Differentiation by integration using orthogonal polynomials, a survey

This survey paper discusses the history of approximation formulas for nn-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 RαRα formulas in connection with discrete smoothing.