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• Author or Editor: Dan Sheng Yu
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# On approximation by rational functions with prescribed numerator degree in L p spaces

Acta Mathematica Hungarica
Authors: Dan Sheng Yu and Song Ping Zhou

## 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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# Simultaneous determination of paeoniflorin and albiflorin in radix paeoniae rubra by HPLC—DAD—ELSD

Acta Chromatographica
Authors: Mei-Xia Zhu, Sheng-Nan Li, Hai-Dan You, Bin Han, Zhi-Ping Wang, Yan-Xi Hu, Jin Li and Yu-Feng Liu

High-performance liquid chromatography coupled with photodiode array detection and evaporative light scattering detection (HPLC—DAD—ELSD) was established to determine paeoniflorin and albiflorin simultaneously in Radix Paeoniae Rubra. The assay was performed on a Diamonsil C18 (4.6 mm × 250 mm, 5 μm) column by a gradient elution program with acetonitrile and aqueous formic acid (0.05% v/v) as mobile phase at a flow rate of 1.0 mL min−1. The detection wavelength of DAD was 230 nm, and the evaporator tube temperature of ELSD was set at 110 °C with the nebulizing gas flow rate of 3 L min−1. The temperature of column was kept at 30 °C. The linear ranges of paeoniflorin and albiflorin were within 0.050–1.510 mg mL−1 and 1.007–5.035 mg mL−1. The recoveries of paeoniflorin and albiflorin were 96.2–102.9% and 95.0–102.4%, respectively, while the relative standard deviation (RSD) of them was 0.2–2.5%. This method was quick, simple, accurate, and specific. It could be used for the quality control of Radix Paeoniae Rubra. The proposed approach was expected as a powerful tool for the quality control of Radix Paeoniae Rubra.

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