2019
Том 71
№ 11

All Issues

Bakan A. G.

Articles: 7
Article (Russian)

On the completeness of algebraic polynomials in the spaces $L_p (ℝ, dμ)$

Bakan A. G.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 3. - pp. 291-301

We prove that the theorem on the incompleteness of polynomials in the space $C^0_w$ established by de Branges in 1959 is not true for the space $L_p (ℝ, dμ)$) if the support of the measure μ is sufficiently dense

Article (Russian)

Supplement to the Mergelyan Theorem on the Denseness of Algebraic Polynomials in the Space $C_w^0$

Bakan A. G.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 7. - pp. 867–878

We give a supplement to the theorem on the denseness of polynomials in the space $C_w^0$ established by Mergelyan in 1956 for the case where algebraic polynomials are dense in $C_w^0$. In the case indicated, we give a complete description of all functions that can be approximated by algebraic polynomials in seminorm.

Article (Russian)

Polynomial Form of de Branges Conditions for the Denseness of Algebraic Polynomials in the Space $C_w^0$

Bakan A. G.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 3. - pp. 305–319

In the criterion for polynomial denseness in the space $C_w^0$ established by de Brange in 1959, we replace the requirement of the existence of an entire function by an equivalent requirement of the existence of a polynomial sequence. We introduce the notion of strict compactness of polynomial sets and establish sufficient conditions for a polynomial family to possess this property.

Brief Communications (Russian)

Criterion for the Denseness of Algebraic Polynomials in the Spaces $L_p \left( {{\mathbb{R}},d {\mu }} \right)$, $1 ≤ p < ∞$

Bakan A. G.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 701-705

The criterion for the denseness of polynomials in the space $L_p \left( {{\mathbb{R}},d {\mu }} \right)$ established by Hamburger in 1921 is extended to the spaces $L_p \left( {{\mathbb{R}},d {\mu }} \right)$, $1 ≤ p < ∞$.

Article (Russian)

Criterion of Polynomial Denseness and General Form of a Linear Continuous Functional on the Space $C_w^0$

Bakan A. G.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 610-622

For an arbitrary function $w:\mathbb{R} \to \left[ {0,1} \right]$, we determine the general form of a linear continuous functional on the space $C_w^0$. The criterion for denseness of polynomials in the space $L_2 \left( {\mathbb{R},d\mu } \right)$ established by Hamburger in 1921 is extended to the spaces $C_w^0$.

Article (Russian)

On sequences that do not increase the number of real roots of polynomials

Bakan A. G., Holub A. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1323–1331

A complete description is given for the sequences $\{λ_k}_{k = 0}^{ ∞}$ such that, for an arbitrary real polynomial $f(t) = \sum\nolimits_{k = 0}^n {a_k t^k }$, an arbitrary $A \in (0, +∞)$, and a fixed $C \in (0,+∞)$, the number of roots of the polynomial $(Tf)(t) = \sum\nolimits_{k = 0}^n {a_k \lambda _k t^k }$ on $[0,C]$ does not exceed the number of roots off $(t)$ on $[0, A]$.

Article (Ukrainian)

Moreau-Rockafellar equality for sublinear functionals

Bakan A. G.

Full text (.pdf)

Ukr. Mat. Zh. - 1989. - 41, № 8. - pp. 1011–1022