The Gibbs phenomenon for the Krawtchouk polynomials

TL;DR

This paper investigates the Gibbs phenomenon in Fourier approximations of the sign function using Krawtchouk polynomials, revealing unique behaviors in the approximation's steepness and Gibbs constant compared to classical orthogonal polynomials.

Contribution

It provides numerical and theoretical analysis showing the Gibbs phenomenon for Krawtchouk polynomials differs from classical cases, including bounded steepness and a distinct Gibbs constant.

Findings

Gibbs phenomenon for Krawtchouk polynomials differs from classical cases.

The steepness of the approximation at zero is bounded and converges to log(4).

The Gibbs constant for Krawtchouk polynomial approximation is different from the classical Gibbs constant.

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 $\mathcal{F}_N'(0)$ of the approximation is bounded by explicitly proving $\lim_{N \to \infty} \mathcal{F}_N'(0) = \log 4$. This is also in contrast to approximations by classical orthogonal polynomials, where the steepness has been shown to be unbounded as the degree increases.