On approximation of the separately continuous functions $2\pi$-periodical in relation to the second variable

Abstract

Using Jackson's and Bernstein's operators we prove that for every topological space $X$ and an arbitrary separately continuous function $f: X \times \mathbb{R}\rightarrow \mathbb{R}$, $2\pi$-periodical in relation to the second variable, there exists such sequence of jointly continuous functions $f_n: X\times \mathbb{R}\rightarrow \mathbb{R}$ such that functions $f_n^x=f_n(x, \cdot): \mathbb{R}\rightarrow \mathbb{R}$ are trigonometric polynomials and $f_n^x\to f^x$ uniformly on $\mathbb{R}$ for every $x\in X$.