Volume 227, Nº 1 (2017)
- Ano: 2017
- Artigos: 8
- URL: https://journals.rcsi.science/1072-3374/issue/view/14867
Article
Functions analytic in a unit ball of bounded L-index in joint variables
Resumo
The concept of boundedness of the L-index in joint variables (see Mat. Stud., 45, 12–26 (2016), dx.doi.org/10.15330/ms.45.1.12-26) is generalized for a function analytic in a ball. The criteria of boundedness of the L-index in joint variables, which describe the local behavior of partial derivatives on the skeleton of a polydisc, are proved.
To the issue of a generalization of the matrix differential-algebraic boundary-value problem
Resumo
We pose a linear matrix differential-algebraic boundary-value problem generalizing the traditional linear boundary-value problems for differential-algebraic equations. We have found the constructive conditions of existence and an algorithm of construction of the solutions of a linear matrix differentialalgebraic boundary-value problem. We propose the construction of a generalized Green operator for the determination of solutions of the linear differential-algebraic boundary-value problem and give some examples of construction of such solutions.
An operator approach to the indefinite Stieltjes moment problem
Resumo
A function f meromorphic on ℂ\ℝ is said to be in the generalized Nevanlinna class Nκ (κ ϵ ℤ+), if f is symmetric with respect to ℝ and the kernel \( {\mathbf{N}}_{\omega }(z)\coloneq \frac{f(z)-\overline{f\left(\omega \right)}}{z-\overline{\omega}} \) has κ negative squares on ℂ+. The generalized Stieltjes class \( {\mathbf{N}}_{\kappa}^k\left(\kappa, k\in {\mathrm{\mathbb{Z}}}_{+}\right) \) is defined as the set of functions f ϵ Nκ such that z f ϵ Nk. The full indefinite Stieltjes moment problem \( {MP}_{\kappa}^k\left(\mathbf{s}\right) \) consists in the following: Given κ, k ϵ ℤ+, and a sequence \( \mathbf{s}={\left\{{s}_i\right\}}_{i=0}^{\infty } \) of real numbers, to describe the set of functions \( f\in {\mathbf{N}}_{\kappa}^k \), which satisfy the asymptotic expansion
for all n big enough. In the present paper, we will solve the indefinite Stieltjes moment problem \( {MP}_{\kappa}^k\left(\mathbf{s}\right) \) within the M. G. Krein theory of u-resolvent matrices applied to a Pontryagin space symmetric operator A[0;N] generated by \( {\mathfrak{J}}_{\left[0;N\right]} \). The u-resolvent matrices of the operator A[0;N] are calculated in terms of generalized Stieltjes polynomials, by using the boundary triple’s technique. Some criteria for the problem \( {MP}_{\kappa}^k\left(\mathbf{s}\right) \) to be solvable and indeterminate are found. Explicit formulae for Padé approximants for the generalized Stieltjes fraction in terms of generalized Stieltjes polynomials are also presented.
The differential-symbol method of solving the problem two-point in time for a nonhomogeneous partial differential equation
Resumo
The solvability of the problem for a nonhomogeneous partial differential equation of the second order in time and, generally, of the infinite order in the spatial variables with local conditions two-point in time in the classes of entire functions is studied. In the case of unique solvability of the problem, a differential-symbol method of construction of its unique solution is proposed. In the case of nonunique solvability of the problem, we construct a partial solution.
Widths of some classes of functions defined by the generalized moduli of continuity ωγ in the space L2
Resumo
The exact values of some widths for the classes of functions \( {H}^{\omega_{\gamma }} \) and \( {W}_{\beta, 2}^{\psi}\left({\omega}_{\gamma },\varPhi \right) \) given by the generalized modulus of continuity ωγ and the majorant Φ satisfying a certain condition are computed. On the indicated classes, the exact values of Fourier coefficients are found.