# On Optimal Regularization Methods for Fractional Differentiation

### Ulrich Tautenhahn

University of Applied Sciences, Zittau, Germany### Rudolf Gorenflo

Freie Universität Berlin, Germany

## Abstract

In this paper we consider the following fractional differentiation problem: given noisy data $f_{δ}L_{2}(R)$ to $f$, approximate the fractional derivative $u=D_{β}f∈L_{2}(R)$ for $β>0$, which is the solution of the integral equation of first kind $(A_{β}u(x)=Γ(β)1 ∫_{–∞}(x–t)_{1–β}u(t)dt =f(x)$. Assuming $∥f–f_{δ}∥_{L_{2}(R)}≤δ$ and $∥u∥_{p}≤E$ (where $∥⋅∥_{p}$ 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 $f_{δ}$. 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