Spectral Properties of Numerical Differentiation

TL;DR

This paper investigates the spectral properties of numerical differentiation formulas for functions on grids with arbitrary nodes, analyzing the spectra of weight coefficients for various cases including infinite points and one-sided approximations.

Contribution

It provides a detailed spectral analysis of differentiation formulas for arbitrary node distributions, including infinite and one-sided cases, which is a novel contribution.

Findings

Spectra of weight coefficients are characterized for various node configurations.

Infinite node formulas have specific spectral properties analyzed in detail.

One-sided approximation formulas are derived and their spectral properties examined.

Abstract

We study the numerical differentiation formulae for functions given in grids with arbitrary number of nodes. We investigate the case of the infinite number of points in the formulae for the calculation of the first and the second derivatives. The spectra of the corresponding weight coefficients sequences are obtained. We examine the first derivative calculation of a function given in odd-number points and analyze the spectra of the weight coefficients sequences in the cases of both finite and infinite number of nodes. We derive the one-sided approximation for the first derivative and examine its spectral properties.