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Inequalities for the derivatives

A. G. Ramm
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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.

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A. G. Ramm. 1999-11-25. Inequalities for the derivatives. https://arxiv.org/abs/math-ph/9911034

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