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Asymptotic property of approximation to x αsgn x by Newman Type Operators

Asymptotic property of approximation to x αsgn x by Newman Type Operators Approximation to the function |x| plays an important role in approximation theory. This paper studies the approximation to the function x α sgn x, which equals |x| if α = 1. We construct a Newman Type Operator r n (x) and prove $$ \mathop {\max }\limits_{\left| x \right| \leqslant 1} \left| {x^\alpha sgn x - r_n } \right| \sim Cn^{\tfrac{1} {4}} e^{ - \tfrac{\pi } {2}} \sqrt {\alpha n} $$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Asymptotic property of approximation to x αsgn x by Newman Type Operators

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Publisher
Springer Journals
Copyright
Copyright © 2010 by Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag GmbH
Subject
Mathematics; Theoretical, Mathematical and Computational Physics; Math Applications in Computer Science; Applications of Mathematics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-007-7147-x
Publisher site
See Article on Publisher Site

Abstract

Approximation to the function |x| plays an important role in approximation theory. This paper studies the approximation to the function x α sgn x, which equals |x| if α = 1. We construct a Newman Type Operator r n (x) and prove $$ \mathop {\max }\limits_{\left| x \right| \leqslant 1} \left| {x^\alpha sgn x - r_n } \right| \sim Cn^{\tfrac{1} {4}} e^{ - \tfrac{\pi } {2}} \sqrt {\alpha n} $$ .

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Apr 23, 2009

References