论文标题
Hardy在有限维度希尔伯特空间中的不平等现象
Hardy's inequalities in finite dimensional Hilbert spaces
论文作者
论文摘要
我们研究了Hardy的不等式中最小常数$ d_n $和$ c_n $的行为d_n \，\ sum_ {k = 1}^{n} a_k^2，\ qquad（a_1，\ ldots，a_n）\ in \ mathbb {r}^n $$和$$ \ int_ {0}^{\ infty} \ big（\ frac {1} {x} {x} \ int \ limits_ {0}^{x} f（t）\，dt \ bigG） f \ in \ Mathcal {h} _n，对于有限维空间$ \ mathbb {r}^n $和$ \ mathcal {h} _n：= \ {f \，：\，\ int_0^x f（t） \ Mathcal {p} _n，p（0）= 0 \} $，其中$ \ Mathcal {p} _n $是一组实现的代数多项式，不超过$ n $。常数$ d_n $和$ c_n $被确定为某些jacobi矩阵的最小特征值，以及$ d_n $和$ d_n $和$ c_n $的双面估计，$ $ 4- \ frac {c} {c} {c} {\ ln n} {\ ln n} n} \ ,, \ qquad c> 0 \，$$已建立。
We study the behaviour of the smallest possible constants $d_n$ and $c_n$ in Hardy's inequalities $$ \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 $$ and $$ \int_{0}^{\infty}\Bigg(\frac{1}{x}\int\limits_{0}^{x}f(t)\,dt\Bigg)^2 dx \leq c_n \int_{0}^{\infty} f^2(x)\,dx, \ \ f\in \mathcal{H}_n, $$ for the finite dimensional spaces $\mathbb{R}^n$ and $\mathcal{H}_n:=\{f\,:\, \int_0^x f(t) dt =e^{-x/2}\,p(x)\ :\ p\in \mathcal{P}_n, p(0)=0\}$, where $\mathcal{P}_n$ is the set of real-valued algebraic polynomials of degree not exceeding $n$. The constants $d_n$ and $c_n$ are identified as the smallest eigenvalues of certain Jacobi matrices and the two-sided estimates for $d_n$ and $c_n$ of the form $$ 4-\frac{c}{\ln n}< d_n, c_n<4-\frac{c}{\ln^2 n}\,,\qquad c>0\, $$ are established.
