On the rates of pointwise convergence for Bernstein polynomials
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910699685412864 |
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| author | Adell, José A. Cárdenas-Morales, Daniel López-Moreno, Antonio J. |
| author_facet | Adell, José A. Cárdenas-Morales, Daniel López-Moreno, Antonio J. |
| contents | Let $f$ be a real function defined on the interval $[0,1]$ which is constant on $(a,b)\subset [0,1]$, and let $B_nf$ be its associated $n$th Bernstein polynomial. We prove that, for any $x\in (a,b)$, $|B_nf(x)-f(x)|$ converges to $0$ as $n\rightarrow \infty $ at an exponential rate of decay. Moreover, we show that this property is no longer true at the boundary of $(a,b)$. Finally, an extension to Bernstein-Kantorovich type operators is also provided |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10135 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the rates of pointwise convergence for Bernstein polynomials Adell, José A. Cárdenas-Morales, Daniel López-Moreno, Antonio J. Classical Analysis and ODEs 41A10, 41A25, 41A36, 60E05 Let $f$ be a real function defined on the interval $[0,1]$ which is constant on $(a,b)\subset [0,1]$, and let $B_nf$ be its associated $n$th Bernstein polynomial. We prove that, for any $x\in (a,b)$, $|B_nf(x)-f(x)|$ converges to $0$ as $n\rightarrow \infty $ at an exponential rate of decay. Moreover, we show that this property is no longer true at the boundary of $(a,b)$. Finally, an extension to Bernstein-Kantorovich type operators is also provided |
| title | On the rates of pointwise convergence for Bernstein polynomials |
| topic | Classical Analysis and ODEs 41A10, 41A25, 41A36, 60E05 |
| url | https://arxiv.org/abs/2411.10135 |