Inequalities for the derivatives
Mathematical Physics
2007-05-23 v1 Classical Analysis and ODEs
math.MP
Numerical Analysis
Abstract
It is proved that one cannot approximate stably the first derivative of a smooth function given noisy values of this function and a bound on this function and its first derivative. Such an approximation is shown to be possible if an a priori bound is known for a fractional derivative of order greater than one. An algorithm is proposed for such a stable approximation and error estimates for the proposed algorithm are given. Under certain assumptions it is proved that this algorithm is best possible among all linear and nonlinear algorithms.
Cite
@article{arxiv.math-ph/9911034,
title = {Inequalities for the derivatives},
author = {A. G. Ramm},
journal= {arXiv preprint arXiv:math-ph/9911034},
year = {2007}
}