Legendre polynomials as a recommended basis for numerical differentiation in the presence of stochastic white noise

In this paper, we consider the problem of estimating the derivative y' of a function y is an element of C-1 [-1, 1] from its noisy version y(delta) ffi contaminated by a stochastic white noise and argue that in certain relevant cases the reconstruction of y' by the derivatives of the partial sums of Fourier-Legendre series of y ffi 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 L-2 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.