Inequalities for the derivatives

TL;DR

The paper investigates the limits of stable approximation of derivatives from noisy data, proposing an algorithm for fractional derivatives and establishing its optimality under certain conditions.

Contribution

It introduces a new algorithm for stable approximation of fractional derivatives and proves its optimality among all algorithms under specific assumptions.

Findings

Stable approximation of first derivatives is impossible with noisy data and bounds.

A new algorithm for fractional derivatives is proposed with error estimates.

The algorithm is proven to be optimal among all linear and nonlinear methods.

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.