JournalszaaVol. 18, No. 2pp. 449–467

On Optimal Regularization Methods for Fractional Differentiation

  • Ulrich Tautenhahn

    University of Applied Sciences, Zittau, Germany
  • Rudolf Gorenflo

    Freie Universität Berlin, Germany
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Abstract

In this paper we consider the following fractional differentiation problem: given noisy data fδL2(R)f^{\delta} L^2(\mathbb R) to ff, approximate the fractional derivative u=DβfL2(R)u = D_{\beta} f \in L^2(\mathbb R) for β>0\beta > 0, which is the solution of the integral equation of first kind (Aβu(x)=1Γ(β)xu(t)dt(xt)1β=f(x)(A_{\beta} u(x) = \frac{1}{\Gamma (\beta)} \int^x_{– \infty} \frac {u(t) dt}{(x–t)^{1– \beta}} = f(x). Assuming ffδL2(R)δ\|f–f^{\delta} \|_{L^2(\mathbb R)} ≤ \delta and upE\| u \|_p ≤ E (where p\| \cdot \|_p denotes the usual Sobolev norm of order p>0p > 0) we answer the question concerning the best possible accuracy for identifying uu from the noisy data fδf^{\delta}. Furthermore, we discuss special regularization methods which realize this best possible accuracy.

Cite this article

Ulrich Tautenhahn, Rudolf Gorenflo, On Optimal Regularization Methods for Fractional Differentiation. Z. Anal. Anwend. 18 (1999), no. 2, pp. 449–467

DOI 10.4171/ZAA/892