| Legendre Polynomials-Based Numerical Differentiation: A Convergence Analysis in a Weighted $L^2$ Space |
| Received:July 19, 2015 Revised:October 21, 2015 |
| Key Words:
Legendre polynomials numerical differentiation Jacobi polynomials weighted $L^2$ space
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| Fund Project:Supported by the National Nature Science Foundation of China (Grant Nos.11301052; 11301045; 11401077; 11271060; 11290143), the Fundamental Research Funds for the Central Universities (Grant No.DUT15RC(3)058) and the Fundamental Research of Civil Aircraft (Grant No.MJ-F-2012-04). |
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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. |
| Citation: |
| DOI:10.3770/j.issn:2095-2651.2016.02.014 |
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