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arXiv · 2603.05408

The Gibbs phenomenon for the Krawtchouk polynomials

John Cullinan·Elisabeth Young
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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.

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John Cullinan, Elisabeth Young. 2026-03-05. The Gibbs phenomenon for the Krawtchouk polynomials. https://arxiv.org/abs/2603.05408

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