Differentiation by integration using orthogonal polynomials, a survey

Details Differentiation by integration using orthogonal polynomials, a survey Differentiation by integration using orthogonal polynomials, a survey

2011-02-25

arxiv.org/abs/1102.5219v2

J. Approx. Theory 164 (2012), 637-667

Mathematics

Classical Analysis and ODEs

35 pages, 3 figures; minor corrections, subsection 3.11 added; accepted by J. Approx. Theory

Scientific paper

10.1016/j.jat.2012.01.003

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_alpha formulas in connection with discrete smoothing.

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