Preprint No.
A-98-14
U. Tautenhahn, R. Gorenflo
Optimal Approximations of Fractional Derivatives
Abstract: In this
paper we consider the following fractional differentiation problem:
given noisy data $f^\delta \in L^2(\rnu)$ to $f$,
determine the fractional derivative $u = D_\beta f \in L^2(\rnu)$
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
$\|f-f^\delta\|_{L^2(\rnus)} \le \delta$
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: regularization, fractional differentiation, Abel integral equations of first kind
Mathematics Subject Classification (MSC91): 65R30 Improperly posed problems
, 65D25 Numerical differentiation
Language: ENG
Available: Pr-A-98-14.ps
Contact: Gorenflo, R., Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (gorenflo@math.fu-berlin.de)
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