Legendre Polynomials-Based Numerical Differentiation: A Convergence Analysis in a Weighted $L^2$ Space
Received:July 19, 2015  Revised:October 21, 2015
Key Word: Legendre polynomials   numerical differentiation   Jacobi polynomials   weighted $L^2$ space
Fund ProjectL: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).
 Author Name Affiliation Qin FANG College of Information and Engineering, Dalian University, Liaoning 116600, P. R. China Haojie LI College of Information and Engineering, Dalian University, Liaoning 116600, P. R. China Min XU School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China
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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.