Average Nikolskii factors for random trigonometric polynomials
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909547195531264 |
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| author | Ling, Yun Geng, Jiaxin Li, Jiansong Wang, Heping |
| author_facet | Ling, Yun Geng, Jiaxin Li, Jiansong Wang, Heping |
| contents | For $1\le p,q\le \infty$, the Nikolskii factor for a trigonometric polynomial $T_{\bf a}$ is defined by $$\mathcal N_{p,q}(T_{\bf a})=\frac{\|T_{\bf a}\|_{q}}{\|T_{\bf a}\|_{p}},\ \ T_{\bf a}(x)=a_{1}+\sum\limits^{n}_{k=1}(a_{2k}\sqrt{2}\cos kx+a_{2k+1}\sqrt{2}\sin kx).$$ We study this average Nikolskii factor for random trigonometric polynomials with independent $N(0,σ^{2})$ coefficients and obtain that the exact order. For $1\leq p<q<\infty$, the average Nikolskii factor is order degree to the 0, as compared to the degree $1/p-1/q$ worst case bound. We also give the generalization to random multivariate trigonometric polynomials. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_16786 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Average Nikolskii factors for random trigonometric polynomials Ling, Yun Geng, Jiaxin Li, Jiansong Wang, Heping Classical Analysis and ODEs 26D05, 42A05 For $1\le p,q\le \infty$, the Nikolskii factor for a trigonometric polynomial $T_{\bf a}$ is defined by $$\mathcal N_{p,q}(T_{\bf a})=\frac{\|T_{\bf a}\|_{q}}{\|T_{\bf a}\|_{p}},\ \ T_{\bf a}(x)=a_{1}+\sum\limits^{n}_{k=1}(a_{2k}\sqrt{2}\cos kx+a_{2k+1}\sqrt{2}\sin kx).$$ We study this average Nikolskii factor for random trigonometric polynomials with independent $N(0,σ^{2})$ coefficients and obtain that the exact order. For $1\leq p<q<\infty$, the average Nikolskii factor is order degree to the 0, as compared to the degree $1/p-1/q$ worst case bound. We also give the generalization to random multivariate trigonometric polynomials. |
| title | Average Nikolskii factors for random trigonometric polynomials |
| topic | Classical Analysis and ODEs 26D05, 42A05 |
| url | https://arxiv.org/abs/2503.16786 |