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# Approximation by rational functions with prescribed numerator degree in Lp spaces for 1

Acta Mathematica Hungarica
Authors: X.F. Mei and F.P. Zhou

## Abstract

The present paper establishes a complete result on approximation by rational functions with prescribed numerator degree in L pspaces for 1 < p < ∞ and proves that if f(x)∈L p [-1,1] changes sign exactly l times in (-1,1), then there exists r(x)∈R n l 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 R n l indicates all rational functions whose denominators consist of polynomials of degree n and numerators polynomials of degree l, and C p , 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.

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# On approximation by rational functions with prescribed numerator degree in L p spaces

Acta Mathematica Hungarica
Authors: Dan Sheng Yu and Song Ping Zhou

## Summary

It is proved that, if \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} $f(x)\in L^p_{[-1,1]}$ \end{document}, \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} $1< p< \infty$ \end{document}, changes sign exactly \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} $l$ \end{document} times, then there exists a real rational function \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} $r(x)\in R_{n}^l$ \end{document} 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} $${\|f-r\|}_{p}\le C_{p,\delta}{(l+1)}^2\omega {(f,n^{-1})}_p,$$ \end{document}
which generalizes a result of Leviatan and Lubinsky in \cite{4}. A weaker similar result in \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} $L^1_{[-1,1]}$ \end{document} is also established.
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# A Turán type inequality for rational functions with prescribed poles

Acta Mathematica Hungarica
Authors: D. S. Yu and S. P. Zhou

Summary By employing a novel idea and simple techniques, we substantially generalize the Turán type inequality for rational functions with real zeros and prescribed poles established by Min  to include L p spaces for 1≤ p ≤ ∞ while loosing the restriction ρ > 2 at the same time.

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# Saturation for Bernstein type rational functions

Acta Mathematica Hungarica
Author: V. Totik
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# Approximation by Bernstein type rational functions. II

Acta Mathematica Hungarica
Authors: Catherine Balázs and J. Szabados
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# Approximation by Bernstein type rational functions

Acta Mathematica Hungarica
Author: Katalin Balázs
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# A Turán type inequality for rational functions with prescribed poles

Acta Mathematica Hungarica
Authors: D. S. Yu and S. P. Zhou
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# Inequalities for the Derivatives of Rational Functions with Real Zeros

Acta Mathematica Hungarica
Author: G. Min
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# Approximation by Bernstein type rational functions on the real axis

Acta Mathematica Hungarica
Author: Katherine Balázs
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# A note on certain next-to-interpolatory rational functions

Acta Mathematica Hungarica
Author: M. A. Bokhari
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