Vol 224, No 4 (2017)

A generalization of the well-known results of M.G. Kreĭn on the description of the self-adjoint contractive extension of a Hermitian contraction is obtained. This generalization concerns the situation where the self-adjoint operator A and extensions e à belong to a Kreĭn space or a Pontryagin space, and their defect operators are allowed to have a fixed number of negative eigenvalues. A result of Yu. L. Shmul’yan on completions of nonnegative block operators is generalized for block operators with a fixed number of negative eigenvalues in a Kreĭn space.

This paper is a natural continuation of S. Hassi’s and author’s recent paper [7].

For G-monogenic mappings taking values in the algebra of complex quaternions, we generalize some analogues of classical integral theorems of the holomorphic function theory of complex variable (the surface and curvilinear Cauchy integral theorems).

The problem of shadow generalized onto domains of the space ℝⁿ, n ≤ 3, is investigated. The problem consists in the determination of the minimal number of balls satisfying some conditions such that every line passing through the given point intersects at least one ball of the collection. We have proved that it is sufficient to have four (two) mutually nonoverlapping closed or open balls in order to generate a shadow at every given point of any domain of the space ℝ³ (ℝ²). They do not include the point, and their centers lie on the domain boundary.

On the classes of functions \( {L}_2^r\left(\mathbb{R}\right) \), where r ∈ℤ₊, for the characteristics of smoothness \( {\Lambda}_k\left(f,t\right)={\left\{\left(1/t\right){\int}_0^t\left\Vert {\varDelta}_h^k(f)\left\Vert {}^2\right. dh\right.\right\}}^{\kern0em 1/2},t\in \left(0,\infty \right),k\in \mathbb{N} \), the exact constants in the Jackson-type inequalities have been obtained in the case of the best mean square approximation by entire functions of the exponential type in the space L₂(ℝ). The exact values of mean ????-widths of the classes of functions defined by Λ_k(f) and the majorants Ψ satisfying some conditions are calculated.