On Optimal Regularization Methods for Fractional Differentiation

Abstract

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 $(A_{\beta }u(x)=\frac{1}{\Gamma (\beta )}{\int }_{–\infty }^x\frac{u(t)dt}{(x–t{)}^{1–\beta }}=f(x)$. Assuming $\Vert f–f^{\delta }{\Vert }_{L^2(\mathbb{R})}\le \delta$ and $\Vert u{\Vert }_p\le E$ (where $\Vert \cdot {\Vert }_p$ 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.