Differentiation by integration with Jacobi polynomials

TL;DR

This paper revisits a Jacobi polynomial-based numerical differentiation method using integration, demonstrating improved bias error and convergence rates for estimating derivatives from noisy data, with stability and effectiveness shown through numerical examples.

Contribution

The paper extends the Jacobi polynomial-based differentiation method to the central case, providing new bias and convergence rate analyses, and validating its stability and effectiveness.

Findings

Bias error is $O(h^{q+2})$ for smooth functions.

Convergence rate is $O( ext{noise level}^{(q+1)/(n+1+q)})$.

Numerical examples confirm stability and effectiveness.

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.