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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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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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Mathematik 3 155 – 191 . [2] Walsh , J. L. , Interpolation and Approximation by Rational Functions in the Complex Domain , Amer. Math. Soc. Colloquium Publications Volume XX ( Providence

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] Walsh , J. L. 1960 Interpolation and Approximation by Rational Functions in the Complex Domain 3 Amer. Math. Soc. Colloquium Publications XX Amer. Math. Soc. Providence .

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313 324 Gao, B., Newman, D. J. and Popov, V. A. , Convex approximation by rational functions, SIAM J. Math. Anal. , 26 (1995), 488

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Walsh, J. L. , Interpolation and Approximation by Rational Functions in the Complex Domain , Amer. Math. Soc. Coll. Publ., Vol. 20, 1960. MR 0218587 36 #1672a

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