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

Abstract.

In this paper, we consider the problem of estimating the derivative article image of a function article image from its noisy version article image contaminated by a stochastic white noise and argue that in certain relevant cases the reconstruction of article image by the derivatives of the partial sums of Fourier–Legendre series of article image 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.

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This journal presents original articles on the theory, numerics and applications of inverse and ill-posed problems. These inverse and ill-posed problems arise in mathematical physics and mathematical analysis, geophysics, acoustics, electrodynamics, tomography, medicine, ecology, financial mathematics etc. Articles on the construction and justification of new numerical algorithms of inverse problem solutions are also published.

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