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Volume 31, Issue 1
$C^p$ Condition and the Best Local Approximation

H. H. Cuenya & D. E. Ferreyra

Anal. Theory Appl., 31 (2015), pp. 58-67.

Published online: 2017-01

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  • Abstract

In this paper, we introduce a condition weaker than the $L^p$ differentiability, which we call $C^p$ condition. We prove that if a function satisfies this condition at a point, then there exists the best local approximation at that point. We also give a necessary and sufficient condition for that a function be $L^p$ differentiable. In addition, we study the convexity of  the set of cluster points of the net of best approximations of $f$, $\{P_\epsilon(f)\}$ as $\epsilon \to 0$.

  • AMS Subject Headings

41A50, 41A10

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COPYRIGHT: © Global Science Press

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@Article{ATA-31-58, author = {}, title = {$C^p$ Condition and the Best Local Approximation}, journal = {Analysis in Theory and Applications}, year = {2017}, volume = {31}, number = {1}, pages = {58--67}, abstract = {

In this paper, we introduce a condition weaker than the $L^p$ differentiability, which we call $C^p$ condition. We prove that if a function satisfies this condition at a point, then there exists the best local approximation at that point. We also give a necessary and sufficient condition for that a function be $L^p$ differentiable. In addition, we study the convexity of  the set of cluster points of the net of best approximations of $f$, $\{P_\epsilon(f)\}$ as $\epsilon \to 0$.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2015.v31.n1.5}, url = {http://global-sci.org/intro/article_detail/ata/4622.html} }
TY - JOUR T1 - $C^p$ Condition and the Best Local Approximation JO - Analysis in Theory and Applications VL - 1 SP - 58 EP - 67 PY - 2017 DA - 2017/01 SN - 31 DO - http://doi.org/10.4208/ata.2015.v31.n1.5 UR - https://global-sci.org/intro/article_detail/ata/4622.html KW - Best $L^p$ approximation, local approximation, $L^p$ differentiability. AB -

In this paper, we introduce a condition weaker than the $L^p$ differentiability, which we call $C^p$ condition. We prove that if a function satisfies this condition at a point, then there exists the best local approximation at that point. We also give a necessary and sufficient condition for that a function be $L^p$ differentiable. In addition, we study the convexity of  the set of cluster points of the net of best approximations of $f$, $\{P_\epsilon(f)\}$ as $\epsilon \to 0$.

H. H. Cuenya & D. E. Ferreyra. (1970). $C^p$ Condition and the Best Local Approximation. Analysis in Theory and Applications. 31 (1). 58-67. doi:10.4208/ata.2015.v31.n1.5
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