2019
Том 71
№ 11

# Bakan A. G.

Articles: 7
Article (Russian)

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

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 $C_w^0$

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 $C_w^0$

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 $L_p \left( {{\mathbb{R}},d {\mu }} \right)$, $1 ≤ p < ∞$

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 $C_w^0$

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

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

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