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Abstract.
In this paper, we consider the problem of estimating the derivative
of a function from its noisy version contaminated by a stochastic white noise and argue that in certain relevant cases the reconstruction of by the derivatives of the partial sums of Fourier–Legendre series of has advantage over some standard approaches. One of the interesting observations made in the paper is that in a Hilbert scale generated by the system of Legendre polynomials the stochastic white noise does not increase, as it might be expected, the loss of accuracy compared to the deterministic noise of the same intensity. We discuss the accuracy of the considered method in the spaces L2 and C and provide a guideline for an adaptive choice of the number of terms in differentiated partial sums (note that this number is playing the role of a regularization parameter). Moreover, we discuss the relation of the considered numerical differentiation scheme with the well-known Savitzky–Golay derivative filters, as well as possible applications in diabetes technology.Keywords: Numerical differentiation; Legendre polynomials; stochastic white noise; adaptive parameter choice; Savitzky–Golay method; diabetes technology
Received: 2012-07-03
Published Online: 2013-04-03
Published in Print: 2013-04-01
© 2013 by Walter de Gruyter Berlin Boston