Том 227, № 1 (2017)

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.

A function f meromorphic on ℂ\ℝ is said to be in the generalized Nevanlinna class N_κ (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 N_κ such that z f ϵ N_k. 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

\( f(z)=-\frac{s_0}{z}-\cdots -\frac{s_2n}{z^{2n+1}}+o\left(\frac{1}{z^{2n+1}}\right)\kern1em \left(z=-y\in {\mathrm{\mathbb{R}}}_{-},y\uparrow \infty \right) \)

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 criteria for continuous and homeomorphic extensions to the boundary of mappings with finite distortion between domains on the Riemann surfaces by prime ends by Carath´eodory are proved.

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 ω_γ and the majorant Φ satisfying a certain condition are computed. On the indicated classes, the exact values of Fourier coefficients are found.