From an Elementary Proof of Error Representation for Hermite Quadrature to a Rediscovery of Legendre Polynomials and Rodrigues Formula

TL;DR

This paper presents a simplified derivation of the error formula for Hermite quadrature, revealing new insights into Legendre polynomials and their relation to Hermite interpolation, applicable to less regular functions.

Contribution

It introduces an elementary derivation of Hermite quadrature error, rediscovering Legendre polynomials and Rodrigues formula, and extends applicability to functions with milder regularity.

Findings

Error formula valid for functions with nth derivative

Legendre polynomials are the error kernels for Hermite quadrature

Rodrigues formula for Legendre polynomials is rediscovered

Abstract

We generalize two-point interpolatory Hermite quadrature to functions with available values and the first (n-1) derivatives at both end points. Armed with integration by parts in the reverse form we provide an elementary derivation of an exact error represenation of Hermite quadrature rule. This approach possesses several advantages over the classical approaches: i) Only integration by parts is needed for the derivation; ii) the error representation requires much milder regularity, namely the existence of nth-order derivative rather than a (2n)th-order derivative of the function under consideration. As a result, our error formula is valid for less regular functions for which the classical ones are not valid; iii) our approach rediscovers Legendre polynomials and more interestingly it provides a surprisingly elegant relation between Legendre polynomial and Hermite interpolation. In particular, Legendre polynomials are precisely the error kernels for interpolatory Hermite quadrature rules; and iv) We also rediscover the Rodrigues formula for Legendre polynomials as part of our findings. For those who are interested in a different proof of the exact error representation for Hermite quadrature rule, we provide an alternative proof using the Peano kernel theorem. We also provide a composite interpolatory Hermite quadrature rule for practical applications.