Mathematics

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.

Invariants on a Set of Reciprocal Iterated Exponential Power Coefficients

Authors: Bulanov A. P.

A chain exponent LB(z) = z · B(z),  having a power sequence {bn}∞n=1, bn ≠ 0, n = 1,2,..., lim n→∞ |bn| < ∞, is defined by a function sequence B(z) = eb1·z·B1(z), B1(z) = eb2·z·B2(z), . . . , Bk−1(z) = ebk·z·Bk(z),. . . (we use the denotation B(z) = ‹ez;b1,b2,...› in the paper).

Quasi-polynomials of Capelli

Authors: Antonov S. Y., Antonova A. V.

This paper deals with the class of Capelli polynomials in free associative algebra F{Z} where F is an arbitrary field and Z is a countable set. The interest to these objects is initiated by assumption that the polynomials (Capelli quasi-polynomials) of some odd degree introduced will be contained in the basis ideal Z2 -graded identities of Z2 -graded matrix algebra M(m,k)(F) when char F = 0. In connection with this assumption the fundamental properties of Capelli quasi-polynomials have been given in the paper.

Correctness of the Local Boundary Value Problem in a Cylindrical Domain for Laplace’s Many-dimensional Equation

Authors: Aldashev S. A.

Correctness of boundary problems in the plane for elliptic equations is well analyzed by analitic function theory of complex variable.
There appear principal difficulties in similar problems when the number of independent variables is more than two. An attractive and suitable method of singular integral equations is less strong because of lock of any complete theory of multidimensional singular integral equations.