On the Sets of Convergence for Sequences of the 𝑞-Bernstein Polynomials with 𝑞>1
| Autor: | Sofiya Ostrovska, Ahmet Yaşar Özban |
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| Jazyk: | angličtina |
| Rok vydání: | 2012 |
| Předmět: | |
| Zdroj: | Abstract and Applied Analysis, Vol 2012 (2012) |
| Druh dokumentu: | article |
| ISSN: | 1085-3375 1687-0409 |
| DOI: | 10.1155/2012/185948 |
| Popis: | The aim of this paper is to present new results related to the convergence of the sequence of the 𝑞-Bernstein polynomials {𝐵𝑛,𝑞(𝑓;𝑥)} in the case 𝑞>1, where 𝑓 is a continuous function on [0,1]. It is shown that the polynomials converge to 𝑓 uniformly on the time scale 𝕁𝑞={𝑞−𝑗}∞𝑗=0∪{0}, and that this result is sharp in the sense that the sequence {𝐵𝑛,𝑞(𝑓;𝑥)}∞𝑛=1 may be divergent for all 𝑥∈𝑅⧵𝕁𝑞. Further, the impossibility of the uniform approximation for the Weierstrass-type functions is established. Throughout the paper, the results are illustrated by numerical examples. |
| Databáze: | Directory of Open Access Journals |
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