Grüss-type and Ostrowski-type inequalities in approximation theory
We discuss the Grass inequalities on spaces of continuous functions defined on a compact metric space. Using the least concave majorant of the modulus of continuity, we obtain a Grass inequality for the functional $L(f) = H(f; x)$, where $H: C[a,b] \rightarrow C[a,b]$ is a positive linear operator...
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| Дата: | 2011 |
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| Автори: | , , , , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2011
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/2758 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| Резюме: | We discuss the Grass inequalities on spaces of continuous functions defined on a compact metric space.
Using the least concave majorant of the modulus of continuity, we obtain a Grass inequality for the functional
$L(f) = H(f; x)$, where $H: C[a,b] \rightarrow C[a,b]$ is a positive linear operator and $x \in [a,b]$ is fixed.
We apply this inequality in the case of known operators, for example, the Bernstein, Hermite-Fejer operator the interpolation operator, convolution-type operators.
Moreover, we derive Grass-type inequalities using Cauchy's mean value theorem, thus generalizing results of Cebysev and Ostrowski.
A Grass inequality on a compact metric space for more than two functions is given, and an analogous Ostrowski-type inequality is obtained.
The latter in turn leads to one further version of Grass' inequality.
In an appendix, we prove a new result concerning the absolute first-order moments of the classical Hermite-Fejer operator. |
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