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arXiv · 2007.10073

Hardy's inequalities in finite dimensional Hilbert spaces

Dimitar K. Dimitrov·Ivan Gadjev·Geno Nikolov·Rumen Uluchev
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Abstract

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.

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Dimitar K. Dimitrov, Ivan Gadjev, Geno Nikolov, Rumen Uluchev. 2020-07-20. Hardy's inequalities in finite dimensional Hilbert spaces. https://arxiv.org/abs/2007.10073

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