Die Lokale L_p — Saturationsklasse des Verfahrens der Integralen Meyer — König und Zeller Operatoren

Abstract

For the linear approximation method (M̂_n) of so — called integrated Meyer -König and Zeller operators [4] on the spaces L_p(I), 1≤p<∞>, I = [0, 1], a local O(n^-1)-saturation theorem will be proved, stating roughly speaking that

$$\begin{array}{*{20}{c}} {{{\left\| {f - {{\hat M}_n}f} \right\|}_{\;p}}\left[ {a,b} \right]{\kern 1pt} \; = \;0({n^{ - 1}})\; \Leftrightarrow \;x{{(1 - x)}^2}f'(x)\; \ne \;C} \\ {{{\left\| {f - {{\hat M}_n}f} \right\|}_{\;p}}\left[ {a,b} \right]{\kern 1pt} \; = \;o({n^{ - 1}})\; \Leftrightarrow \;x{{(1 - x)}^2}f'(x)\; = \;C{\kern 1pt} (C \in \mathbb{R},x \in [a,b],0 < a < b < 1)} \\ \end{array}$$

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