Mathematics

Classical solution by the Fourier method of mixed problems with minimum requirements on the initial data

Authors: Khromov A. P., Burlutskaya M. S.

The article gives a new short proof the V. A. Chernyatin theorem about the classical solution of the Fourier method of the mixed problem for the wave equation with fixed ends with minimum requirements on the initial data. Next, a similar problem for the simplest functional differential equation of the first order with involution in the case of the fixed end is considered, and also obtained definitive results. These results are due to a significant use of ideas A. N. Krylova to accelerate the convergence of series, like Fourier series.

Everywhere divergence of Lagrange processes on the unit circle

Authors: Tyshkevich S. V., Shatalina A. V.

We study the convergence of Lagrange interpolation processes in the closed unit disk. Choosing a matrix with a certain distribution of interpolation nodes allowed to construct the set, completely covering the unit circle, and the function for which the process diverges everywhere on this set.

The new approach to solving the Riemann boundary value problem with infinite index

Authors: Salimov R. B., Karabasheva E. N.

This research considers Riemann–Hilbert boundary value problem with infinite index where edge condition of problem is established by the real axis. To solve this problem the approach based on the removal of the infinite discontinuity of the argument of boundary condition coefficient is used. The approach is analogous to the one which, in the context of the finite index of the problem in researches by F. D. Gakhov, helps to remove a discontinuity of initial genre of boundary condition coefficient with specially created functions, different from the ones in this research.

On Poisson Customary Polynomial Identities

Authors: Ratseev S. M.
We study Poisson customary and Poisson extended customary polynomials. We show that the sequence of codimensions {rn(V )}n¸1 of every extended customary space of variety V of Poisson algebras over an arbitrary field is either bounded by a polynomial or at least exponential. Furthermore, if this sequence is bounded by polynomial then there is a polynomial R(x) with rational coefficients such that rn(V ) = R(n) for all sufficiently large n. We present lower and upper bounds for the polynomials R(x) of an arbitrary fixed degree.

One counterexample of shape-preserving approximation

Authors: Pleshakov M. G., Tyshkevich S. V.

Let 2s points yi=−π≤y2s<… Example. For each k∈N, k>2, and n∈N there a function f(x):=f(x;s,Y,n,k) exists, such that f∈Δ(1)(Y) and

About reversibility states of linear differential operators with periodic unbounded operator coefficients

Authors: Didenko V. B.

For investigated linear differential operator (equation) with unbounded periodic operator coefficients defined at one of the Banach space of vector functions defined on all real axis difference operator (equation) with constant operator coefficient defined at appropriate Banach space of two-side vector sequences is considered. For differential and difference operators propositions about kernel and co-image dimensions coincidence, simultaneous complementarity of kernels and images, simultaneous reversibility, spectrum interrelation are proved.

Weighted Integrability of Sums of Series with Respect to Multiplicative Systems

Authors: Fadeev R. N., Volosivets S. S.
A necessary and sufficient condition for Lp-integrability with power weight of a function f represented by the series with respect to multiplicative systems with generalized monotone coefficients is obtained. The integrability of the majorant of partial sums of a representing series is also described by the same conditions. In addition we study the integrability of difference quotient (f(x) − f(0))/x.

On Subvariety of Variety Generated by a Simple Infinite Lie Algebra of Cartan Type General Series W2

Authors: Bogdanchuk O. A.
We consider numerical characteristics of Lie algebras variety over a field of characteristic zero, basically, the exponent of variety.
Here, was constructed the infinite series of varieties of Lie algebras with different fractional exponents, which belong to variety
generated by a simple infinite Lie algebra of Cartan type general series W2.