Preprint No. A-98-14

U. Tautenhahn, R. Gorenflo

Optimal Approximations of Fractional Derivatives

Abstract: In this paper we consider the following fractional differentiation problem: given noisy data $f^\delta \in L^2(\rnu)$ to $f$, determine the fractional derivative $u = D_\beta f \in L^2(\rnu)$ for $\beta > 0$, which is the solution of the integral equation of first kind $(A_\beta u) (x) = {1 \over \Gamma (\beta)} \int_{-\infty}^x {u(t)\, dt \over (x-t)^{1-\beta}} = f(x)$. Assuming $\|f-f^\delta\|_{L^2(\rnus)} \le \delta$ and $\|u\|_p \le E$ (where $\|\cdot \|_p$ denotes the usual Sobolev norm of order $p > 0$) we answer the question concerning the best possible accuracy for identifying $u$ from the noisy data $f^\delta $. Furthermore, we discuss special regularization methods which realize this best possible accuracy.

Keywords: regularization, fractional differentiation, Abel integral equations of first kind

Mathematics Subject Classification (MSC91): 65R30 Improperly posed problems , 65D25 Numerical differentiation

Language: ENG


Contact: Gorenflo, R., Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (

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