Filtered Legendre expansion method for numerical differentiation at the boundary point with application to blood glucose predictions

• Published in: Applied mathematics and computation Vol. 224; pp. 835 - 847
• Main Authors: Mhaskar, Hrushikesh N., Naumova, V., Pereverzyev, S.V.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.11.2013
• Subjects: Approximation; Blood; Blood glucose prediction; Boundaries; Estimates; Glucose; Legendre polynomials; Mathematical analysis; Mathematical models; Noise; Numerical differentiation; Blood glucose prediction; Legendre polynomials; Numerical differentiation
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2013.09.015

Let f:[-1,1]→R be continuously differentiable. We consider the question of approximating f′(1) from given data of the form (tj,f(tj))j=1M where the points tj are in the interval [-1,1]. It is well known that the question is ill-posed, and there is very little literature on the subject known to us. We consider a summability operator using Legendre expansions, together with high order quadrature formulas based on the points tj’s to achieve the approximation. We also estimate the effect of noise on our approximation. The error estimates, both with or without noise, improve upon those in the existing literature, and appear to be unimprovable. The results are applied to the problem of short term prediction of blood glucose concentration, yielding better results than other comparable methods.