Orthogonal polynomials and regression-based symmetric derivatives (Q2464627)

A typical method for constructing difference quotients involves algebraic manipulation of Taylor polynomial approximations of \(f\) at \(x\). From a statistical perspective here is shown that certain types of symmetric derivatives originate from a simple least-square regression problem involving Chebyshev polynomials. If the number of data points used in this regression tends to infinity, the resulting integrals involving Legendre polynomials tend to Lanczos derivatives, a result that demonstrate that this is a continuous version of the symmetric derivative.