Average Nikolskii factors for random diffusion polynomials on closed Riemannian manifolds
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866912472876711936 |
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| author | Ling, Yun Wang, Heping |
| author_facet | Ling, Yun Wang, Heping |
| contents | For $1\le p,q\le \infty$, the Nikolskii factor for a diffusion polynomial $P_{\bf a}$ of degree at most $n$ is defined by $$N_{p,q}(P_{\bf a})=\frac{\|P_{\bf a}\|_{q}}{\|P_{\bf a}\|_{p}},\ \ P_{\bf a}({\bf x})=\sum_{k:λ_{k}\leq n}a_{k}ϕ_{k}({\bf x}),$$ where ${\bf a}=\{a_k\}_{λ_k\le n}$, and $\{(ϕ_k,-λ_k^2)\}_{k=0}^\infty$ are the eigenpairs of the Laplace-Beltrami operator $Δ_{\mathbb M}$ on a closed smooth Riemannian manifold $\mathbb M$ with normalized Riemannian measure. We study this average Nikolskii factor for random diffusion polynomials with independent $N(0,σ^{2})$ coefficients and obtain the exact orders. For $1\leq p<q<\infty$, the average Nikolskii factor is of order $n^{0}$ (i.e., constant), as compared to the worst case bound of order $n^{d(1/p-1/q)}$, and for $1\leq p<q=\infty$, the average Nikolskii factor is of order $(\ln n)^{1/2}$ as compared to the worst case bound of order $n^{d/p}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_06505 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Average Nikolskii factors for random diffusion polynomials on closed Riemannian manifolds Ling, Yun Wang, Heping Probability Classical Analysis and ODEs 26D05, 42A05 For $1\le p,q\le \infty$, the Nikolskii factor for a diffusion polynomial $P_{\bf a}$ of degree at most $n$ is defined by $$N_{p,q}(P_{\bf a})=\frac{\|P_{\bf a}\|_{q}}{\|P_{\bf a}\|_{p}},\ \ P_{\bf a}({\bf x})=\sum_{k:λ_{k}\leq n}a_{k}ϕ_{k}({\bf x}),$$ where ${\bf a}=\{a_k\}_{λ_k\le n}$, and $\{(ϕ_k,-λ_k^2)\}_{k=0}^\infty$ are the eigenpairs of the Laplace-Beltrami operator $Δ_{\mathbb M}$ on a closed smooth Riemannian manifold $\mathbb M$ with normalized Riemannian measure. We study this average Nikolskii factor for random diffusion polynomials with independent $N(0,σ^{2})$ coefficients and obtain the exact orders. For $1\leq p<q<\infty$, the average Nikolskii factor is of order $n^{0}$ (i.e., constant), as compared to the worst case bound of order $n^{d(1/p-1/q)}$, and for $1\leq p<q=\infty$, the average Nikolskii factor is of order $(\ln n)^{1/2}$ as compared to the worst case bound of order $n^{d/p}$. |
| title | Average Nikolskii factors for random diffusion polynomials on closed Riemannian manifolds |
| topic | Probability Classical Analysis and ODEs 26D05, 42A05 |
| url | https://arxiv.org/abs/2507.06505 |