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Zeitschrift für Analysis und ihre Anwendungen

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Volume 18, Issue 2, 1999, pp. 449–467
DOI: 10.4171/ZAA/892

Published online: 1999-06-30

On Optimal Regularization Methods for Fractional Differentiation

Ulrich Tautenhahn and Rudolf Gorenflo[1]

(1) Freie Universität Berlin, Germany

In this paper we consider the following fractional differentiation problem: given noisy data $f^{\delta} L^2(\mathbb R)$ to $f$, approximate the fractional derivative $u = D_{\beta} f \in L^2(\mathbb R)$ for $\beta > 0$, which is the solution of the integral equation of first kind $(A_{\beta} u(x) = \frac{1}{\Gamma (\beta)} \int^x_{– \infty} \frac {u(t) dt}{(x–t)^{1– \beta}} = f(x)$. Assuming $\|f–f^{\delta} \|_{L^2(\mathbb R)} ≤ \delta$ and $\| u \|_p ≤ E$ (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: Ill-posed problems, fractional differentiation, regularization methods, optimal error bounds

Tautenhahn Ulrich, Gorenflo Rudolf: On Optimal Regularization Methods for Fractional Differentiation. Z. Anal. Anwend. 18 (1999), 449-467. doi: 10.4171/ZAA/892