions on a Finite Interval

Authors: Lomov I. S.

The paper discusses the basics of the spectral method of V. A. Il’in on an example of a simple second order differential operator on a segment of the number line. The first theorem of Il’in on the unconditional basis property is stated. Its detailed proof is given. A chain of generalizations of this theorem is traced. A recently established a theorem on the unconditional basis property for the differential operators with general integral boundary conditions is formulated.

Necessary and Sufficient Condition for an Orthogonal Scaling Function on Vilenkin Groups

Authors: Berdnikov G. S.

There are several approaches to the problem of construction of an orthogonal MRA on Vilenkin groups, but all of them are reduced to the search of the so-called scaling function. In 2005 Yu. Farkov used the so-called “blocked sets” in order to find all possible band-limited scaling functions with compact support for each set of certain parameters and his conditions are necessary and sufficient. S. F. Lukomskii, Iu. S. Kruss and G. S.

Nonlocal Boundary-Value Problems in the Cylindrical Domain for the Multidimensional Laplace Equation

Authors: Aldashev S. A.

Correct statements of boundary value problems on the plane for elliptic equations by the method of analytic function theory of a complex variable. Investigating similar questions, when the number of independent variablesis greater than two, problems of a fundamental nature arise. Avery attractive and convenient method of singular integral equations loses its validity due to the absence of any complete theory of multidimensional singular integral equations. The author has previously studied local boundary value problems in a cylindrical domain for multidimensional elliptic equations.

Approximation of Continuous 2 p-Periodic Piecewise Smooth Functions by Discrete Fourier Sums

Authors: Akniev G. G.

Let N be a natural number greater than 1. Select N uniformly distributed points tk = 2πk/N + u (0 6 k 6 N − 1), and denote by Ln,N(f) = Ln,N(f,x) (1 6 n 6 N/2) the trigonometric polynomial of order n possessing the least quadratic deviation from f with respect to the system {tk}N−1 k=0 . Select m + 1 points −π = a0 < a1 < ... < am−1 < am = π, where m > 2, and denote Ω = {ai}m i=0. Denote by Cr Ω a class of 2π-periodic continuous functions f, where f is r-times differentiable on each segment ∆i = [ai,ai+1] and f(r) is absolutely continuous on ∆i.