On One Counterexample in Convex Approximation
We prove the existence of a function fcontinuous and convex on [−1, 1] and such that, for any sequence {p n} n= 1 ∞of algebraic polynomials p nof degree ≤ nconvex on [−1, 1], the following relation is true: \(\begin{array}{*{20}c} {\lim \sup } \\ {n \to \infty } \\ \end{array} \begin{array}{*{20}c...
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| Datum: | 2000 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2000
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/4576 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | We prove the existence of a function fcontinuous and convex on [−1, 1] and such that, for any sequence {p n} n= 1 ∞of algebraic polynomials p nof degree ≤ nconvex on [−1, 1], the following relation is true: \(\begin{array}{*{20}c} {\lim \sup } \\ {n \to \infty } \\ \end{array} \begin{array}{*{20}c} {\max } \\{x \in [ - 1,1]} \\ \end{array} \frac{{|f(x) - p_n (x)|}}{{\omega _4 (\rho _n (x),f)}} = \infty \) , where ω4(t, f) is the fourth modulus of continuity of the function fand \(\rho _n \left( x \right): = \frac{1}{{n^2 }} + \frac{1}{n}\sqrt {1 - x^2 } \) . We generalize this result to q-convex functions. |
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