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Vol 222, No 6 (2017)
- Year: 2017
- Articles: 8
- URL: https://journal-vniispk.ru/1072-3374/issue/view/14822
Article
Darboux transformation with parameter of generalized Jacobi matrices
Abstract
A monic generalized Jacobi matrix \( \mathfrak{J} \) is factorized into upper and lower triangular two-diagonal block matrices of special forms so that J = UL. It is shown that such factorization depends on a free real parameter d(∈ ℝ). As the main result, it is shown that the matrix \( {\mathfrak{J}}^{\left(\mathbf{d}\right)}= LU \) is also a monic generalized Jacobi matrix. The matrix \( {\mathfrak{J}}^{\left(\mathbf{d}\right)} \) is called the Darboux transform of \( \mathfrak{J} \) with parameter d. An analog of the Geronimus formula for polynomials of the first kind of the matrix \( {\mathfrak{J}}^{\left(\mathbf{d}\right)} \) is proved, and the relations between m-functions of J and \( {\mathfrak{J}}^{\left(\mathbf{d}\right)} \) are found.
703-722
On the removal of singularities of the Orlicz–Sobolev classes
Abstract
We study the local behavior of closed-open discrete mappings of the Orlicz–Sobolev classes in ℝn; n ≥ 3: It is proved that the indicated mappings have continuous extensions to an isolated boundary point x0 of a domain D/{x0}, whenever its inner dilatation of order p ∈ (n − 1; n] has FMO (finite mean oscillation) at this point, and, in addition, the limit sets of f at x0 and on ∂D are disjoint. Another sufficient condition for the possibility of a continuous extension can be formulated as a condition of divergence of a certain integral.
723-740
741-749
The best M-term trigonometric approximations of the classes of periodic multivariate functions with bounded generalized derivative in the space Lq
Abstract
The order estimates of the best M-term trigonometric approximations of functions \( {D}_{\beta}^{\psi } \) and the classes of (ψ , β)-differentiable periodic multivariate functions in the space Lq, 2 ≤ q < ∞ are obtained. It is shown that, under certain conditions imposed on the parameter q; the best M-term trigonometric approximations \( {e}_M{\left({L}_{\beta, 1}^{\psi}\right)}_q \) have better order than the best orthogonal trigonometric approximations \( e\frac{1}{M}{\left({L}_{\beta, 1}^{\psi}\right)}_q \).
750-761
On the recursive sequence \( {x}_{n+1}=\frac{x_{n-\left(4k+3\right)}}{1+\prod_{t=0}^2{x}_{n-\left(k+1\right)t-k}} \)
Abstract
The solution of the difference equation
where x−(4k+3), x−(4k+2), . . . , x−1, x0 ∈ (0, ∞) and k = 0, 1, . . . , is studied.
762-771
772-786
Constructive sparse trigonometric approximations for the functions with generalized mixed smoothness
Abstract
The order bounds (in the case of uniform metrics) and exact order bounds (in the case of integral metrics) for the best m-term trigonometric approximation of periodic functions with generalized mixed smoothness from classes close to the Nikol’skii–Besov-type ones are obtained. Every of the upper bounds is realized by a constructive method based on greedy algorithms.
787-796
797-818