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Approximation by rational functions with prescribed numerator degree in L p spaces for 1

Acta Mathematica Hungarica
Volume/Issue:
Volume 102: Issue 4
DOI:
https://doi.org/10.1023/b:amhu.0000024682.31751.80
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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On L 1-convergence of Fourier series of complex valued functions under the GM7 condition

Acta Mathematica Hungarica
Volume/Issue:
Volume 133: Issue 1-2
DOI:
https://doi.org/10.1007/s10474-010-0062-2
Authors: F. J. Feng and S. P. Zhou

Abstract

Let f∊L 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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The asymptotic property of approximation to |x| by Newman's rational operators

Acta Mathematica Hungarica
Volume/Issue:
Volume 103: Issue 4
DOI:
https://doi.org/10.1023/b:amhu.0000028831.89374.d3
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
Volume/Issue:
Volume 67: Issue 1-2
DOI:
https://doi.org/10.1007/bf01874526
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
Volume/Issue:
Volume 69: Issue 1-2
DOI:
https://doi.org/10.1007/bf01874615
Authors: T. F. Xie and S. P. Zhou
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