The Gibbs phenomenon for the Krawtchouk polynomials
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910042581630976 |
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| author | Cullinan, John Young, Elisabeth |
| author_facet | Cullinan, John Young, Elisabeth |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_05408 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Gibbs phenomenon for the Krawtchouk polynomials Cullinan, John Young, Elisabeth Mathematical Physics 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. |
| title | The Gibbs phenomenon for the Krawtchouk polynomials |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2603.05408 |