arXiv 1605.025082016-05-09math.CA

关于带Laguerre权的$L_2$马氏不等式

On the $L_2$ Markov Inequality with Laguerre Weight

Geno Nikolov, Alexei Shadrin

备注：: 17 pages, one figure

摘要：

令$w_α(t)=t^α\,e^{-t}$、$α>-1$是拉盖尔权重函数，并且$|\cdot|_{w_α}$表示相关联的$L_2$范数，即，$$ | f|_{w_α}:=\Big(\int_{0}^{\infty}w_α(t)| f(t)|^2\,dt\Big)^{1/2}. $$用${\cal P}_n$表示次数不超过$n$的代数多项式的集合。本文研究了马氏不等式中的最佳常数$c_n(α)$，$$ | p^{\prime}|_{w_α}\leq c_n(α)\,| p|_{w_α}\,,\quad p\in {\cal P}_n\,, $$，即常数$$ c_{n}(α)=\sup_{\mathop{}^{p\in {\cal P}_n}_{p\ne 0}}\frac{| p^{\prime}|_{w_α}}{| p|_{w_α}}\,, $$及其渐近值$$ c(α)=\lim_{n\rightarrow\infty}\frac{c_{n}(α)}{n}\,. $$，得到了$c_{n}(α)$和$c(α)$的上下界.注意，根据P. Dörfler在2002年的一个结果，$c(α)=[j_{(α-1)/2,1}]^{-1}$，$j_{ν,1}$是贝塞尔函数$J_ν(z)$的第一个正零点，因此我们对$c(α)$的边界也意味着对$j_{(α-1)/2,1}$的边界。

英文摘要：

Let $w_α(t)=t^α\,e^{-t}$, $α>-1$, be the Laguerre weight function, and $|\cdot|_{w_α}$ denote the associated $L_2$-norm, i.e., $$ | f|_{w_α}:=\Big(\int_{0}^{\infty}w_α(t)| f(t)|^2\,dt\Big)^{1/2}. $$ Denote by ${\cal P}_n$ the set of algebraic polynomials of degree not exceeding $n$. We study the best constant $c_n(α)$ in the Markov inequality in this norm, $$ | p^{\prime}|_{w_α}\leq c_n(α)\,| p|_{w_α}\,,\quad p\in {\cal P}_n\,, $$ namely the constant $$ c_{n}(α)=\sup_{\mathop{}^{p\in {\cal P}_n}_{p\ne 0}}\frac{| p^{\prime}|_{w_α}}{| p|_{w_α}}\,, $$ and we are also interested in its asymptotic value $$ c(α)=\lim_{n\rightarrow\infty}\frac{c_{n}(α)}{n}\,. $$ In this paper we obtain lower and upper bounds for both $c_{n}(α)$ and $c(α)$. % Note that according to a result of P. Dörfler from 2002, $c(α)=[j_{(α-1)/2,1}]^{-1}$, with $j_{ν,1}$ being the first positive zero of the Bessel function $J_ν(z)$, hence our bounds for $c(α)$ imply bounds for $j_{(α-1)/2,1}$ as well.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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