The Gibbs phenomenon for series of orthogonal polynomials

Abstract

This note considers the four classes of orthogonal polynomials – Chebyshev, Hermite, Laguerre, Legendre – and investigates the Gibbs phenomenon at a jump discontinuity for the corresponding orthogonal polynomial series expansions. The perhaps unexpected thing is that the Gibbs constant that arises for each class of polynomials appears to be the same as that for Fourier series expansions. Each class of polynomials has features which are interesting numerically. Finally a plausibility argument is included showing that this phenomenon for the Gibbs constants should not have been unexpected. These findings suggest further investigations suitable for undergraduate research projects or small group investigations.

Notes

¹More and deeper information and further references about each orthogonal class can be found in Eric Weinstein's world of MATHEMATICS at http://www.mathworld.wolfram.com and at the engineering fundamentals website http://www.efunda.com.

²The formula given on the website http://www.efunda.com/math/Laguerre is incorrect.