Differentiation by integration with Jacobi polynomials

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Abstract

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∈N) order derivative from noisy data of a smooth function belonging to at least Cn+1+q(q∈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(hq+2) where h is the integration window length for f∈Cn+q+2 in the noise free case and the corresponding convergence rate is O(δq+1/n+1+q) where δ is the noise level for a well-chosen integration window length. Numerical examples show that this proposed method is stable and effective. © 2010 Elsevier B.V. All rights reserved.

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Liu, D. Y., Gibaru, O., & Perruquetti, W. (2011). Differentiation by integration with Jacobi polynomials. Journal of Computational and Applied Mathematics, 235(9), 3015–3032. https://doi.org/10.1016/j.cam.2010.12.023

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