球多项式的渐近 Nikolskii 常数估计
经典分析与常微分方程
2019-07-10 v1
摘要
令 Π n d \Pi_n^d Π n d S d ⊂ R d + 1 \mathbb{S}^d\subset \mathbb{R}^{d+1} S d ⊂ R d + 1 d σ d\sigma d σ ∫ S d d σ ( x ) = 1 \int_{\mathbb{S}^d} \, d\sigma(x)=1 ∫ S d d σ ( x ) = 1 L ∗ ( d ) : = lim n → ∞ 1 dim Π n d sup f ∈ Π n d ∥ f ∥ L ∞ ( S d ) ∥ f ∥ L 1 ( S d ) , \mathcal{L}^\ast(d):=\lim_{n\to \infty} \frac 1 {\dim \Pi_n^d} \sup_{f\in \Pi_n^d} \frac { \|f\|_{L^\infty(\mathbb{S}^d)}}{\|f\|_{L^1(\mathbb{S}^d)}}, L ∗ ( d ) := n → ∞ lim dim Π n d 1 f ∈ Π n d sup ∥ f ∥ L 1 ( S d ) ∥ f ∥ L ∞ ( S d ) , I α : = inf a k ∥ j α + 1 ( t ) − ∑ k = 1 ∞ a k j α ( q α + 1 , k t / q α + 1 , 1 ) ∥ L ∞ ( R + ) \mathcal{I}_\alpha:=\inf_{a_k} \Bigl\| j_{\alpha+1} (t)- \sum_{k=1}^\infty a_k j_{\alpha} \bigl( q_{\alpha+1,k}t/q_{\alpha+1,1}\bigr)\Bigr\|_{L^\infty(\mathbb{R}_+)} I α := a k inf j α + 1 ( t ) − k = 1 ∑ ∞ a k j α ( q α + 1 , k t / q α + 1 , 1 ) L ∞ ( R + ) R + \mathbb{R}_+ R + { a k } k = 1 ∞ ⊂ R \{a_k\}_{k=1}^\infty\subset \mathbb{R} { a k } k = 1 ∞ ⊂ R j α j_\alpha j α j α ( 0 ) = 1 j_\alpha(0)=1 j α ( 0 ) = 1 { q α + 1 , k } k = 1 ∞ \{q_{\alpha+1,k}\}_{k=1}^\infty { q α + 1 , k } k = 1 ∞ j α + 1 j_{\alpha+1} j α + 1 α ≥ − 0.272 \alpha\ge -0.272 α ≥ − 0.272 I α = ∫ 0 q α + 1 , 1 j α + 1 ( t ) t 2 α + 1 d t ∫ 0 q α + 1 , 1 t 2 α + 1 d t = 1 F 2 ( α + 1 ; α + 2 , α + 2 ; − q α + 1 , 1 2 4 ) . \mathcal{I}_\alpha= \frac{\int_{0}^{q_{\alpha+1,1}}j_{\alpha+1}(t)t^{2\alpha+1}\,dt}{\int_{0}^{q_{\alpha+1,1}}t^{2\alpha+1}\,dt}= {}_{1}F_{2}\Bigl(\alpha+1;\alpha+2,\alpha+2;-\frac{q_{\alpha+1,1}^{2}}{4}\Bigr). I α = ∫ 0 q α + 1 , 1 t 2 α + 1 d t ∫ 0 q α + 1 , 1 j α + 1 ( t ) t 2 α + 1 d t = 1 F 2 ( α + 1 ; α + 2 , α + 2 ; − 4 q α + 1 , 1 2 ) . L ∗ ( d ) \mathcal{L}^\ast(d) L ∗ ( d ) d → ∞ d\to\infty d → ∞ 0.5 d ≤ L ∗ ( d ) ≤ ( 0.857 ⋯ ) d ( 1 + ε d ) with ε d = O ( d − 2 / 3 ) . 0.5^d\le \mathcal{L}^{*}(d)\le (0.857\cdots)^{d\,(1+\varepsilon_d)} \ \ \ \ \ \text{with $\varepsilon_d =O(d^{-2/3})$.} 0. 5 d ≤ L ∗ ( d ) ≤ ( 0.857 ⋯ ) d ( 1 + ε d ) with ε d = O ( d − 2/3 ) .
引用
@article{arxiv.1907.03832,
title = {Estimates of the asymptotic Nikolskii constants for spherical polynomials},
author = {Feng Dai and Dmitry Gorbachev and Sergey Tikhonov},
journal= {arXiv preprint arXiv:1907.03832},
year = {2019}
}
备注
27 pages