Differentiation by integration with Jacobi polynomials

  • Authors:

  • Da-yan Liu;Olivier Gibaru;Wilfrid Perruquetti

  • Affiliations:

  • íquipe Projet ALIEN, INRIA Lille-Nord Europe, Parc Scientifique de la Haute Borne 40, avenue Halley Bít.A, Park Plaza, 59650 Villeneuve d'Ascq, France and Paul Painlevé (CNRS, UMR 8 ...;íquipe Projet ALIEN, INRIA Lille-Nord Europe, Parc Scientifique de la Haute Borne 40, avenue Halley Bít.A, Park Plaza, 59650 Villeneuve d'Ascq, France and Arts et Métiers ParisTech ...;íquipe Projet ALIEN, INRIA Lille-Nord Europe, Parc Scientifique de la Haute Borne 40, avenue Halley Bít.A, Park Plaza, 59650 Villeneuve d'Ascq, France and LAGIS (CNRS, UMR 8146), íc ...

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2011

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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 C^n^+^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(h^q^+^2) where h is the integration window length for f@?C^n^+^q^+^2 in the noise free case and the corresponding convergence rate is O(@d^q^+^1^n^+^1^+^q) where @d is the noise level for a well-chosen integration window length. Numerical examples show that this proposed method is stable and effective.