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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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# The asymptotic property of approximation to |x| by Newman's rational operators

Acta Mathematica Hungarica
Authors: T.F. Xie and S.P. Zhou

## Abstract

Let
\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} $$a = e^{ - 1/\sqrt n } ,p(x) = \Pi _{k = 1}^{n - 1} (a^k + x),r_n (x) = x\frac{{p(x) - p( - x)}} {{p(x) + p( - x)}}$$ \end{document}
. The present note gives the asymptotoc formula of max
\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} $$\mathop {\max }\limits_{|x| \leqq 1} \left| {|x| - r_n (x)} \right|$$ \end{document}
.
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# A remark on approximation by monotone sequences of polynomials

Acta Mathematica Hungarica
Authors: T. F. Xie and S. P. Zhou
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# On simultaneous approximation to a differentiable function and its derivative by Pál-type interpolation polynomials

Acta Mathematica Hungarica
Authors: T. F. Xie and S. P. Zhou
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# On L 1-convergence of Fourier series of complex valued functions under the GM7 condition

Acta Mathematica Hungarica
Authors: F. J. Feng and S. P. Zhou

## Abstract

Let fL 2π be a real-valued even function with its Fourier series , and let S n(f,x) be the nth partial sum of the Fourier series, n≧1. The classical result says that if the nonnegative sequence {a n} is decreasing and , then if and only if . Later, the monotonicity condition set on {a n} is essentially generalized to MVBV (Mean Value Bounded Variation) condition. Very recently, Kórus further generalized the condition in the classical result to the so-called GM7 condition in real space. In this paper, we give a complete generalization to the complex space.

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# Competitive inhibition of phytic acid on enzymatic browning of chestnut (Castanea mollissima Blume)

Acta Alimentaria
Authors: G.P. Li, D. Zhou, L.N. Kan, Y.W. Wu, J.F. Fan, and J. Ouyang

The inhibitory effects of phytic acid (PA) on the browning of fresh-cut chestnuts and the associated mechanisms of PA on polyphenol oxidase (PPO) and peroxidase (POD) activities were investigated. The enzymatic browning of chestnut surfaces and interiors was suppressed by soaking shelled and sliced chestnuts in a PA solution. The specific activities of PPO and POD extracted from chestnuts declined due to inhibition by PA. PA was determined to be a competitive inhibitor of both PPO and POD by Lineweaver-Burk plots. The binding modes of PA with PPO and POD were analysed by AutoDock 4.2.

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