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Global and Pointwise Estimates for Approximation by Rational Functions with Polynomials of Positive Coefficients as the Denominators

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Abstract:

For non-negative continuous function f(x) defined on [0,1], and f≠0, the present paper proves that, there is a polynomial P_n(x)∈Π_n(+), such that

$\left|f(x)-\frac{1}{P_n(x)}\right|\leq C\omega_{\varphi^\lambda}\big(f,n^{-1/2}A_n^{1-\lambda}(x)\big),$
where A_n(x)=\sqrt{x(1-x)}+1/\sqrt{n},\;0\leq \lambda\leq 1,

$and$ \Pi_n(+)$ indicates the set of all polynomials of degree n with positive coefficients. When

$f(x)$ has exact

$l$ sign change points in (0, 1), we also construct a rational function

$r(x)\in R_n^l(+)$ such that

$\left|f(x)-r(x)\right|\leqC(l+1)^{2}\omega_{\varphi^\lambda}\big(f,n^{-1/2}A_n^{1-\lambda}(x)\big).$

Key words: Polynomials of positive coefficients, Rational functions, Approximation rate, Global estimates, Pointwise estimates

CLC Number:

41A20

YU Dan-Sheng, ZHOU Song-Ping. Global and Pointwise Estimates for Approximation by Rational Functions with Polynomials of Positive Coefficients as the Denominators[J].Acta mathematica scientia,Series A, 2011, 31(2): 305-319.

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