| 基于Legendre多项式的数值微分: 加权$L^2$空间中的收敛性分析 |
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| 引用本文： | 方芩,李豪杰,徐敏.基于Legendre多项式的数值微分: 加权$L^2$空间中的收敛性分析[J].数学研究及应用,2016,36(2):247-252. |
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| 作者姓名： | 方芩 李豪杰 徐敏 |
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| 作者单位： | 大连大学信息工程学院, 辽宁 大连 116622,大连大学信息工程学院, 辽宁 大连 116622,大连理工大学数学科学学院, 辽宁 大连 116024 |
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| 基金项目： | 国家自然科学基金(Grant Nos.11301052; 11301045; 11401077;11271060; 11290143), 中央高校基本科研专项资金(Grant No.DUT15RC(3)058), 民用飞机基础研究(Grant No.MJ-F-2012-04). |
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| 摘 要： | We consider the problem of estimating the derivative of a function f from its noisy version fδby using the derivatives of the partial sums of Fourier-Legendre series of f~δ. Instead of the observation L~2 space, we perform the reconstruction of the derivative in a weighted L~2 space. This takes full advantage of the properties of Legendre polynomials and results in a slight improvement on the convergence order. Finally, we provide several numerical examples to demonstrate the efficiency of the proposed method.
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| 关 键 词： | Legendre多项式 数值微分 Jacobi多项式 加权$L^2$空间 |
| 收稿时间： | 2015/7/19 0:00:00 |
| 修稿时间： | 2015/10/21 0:00:00 |
Legendre Polynomials-Based Numerical Differentiation: A Convergence Analysis in a Weighted $L^2$ Space |
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| Qin FANG,Haojie LI and Min XU.Legendre Polynomials-Based Numerical Differentiation: A Convergence Analysis in a Weighted $L^2$ Space[J].Journal of Mathematical Research with Applications,2016,36(2):247-252. |
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| Authors: | Qin FANG Haojie LI and Min XU |
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| Institution: | College of Information and Engineering, Dalian University, Liaoning 116600, P. R. China,College of Information and Engineering, Dalian University, Liaoning 116600, P. R. China and School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China |
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| Abstract: | We consider the problem of estimating the derivative of a function $f$ from its noisy version $f^{\delta}$ by using the derivatives of the partial sums of Fourier-Legendre series of $f^{\delta}$. Instead of the observation $L^2$ space, we perform the reconstruction of the derivative in a weighted $L^2$ space. This takes full advantage of the properties of Legendre polynomials and results in a slight improvement on the convergence order. Finally, we provide several numerical examples to demonstrate the efficiency of the proposed method. |
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| Keywords: | Legendre polynomials numerical differentiation Jacobi polynomials weighted $L^2$ space |
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