Differentiation by integration using orthogonal polynomials, a survey

TL;DR

This survey reviews the history and advances in approximation formulas for derivatives using orthogonal polynomials, unifying continuous and discrete cases and exploring related topics like wavelets and Fourier-Bessel functions.

Contribution

It provides a more general framework for derivative approximation formulas and unifies the continuous and discrete cases, expanding on existing literature.

Findings

Unified continuous and discrete approximation formulas

Extended results beyond previous literature

Connections to wavelets and Fourier-Bessel functions

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_alpha formulas in connection with discrete smoothing.