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Transactions of the American Mathematical Society

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Quantitative Korovkin theorems for positive linear operators on $ L\sb{p}$-spaces

Authors: H. Berens and R. DeVore
Journal: Trans. Amer. Math. Soc. 245 (1978), 349-361
MSC: Primary 41A35; Secondary 41A36
MathSciNet review: 511414
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Abstract: Let $ ({L_n})$ be a sequence of positive linear operators on $ {L_p}(\Omega )$, $ 1 \leqslant p < \infty $ or $ C(\Omega )$ with $ \Omega \subseteq {R^m}$. For suitable $ \Omega $, the functions $ ({\varphi _i})_{i = 0}^{m + 1}$ given by $ {\varphi _0}(x) \equiv 1$, $ {\varphi _i}(x) \equiv {x_i}$, $ 1 \leqslant i \leqslant m$,and $ {\varphi _{m + 1}}(x) \equiv {\left\vert x \right\vert^2}$ form a test set for $ {L_p}(\Omega )$. That is, if $ {L_n}({\varphi _i})$ converges to $ {\varphi _i}$ in $ {\left\Vert \cdot \right\Vert _p}$ for each $ i = 0,1, \ldots ,m + 1$, then $ {L_n}(f)$ converges to f in $ {\left\Vert \cdot \right\Vert _p}$ for each $ f \in {L_p}(\Omega )$. We give here quantitative versions of this result. Namely, we estimate $ {\left\Vert {f - {L_n}f} \right\Vert _p}$ in terms of the error $ {\left\Vert {{\varphi _i} - {L_n}{\varphi _i}} \right\Vert _p}$, $ 0 \leqslant i \leqslant m + 1$,and the smoothness of the function f.

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Keywords: Positive linear operators, quantitative estimates, degree of approximation, Korovkin theorems
Article copyright: © Copyright 1978 American Mathematical Society