Sharp sign uncertainty for trigonometric polynomials
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866910281945317376 |
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| author | Ismoilov, Tolibjon |
| author_facet | Ismoilov, Tolibjon |
| contents | We study sign uncertainty principles for trigonometric polynomials of prescribed degree $N$ with respect to a symmetric Borel measure $μ$ on the unit circle $\mathbb{R}/\mathbb{Z}$. For each such measure, we determine the smallest radius of the last sign change for trigonometric polynomials with non-positive $μ$-integral. We further extend these results to polar measures on higher-dimensional spheres $\mathbb{S}^d$, showing that the extremal problem reduces to the one-dimensional case via the polar part of the measure, and we establish a polynomial analogue on $[0,1]$ using orthogonal polynomials on the real line. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_02299 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp sign uncertainty for trigonometric polynomials Ismoilov, Tolibjon Classical Analysis and ODEs 42A16, 41A55, 42C05, 42A05, 46E22 We study sign uncertainty principles for trigonometric polynomials of prescribed degree $N$ with respect to a symmetric Borel measure $μ$ on the unit circle $\mathbb{R}/\mathbb{Z}$. For each such measure, we determine the smallest radius of the last sign change for trigonometric polynomials with non-positive $μ$-integral. We further extend these results to polar measures on higher-dimensional spheres $\mathbb{S}^d$, showing that the extremal problem reduces to the one-dimensional case via the polar part of the measure, and we establish a polynomial analogue on $[0,1]$ using orthogonal polynomials on the real line. |
| title | Sharp sign uncertainty for trigonometric polynomials |
| topic | Classical Analysis and ODEs 42A16, 41A55, 42C05, 42A05, 46E22 |
| url | https://arxiv.org/abs/2606.02299 |