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Hardy’s inequalities in finite dimensional Hilbert spaces


Authors: Dimitar K. Dimitrov, Ivan Gadjev, Geno Nikolov and Rumen Uluchev
Journal: Proc. Amer. Math. Soc. 149 (2021), 2515-2529
MSC (2020): Primary 26D10, 26D15; Secondary 33C45, 15A42
DOI: https://doi.org/10.1090/proc/15467
Published electronically: March 26, 2021
MathSciNet review: 4246802
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Abstract: We study the behaviour of the smallest possible constants $d_n$ and $c_n$ in Hardy’s inequalities \begin{equation*} \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 \end{equation*} and \begin{equation*} \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, \end{equation*} 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 \begin{equation*} 4-\frac {c}{\ln n}< d_n, c_n<4-\frac {c}{\ln ^2 n} ,\qquad c>0 \end{equation*} are established.


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Additional Information

Dimitar K. Dimitrov
Affiliation: Departamento de Matemática, IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP, Brazil
MR Author ID: 308699
Email: d_k_dimitrov@yahoo.com

Ivan Gadjev
Affiliation: Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria
MR Author ID: 1090555
ORCID: 0000-0002-4444-9921
Email: gadjev@fmi.uni-sofia.bg

Geno Nikolov
Affiliation: Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria
MR Author ID: 131505
ORCID: 0000-0001-5608-2488
Email: geno@fmi.uni-sofia.bg

Rumen Uluchev
Affiliation: Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria
MR Author ID: 175915
ORCID: 0000-0002-9122-7088
Email: rumenu@fmi.uni-sofia.bg

Received by editor(s): July 20, 2020
Received by editor(s) in revised form: September 14, 2020, and September 14, 2020
Published electronically: March 26, 2021
Additional Notes: Research supported by the Brazilian Science Foundations FAPESP under Grants 2016/09906-0 and 2016/10357-1 and CNPq under Grant 306136/2017-1 and the Bulgarian National Research Fund through Contract DN 02/14.
Article copyright: © Copyright 2021 American Mathematical Society