Differentiation by integration with Jacobi polynomials

Da-Yan Liu 1, 2, 3 Olivier Gibaru 3, 4 Wilfrid Perruquetti 3, 5
3 NON-A - Non-Asymptotic estimation for online systems
Inria Lille - Nord Europe, CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL) - UMR 9189
5 SyNeR - Systèmes Non Linéaires et à Retards
CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL) - UMR 9189
Abstract : In this paper, the numerical differentiation by integration method based on Jacobi polynomials originally introduced by Mboup, Fliess and Join is revisited in the central case where the used integration window is centered. Such method based on Jacobi polynomials was introduced through an algebraic approach and extends the numerical differentiation by integration method introduced by Lanczos. The here proposed method is used to estimate the $n^{th}$ ($n \in \mathbb{N}$) order derivative from noisy data of a smooth function belonging to at least $C^{n+1+q}$ $(q \in \mathbb{N})$. In the recent paper of Mboup, Fliess and Join , where the causal and anti-causal case 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 $f\in C^{n+q+2}$ in the noise free case and the corresponding convergence rate is $O(\delta^{\frac{q+1}{n+1+q}})$ where $\delta$ is the noise level for a well chosen integration window length. Numerical examples show that this proposed method is stable and effective.
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Article dans une revue
Journal of Computational and Applied Mathematics, Elsevier, 2011, 235 (9), pp.Pages 3015-3032. 〈10.1016/j.cam.2010.12.023〉
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Da-Yan Liu, Olivier Gibaru, Wilfrid Perruquetti. Differentiation by integration with Jacobi polynomials. Journal of Computational and Applied Mathematics, Elsevier, 2011, 235 (9), pp.Pages 3015-3032. 〈10.1016/j.cam.2010.12.023〉. 〈inria-00550160v2〉

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