On Optimal Regularization Methods for Fractional Differentiation

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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.