Gibbs phenomena for some classical orthogonal polynomials

Abstract

The Gibbs phenomena associated to partial sums of Fourier series are now well understood. In this paper, we show that Gibbs phenomena also occur for expansions of functions in terms of members of one of several general classes of orthogonal polynomials, in particular treating in a relatively unified manner expansions in either Legendre (more generally Jacobi), Hermite, or Laguerre polynomials. Noteworthy is the fact that the Gibbs constants associated to all of these expansions have the same value, approximately 0.0893 or more precisely 1π∫0πsin⁡ttdt−12.