ON THE KRAWTCHOUK EXPANSION OF THE SIGN FUNCTION

We study the Fourier approximation \(\mathcal {F}_N\) ${\mathcal{F}}_N$ of the sign function by the Krawtchouk polynomials. We show that the steepness \(\mathcal {F}_N'(0)\) ${\mathcal{F}}_N^{\prime }(0)$ of the approximation is bounded by explicitly proving \(\lim _{N \rightarrow \infty } \mathcal {F}_N'(0) = \log 4\) ${lim}_{N\to \infty }{\mathcal{F}}_N^{\prime }(0)=log4$ . This is in contrast to approximations by classical orthogonal polynomials, where the steepness has been shown to be unbounded as the degree increases. We also give numerical evidence that the Gibbs phenomenon of the approximation differs from the classical Gibbs constant; this is also in contrast to other families of orthogonal polynomials.