Hardy’s inequalities in finite dimensional Hilbert spaces Article Swipe
Dimitar K. Dimitrov
,
Ivan Gadjev
,
Geno Nikolov
,
Rumen Uluchev
·
YOU?
·
· 2021
· Open Access
·
· DOI: https://doi.org/10.1090/proc/15467
YOU?
·
· 2021
· Open Access
·
· DOI: https://doi.org/10.1090/proc/15467
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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- en
- Landing Page
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- 20
- OpenAlex ID
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Hardy’s inequalities in finite dimensional Hilbert spacesWork title
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preprintOpenAlex work type
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enPrimary language
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2021Year of publication
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2021-01-20Full publication date if available
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Dimitar K. Dimitrov, Ivan Gadjev, Geno Nikolov, Rumen UluchevList of authors in order
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https://doi.org/10.1090/proc/15467Publisher landing page
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greenOpen access status per OpenAlex
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https://arxiv.org/pdf/2007.10073Direct OA link when available
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0Total citation count in OpenAlex
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| primary_location.landing_page_url | https://doi.org/10.1090/proc/15467 |
| publication_date | 2021-01-20 |
| publication_year | 2021 |
| referenced_works | https://openalex.org/W1493135876, https://openalex.org/W2980671223, https://openalex.org/W2066213190 |
| referenced_works_count | 3 |
| abstract_inverted_index.\ | 31, 32 |
| abstract_inverted_index.$$ | 15, 22, 24, 35, 92, 99 |
| abstract_inverted_index.:\ | 48 |
| abstract_inverted_index.We | 0 |
| abstract_inverted_index.as | 73 |
| abstract_inverted_index.dt | 46 |
| abstract_inverted_index.dx | 26 |
| abstract_inverted_index.in | 12 |
| abstract_inverted_index.is | 54 |
| abstract_inverted_index.of | 4, 57, 61, 77, 89 |
| abstract_inverted_index.The | 66 |
| abstract_inverted_index.\in | 20 |
| abstract_inverted_index.and | 10, 23, 42, 69, 81, 87 |
| abstract_inverted_index.are | 71, 100 |
| abstract_inverted_index.c_n | 28 |
| abstract_inverted_index.for | 36, 85 |
| abstract_inverted_index.not | 63 |
| abstract_inverted_index.set | 56 |
| abstract_inverted_index.the | 2, 5, 37, 55, 74, 82, 90 |
| abstract_inverted_index.$n$. | 65 |
| abstract_inverted_index.\leq | 27 |
| abstract_inverted_index.d_n, | 95 |
| abstract_inverted_index.f(t) | 45 |
| abstract_inverted_index.f\in | 33 |
| abstract_inverted_index.form | 91 |
| abstract_inverted_index.p\in | 49 |
| abstract_inverted_index.$c_n$ | 11, 70, 88 |
| abstract_inverted_index.$d_n$ | 9, 68, 86 |
| abstract_inverted_index.study | 1 |
| abstract_inverted_index.where | 52 |
| abstract_inverted_index.Jacobi | 79 |
| abstract_inverted_index.\qquad | 18 |
| abstract_inverted_index.degree | 62 |
| abstract_inverted_index.finite | 38 |
| abstract_inverted_index.n}< | 94 |
| abstract_inverted_index.spaces | 40 |
| abstract_inverted_index.Hardy's | 13 |
| abstract_inverted_index.certain | 78 |
| abstract_inverted_index.\int_0^x | 44 |
| abstract_inverted_index.c>0\, | 98 |
| abstract_inverted_index.matrices | 80 |
| abstract_inverted_index.possible | 7 |
| abstract_inverted_index.smallest | 6, 75 |
| abstract_inverted_index.algebraic | 59 |
| abstract_inverted_index.behaviour | 3 |
| abstract_inverted_index.constants | 8, 67 |
| abstract_inverted_index.estimates | 84 |
| abstract_inverted_index.exceeding | 64 |
| abstract_inverted_index.two-sided | 83 |
| abstract_inverted_index.identified | 72 |
| abstract_inverted_index.p(0)=0\}$, | 51 |
| abstract_inverted_index.dimensional | 39 |
| abstract_inverted_index.eigenvalues | 76 |
| abstract_inverted_index.f^2(x)\,dx, | 30 |
| abstract_inverted_index.n}\,,\qquad | 97 |
| abstract_inverted_index.polynomials | 60 |
| abstract_inverted_index.real-valued | 58 |
| abstract_inverted_index.\mathbb{R}^n | 21 |
| abstract_inverted_index.established. | 101 |
| abstract_inverted_index.inequalities | 14 |
| abstract_inverted_index.$\mathbb{R}^n$ | 41 |
| abstract_inverted_index.4-\frac{c}{\ln | 93 |
| abstract_inverted_index.\mathcal{H}_n, | 34 |
| abstract_inverted_index.\mathcal{P}_n, | 50 |
| abstract_inverted_index.$\mathcal{P}_n$ | 53 |
| abstract_inverted_index.(a_1,\ldots,a_n) | 19 |
| abstract_inverted_index.=e^{-x/2}\,p(x)\ | 47 |
| abstract_inverted_index.\int_{0}^{\infty} | 29 |
| abstract_inverted_index.c_n<4-\frac{c}{\ln^2 | 96 |
| abstract_inverted_index.$\mathcal{H}_n:=\{f\,:\, | 43 |
| abstract_inverted_index.d_n\,\sum_{k=1}^{n}a_k^2, | 17 |
| abstract_inverted_index.\sum_{k=1}^{n}\Big(\frac{1}{k}\sum_{j=1}^{k}a_j\Big)^2\leq | 16 |
| abstract_inverted_index.\int_{0}^{\infty}\Bigg(\frac{1}{x}\int\limits_{0}^{x}f(t)\,dt\Bigg)^2 | 25 |
| cited_by_percentile_year | |
| countries_distinct_count | 2 |
| institutions_distinct_count | 4 |
| citation_normalized_percentile.value | 0.02136523 |
| citation_normalized_percentile.is_in_top_1_percent | False |
| citation_normalized_percentile.is_in_top_10_percent | False |