分母为正系数多项式的有理函数逼近的整体和点态估计

数学物理学报 ›› 2011, Vol. 31 ›› Issue (2): 305-319.

分母为正系数多项式的有理函数逼近的整体和点态估计

虞旦盛¹|周颂平²

1.杭州师范大学数学系杭州 310036|2.浙江理工大学数学研究所杭州 310028

收稿日期:2008-11-08 修回日期:2009-12-06 出版日期:2011-04-25 发布日期:2011-04-25
基金资助:
国家自然科学基金(10901044)和浙江省钱江人才计划项目资助

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

YU Dan-Sheng¹, ZHOU Song-Ping²

1.Departement of Mathematics, Hangzhou Normal University, Hangzhou 310036|2.Institute of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310028

Received:2008-11-08 Revised:2009-12-06 Online:2011-04-25 Published:2011-04-25
Supported by:
国家自然科学基金(10901044)和浙江省钱江人才计划项目资助

摘要/Abstract

摘要：

对任意定义在[0,1]上的非负连续函数f(x)(f≠0), 该文证得: 存在一个正系数多项式P_n(x)∈Π_n(+),使得

\[\left|f(x)-\frac{1}{P_n(x)}\right|\leq C\omega_{\varphi^\lambda}\big(f,n^{-1/2}A_n^{1-\lambda}(x)\big),\]
其中A_n(x)=\sqrt{x(1-x)}+1/\sqrt{n},\;0≤λ≤ 1, 而Π_n(+)表示由所有次数不超过n的正系数多项式构成的集合. 当f(x)在(0, 1)内恰好改变l次符号时, 该文构造了有理函数r(x)\in R_n^l(+),使得

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

关键词: 正系数多项式, 有理函数, 逼近阶, 整体估计, 点态估计

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

中图分类号:

41A20

虞旦盛, 周颂平. 分母为正系数多项式的有理函数逼近的整体和点态估计[J]. 数学物理学报, 2011, 31(2): 305-319.

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.

参考文献

[1] Cao F L. Pointwise and global estimates for reciprocals of polynomials with positive coffecients. Math Appl, 2003, 16: 65--69

[2] Devore R A. The Approximation of Continuous Functions by Positive Linear Operators. New York: Springer-Verlag, 1972

[3] Devore R A, Leviatan D, Yu X M. L^p approximation by reciprocals of trigonometric and algebraic polynomials. Canad Math Bull, 1990, 33: 460--469

[4] Ditzian Z, Totik V. Moduli of Smoothness. New York: Springer-Verlag, 1987

[5] Ditzian Z, Jiang D. Approximation of functions by polynomials in C_[-1,1] Canad J Math, 1992, 44: 924--940

[6] Leviatan D, Levin A L, Saff E B. On approximation in the L^p-norm by reciprocals of polynomials. J Approx Theory, 1989, 57: 322--331

[7] Leviatan D, Lubinsky D S. Degree of approximation by rational functions with prescribed numerator degree. Canad J Math, 1994, 46: 619--633

[8] Levin A L, Saff E B. Degree of approximation of real functions by reciprocals of real and complex polynomials. SIAM J Math Anal, 1988, 18: 233--245

[9] Lorentz G G. The degree of approximation by polynomials with positive coefficients. Math Annalen, 1963, 151: 239--251

[10] Mei X F, Zhou S P. Approximation by reciprocals of polynomials in L^p_[-1,1](1102: 321--336

[11] Xu G Q. The degree of approximation of real functions by reciprocals of polynomials with positive coefficients. J Engin Math, 1996, 13: 112--116

[12] Yu D S, Zhou S P. Approximation by rational functions with polynomials of positive coefficients as the denominators. Anal Theory Appl, 2006, 22: 1--7

[13] Yu D S, Zhou S P. Recent development in rational approximation (II)-A survey on approximation by reciprocals of polynomials. Adv Math (Beijing), 2005, 34: 269--280

[14] Yu D S, Zhou S P. Approximation by rational functions with polynomials of positive coefficients as the denominators in L^p spaces. Acta Math Hungar, 2006, 111: 221--236

[15] Zhao Y, Zhou S P. Approximation by reciprocals of polynomials with positive coefficients in L^p spaces. Acta Math Hungar, 2001, 92: 205--217