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] and [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] and [20] and extends the numerical differentiation by integration method introduced by Lanczos (1956) [21]. The method proposed here, rooted in [19] and [20], is used to estimate the nnth (n∈Nn∈N) order derivative from noisy data of a smooth function belonging to at least Cn+1+q(q∈N)Cn+1+q(q∈N). In [19] and [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)O(hq+2) where hh is the integration window length for f∈Cn+q+2f∈Cn+q+2 in the noise free case and the corresponding convergence rate is O(δq+1n+1+q) where δδ is the noise level for a well-chosen integration window length. Numerical examples show that this proposed method is stable and effective.