The Gibbs phenomenon for the Krawtchouk polynomials

Abstract

We study the Fourier approximation ${\mathcal{F}}_N$ of the sign function by the Krawtchouk polynomials. We give numerical evidence that the Gibbs phenomenon of the approximation differs from the classical Gibbs constant; this is in contrast to other families of orthogonal polynomials. We also show that the steepness ${\mathcal{F}}_N^{\prime }(0)$ of the approximation is bounded by explicitly proving ${\lim }_{N\to \infty }{\mathcal{F}}_N^{\prime }(0)=\log 4$. This is also in contrast to approximations by classical orthogonal polynomials, where the steepness has been shown to be unbounded as the degree increases.

16 pages