Differentiation by integration with Jacobi polynomials

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 n th ( 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 ( δ q + 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.