On Optimal Regularization Methods for Fractional Differentiation

U. Tautenhahn and R. Gorenflo

Abstract: In this paper we consider the following fractional differentiation problem: given noisy data $f^\delta \in L^2(\R)$ to $f$, approximate the fractional derivative $u = D_\beta f \in L^2(\R)$ 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 $\|f-f^\delta\|_{L^2(\R)} \le \delta$ 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: ill-posed problems, fractional differentiation, regularization methods, optimal error bounds

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