Towards a Universal Gibbs Constant

TL;DR

This paper demonstrates that various families of orthogonal polynomials, including Gegenbauer, Laguerre, and Hermite, exhibit the Gibbs Phenomenon, extending previous findings and providing elementary proofs and numerical insights.

Contribution

It shows that the Gibbs Phenomenon occurs in generalized Laguerre and Hermite polynomials, broadening the understanding of this effect across different orthogonal polynomial families.

Findings

Gibbs Phenomenon observed in Laguerre and Hermite polynomials

Elementary methods used for proof

Numerical example illustrating convergence rate

Abstract

In this paper we build on the work of \cite{kaber} where it was shown that the one-parameter family of Gegenbauer Polynomials (GP) exhibit a Gibbs Phenomenon at a jump discontinuity. We show that the one-parameter family of Generalized Laguerre Polynomials (GLP) also exhibit a Gibbs Phenomenon. Among many differences, a major one is that the GLP are orthogonal on a non-compact subset of $\R$, while the GP are orthogonal on $[-1,1]$. Our strategy follows that of \cite{kaber} and we use entirely elementary methods to arrive at our result. As a special case we show that the Hermite Polynomials also possess a Gibbs Phenomenon. We conclude with a numerical example exhibiting the rate of convergence to the Gibbs constant and a conjectured identity for special values of the GLP.