# Approximation by Rational Functions with Prescribed Numerator Degree in $L^p$ Spaces for $1<p< \infty$

Mei, X. F. and Zhou, Songping (2002) Approximation by Rational Functions with Prescribed Numerator Degree in $L^p$ Spaces for $1<p< \infty$. [Preprint]

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The present paper establishes a complete result on approximation by rational functions with prescribed numerator degree in $L^{p}$ spaces for $1<p<\infty$ and proves that if $f(x)\in L_{[-1,1]}^{p}$ changes sign exact $l$ times in $(-1,1)$, then there exist a $r(x)\in R_{n}^{l}$ such that $$\|f(x)-r(x)\|_{L^{p}} \leq C_{p, l, b} \omega(f,n^{-1})_{L^{p}},$$ 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 only depending on $p$, $l$ and $b$ which relates to the distance among the sign change points of $f(x)$ and will be give in \S 3.