Hardy’s inequalities in finite dimensional Hilbert spaces,Proceedings of the American Mathematical Society

当前位置： X-MOL 学术 › Proc. Am. Math. Soc. › 论文详情

Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)

Hardy’s inequalities in finite dimensional Hilbert spaces
Proceedings of the American Mathematical Society ( IF 0.8 ) Pub Date : 2021-03-26 , DOI: 10.1090/proc/15467
Dimitar K. Dimitrov , Ivan Gadjev , Geno Nikolov , Rumen Uluchev

Abstract:We study the behaviour of the smallest possible constants

and

in Hardy's inequalities

$\displaystyle \sum _{k=1}^{n}\Big (\frac {1}{k}\sum _{j=1}^{k}a_j\Big )^2\leq d_n\,\sum _{k=1}^{n}a_k^2, \qquad (a_1,\ldots ,a_n) \in \mathbb{R}^n$

and

$\displaystyle \int _{0}^{\infty }\Bigg (\frac {1}{x}\int _{0}^{x}f(t)\,dt\Bigg )^2 dx \leq c_n \int _{0}^{\infty }f^2(x)\,dx,\qquad f\in \mathcal {H}_n,$

for the finite dimensional spaces $\mathbb{R} ^n$ and $\mathcal {H}_n\colonequals \{f\,:\, \int _0^x f(t) dt =e^{-x/2}\,p(x)\ :\ p\in \mathcal {P}_n, p(0)=0\}$ , where $\mathcal {P}_n$ is the set of real-valued algebraic polynomials of degree not exceeding

. The constants

and

are identified to be expressed in terms of the smallest zeros of the so-called continuous dual Hahn polynomials and the two-sided estimates for

and

of the form

$\displaystyle 4-\frac {c}{\ln n}< d_n, c_n<4-\frac {c}{\ln ^2 n}\,,\qquad c>0\,$

are established.

中文翻译：

有限维希尔伯特空间中的Hardy不等式

摘要：我们研究了最小可能常数

和

Hardy不等式的行为

$\ displaystyle \ sum _ {k = 1} ^ {n} \ Big（\ frac {1} {k} \ sum _ {j = 1} ^ {k} a_j \ Big）^ 2 \ leq d_n \，\ sum _ {k = 1} ^ {n} a_k ^ 2，\ qquad（a_1，\ ldots，a_n）\ in \ mathbb {R} ^ n$

和

$\ displaystyle \ int _ {0} ^ {\ infty} \ Bigg（\ frac {1} {x} \ int _ {0} ^ {x} f（t）\，dt \ Bigg）^ 2 dx \ leq c_n \ int _ {0} ^ {\ infty} f ^ 2（x）\，dx，\ qquad f \ in \ mathcal {H} _n，$

对于有限维空间和，其中的度数的实值代数多项式的集合不超过。常数和被确定为用所谓的连续对偶Hahn多项式的最小零点以及形式为和的两侧估计来表示 $\ mathbb {R} ^ n$ $\ mathcal {H} _n \ colonequals \ {f \，：\，\ int _0 ^ xf（t）dt = e ^ {-x / 2} \，p（x）\：\ p \ in \ mathcal { P} _n，p（0）= 0 \}$ $\ mathcal {P} _n$

$\ displaystyle 4- \ frac {c} {\ ln n} <d_n，c_n <4- \ frac {c} {\ ln ^ 2 n} \ ,, \ qquad c> 0 \，$

被建立。

更新日期：2021-04-22

点击分享查看原文

点击收藏

公开下载

阅读更多本刊最新论文本刊介绍/投稿指南11