Differentiation by integration using orthogonal polynomials, a survey

This survey paper discusses the history of approximation formulas for n-th order derivatives by integrals involving orthogonal polynomials. There is a large but rather disconnected corpus of literature on such formulas. We give some results in greater generality than in the literature. Notably we un...

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Bibliographic Details
Published inJournal of approximation theory Vol. 164; no. 5; pp. 637 - 667
Main Authors Diekema, Enno, Koornwinder, Tom H.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.05.2012
Subjects
Approximated higher order derivative
Filter
Least-square approximation
Orthogonal polynomial
Smoothing
Wavelet
Wavelet
Approximated higher order derivative
Least-square approximation
Filter
Smoothing
Orthogonal polynomial
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Summary:This survey paper discusses the history of approximation formulas for n-th order derivatives by integrals involving orthogonal polynomials. There is a large but rather disconnected corpus of literature on such formulas. We give some results in greater generality than in the literature. Notably we unify the continuous and discrete case. We make many side remarks, for instance on wavelets, Mantica’s Fourier–Bessel functions and Greville’s minimum Rα formulas in connection with discrete smoothing.
ISSN:0021-9045
1096-0430
DOI:10.1016/j.jat.2012.01.003