On approximation of the separately continuous functions $2\pi$-periodical in relation to the second variable
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 f...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Vasyl Stefanyk Precarpathian National University
2013-01-01
|
| Series: | Karpatsʹkì Matematičnì Publìkacìï |
| Online Access: | http://journals.pu.if.ua/index.php/cmp/article/view/34 |
| Summary: | 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 $x\in X$. |
|---|---|
| ISSN: | 2075-9827 2313-0210 |