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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 $ d_n$ and $ c_n$ 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 $ n$. The constants $ d_n$ and $ c_n$ 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 $ d_n$ and $ c_n$ 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$ d_n $$ c_n $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 $

$ \ 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 $$ n $$ d_n $$ c_n $$ d_n $$ c_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
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