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Convergence Rate of the Causal Jacobi Derivative Estimator
Da-Yan Liu1, 2, 3, Olivier Gibaru1, 4, Wilfrid Perruquetti1, 2

Numerical causal derivative estimators from noisy data are essential for real time applications especially for control applications or fluid simulation so as to address the new paradigms in solid modeling and video compression. By using an analytical point of view due to Lanczos \cite{C. Lanczos} to this causal case, we revisit $n^{th}$\ order derivative estimators originally introduced within an algebraic framework by Mboup, Fliess and Join in \cite{num,num0}. Thanks to a given noise level $\delta$ and a well-suitable integration length window, we show that the derivative estimator error can be $\mathcal{O}(\delta ^{\frac{q+1}{n+1+q}})$ where $q$\ is the order of truncation of the Jacobi polynomial series expansion used. This so obtained bound helps us to choose the values of our parameter estimators. We show the efficiency of our method on some examples.
1:  INRIA Lille - Nord Europe - Non-A
2:  LAGIS - Laboratoire d'Automatique, Génie Informatique et Signal
3:  Laboratoire de Mathématiques Paul Painlevé
4:  L2MA - Laboratoire de Métrologie et de Mathématiques Appliquées
Numerical differentiation – Ill-posed problems – Jacobi orthogonal series