Acta Mathematica Hungarica
Volume/Issue:
Volume 102: Issue 4
Approximation by rational functions with prescribed numerator degree in Lp spaces for 1
Authors: X.F. Mei 1 and F.P. Zhou 2
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  • 1 Zhejiang Education College Hangzhou Department of Mathematics Zhejiang 310012 China Zhejiang 310012 China
  • 2 Zhejiang Institute of Science and Technology Hangzhou Institute of Mathematics Zhejiang 310018 China Zhejiang 310018 China
DOI:
https://doi.org/10.1023/b:amhu.0000024682.31751.80
Page Count:
321–336
Publication Date:
03 Mar 2004
Online Publication Date:
02 Nov 2004
Article Category:
Research Article
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Abstract  

The present paper establishes a complete result on approximation by rational functions with prescribed numerator degree in Lpspaces for 1 < p < ∞ and proves that if f(x)∈Lp[-1,1] changes sign exactly l times in (-1,1), then there exists r(x)∈Rnl such that

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\| {f(x) - r(x)} \right\|_{L^p } \leqq C_{p,l,b} \omega (f,n^{ - 1} )_{L^p } ,$$ \end{document}
where Rnl indicates all rational functions whose denominators consist of polynomials of degree n and numerators polynomials of degree l, and Cp, l,b is a positive constant depending only on p, l and b which relates to the distance among the sign change points of f(x) and will be given in 3.

Cover Acta Mathematica Hungarica
Acta Mathematica Hungarica
Print ISSN:
0236-5294
Online ISSN:
1588-2632
Keywords:
rational function; prescribed numerator; Steklov function; modified Jackson kernel

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