Orthogonal Polynomials and Regression-Based Symmetric Derivatives

Abstract

We demonstrate how certain types of symmetric derivatives originate from a simple least-squares regression problem involving discrete Chebyshev polynomials. As the number of data points used in this regression tends to infinity, the resulting integrals, which involve Legendre polynomials, lead to Lanczos derivatives, a result that demonstrates how this latter entity is merely a continuous version of the symmetric derivative.