2019
Том 71
№ 11

# Voloshyn H. A.

Articles: 2
Article (Ukrainian)

### Haar’s condition and joint polynomiality of separate polynomial functions

Ukr. Mat. Zh. - 2017. - 69, № 1. - pp. 17-27

For systems of functions $F = \{ f_n \in K^X : n \in N\}$ and $G = \{ g_n \in K^Y : n \in N\}$ we consider an $F$ -polynomial $f = \sum^n_{k=1}\lambda_k f_k$, a $G$-polynomial $h = \sum^n_{k,j=1} \lambda_{k,j} f_k \otimes g_j$, and an $F \otimes G$-polynomial $(f_k\otimes g_j)(x, y) = = f_k(x)g_j(y)$, where $(f_k\otimes g_j)(x, y) = f_k(x)g_j(y)$. By using the well-known Haar’s condition from the approximation theory we study the following question: under what assumptions every function $h : X \times Y \rightarrow K$, such that all $x$-sections $h^x = h(x, \cdot )$ are $G$-polynomials and all $y$-sections $h_y = h(\cdot , y)$ are $F$ -polynomials, is an $F \otimes G$-polynomialy. A similar problem is investigated for functions of $n$ variables.

Article (Ukrainian)

### Sequential closure of the space of jointly continuous functions in the space of separately continuous functions

Ukr. Mat. Zh. - 2016. - 68, № 2. - pp. 156-161

Given compact spaces $X$ and $Y$, we study the space $S(X \times Y )$ of separately continuous functions $f : X \times Y \rightarrow R$ endowed with the locally convex topology generated by the seminorms $|| f||^x = \mathrm{max}_{y \in Y} |f(x, y)|,\; x \in X$, and $|| f||_y = \mathrm{max}_{x \in X} |f(x, y)|,\; y \in Y$. Under the assumption that the compact space $X$ is metrizable, we prove that a separately continuous function $f : X \times Y \rightarrow R$ is the limit of a sequence $(f_n)^{\infty}_{n=1}$ of jointly continuous function $f_n : X \times Y \rightarrow R$ in $S(X \times Y )$ provided that the set $D(f)$ of discontinuity points of $f$ has countable projections on $X$.