Full length article: Differentiation by integration using orthogonal polynomials, a survey

Authors:
Enno Diekema;Tom H. Koornwinder
Affiliations:
P.O.Box 2439, 7302 EP Apeldoorn, The Netherlands;Korteweg-de Vries Institute, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands
Venue:
Journal of Approximation Theory
Year:
2012

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Abstract

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 R"@a formulas in connection with discrete smoothing.