Differentiation by integration with Jacobi polynomials

作者：Liu Da yan; Gibaru Olivier^*; Perruquetti Wilfrid

来源：Journal of Computational and Applied Mathematics, 2011, 235(9): 3015-3032.

DOI：10.1016/j.cam.2010.12.023

• Numerical differentiation
• Ill-posed problems
• Jacobi orthogonal polynomials
• Orthogonal series

In this paper, the numerical differentiation by integration method based on Jacobi polynomials originally introduced by Mboup et al. [19,20] is revisited in the central case where the used integration window is centered. Such a method based on Jacobi polynomials was introduced through an algebraic approach [19,20] and extends the numerical differentiation by integration method introduced by Lanczos (1956) [21]. The method proposed here, rooted in [19,20] is used to estimate the nth (n is an element of N) order derivative from noisy data of a smooth function belonging to at least Cn+1+q(q is an element of N). In [19,20], where the causal and anti-causal cases were investigated, the mismodelling due to the truncation of the Taylor expansion was investigated and improved allowing a small time-delay in the derivative estimation. Here, for the central case, we show that the bias error is O(h(q+2)) where h is the integration window length for f is an element of Cn+q+2 in the noise free case and the corresponding convergence rate is O(delta(q+1/n+1+q)) where delta is the noise level for a wellchosen integration window length. Numerical examples show that this proposed method is stable and effective.

• 出版日期2011-3-1
• 单位INRIA