Shape-preserving kolmogorov widths of classes of s-monotone integrable functions
Let $s ∈ ℕ$ and $Δ^s_{+}$ be a set of functions $x$ which are defined on a finite interval $I$ and are such that, for all collections of $s + 1$ pairwise different points $t_0,..., t_s \in I$, the corresponding divided differences $[x; t_0,..., t_s ]$ of order $s$ are nonnegative. Let $\Delta^s_{+}...
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| Datum: | 2004 |
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| Format: | Artikel |
| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2004
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3808 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509942369222656 |
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| author | Konovalov, V. N. Коновалов, В. Н. Коновалов, В. Н. |
| author_facet | Konovalov, V. N. Коновалов, В. Н. Коновалов, В. Н. |
| author_sort | Konovalov, V. N. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:11:04Z |
| description | Let $s ∈ ℕ$ and $Δ^s_{+}$ be a set of functions $x$ which are defined on a finite interval $I$ and are such that, for all collections of $s + 1$ pairwise different points $t_0,..., t_s \in I$, the corresponding divided differences
$[x; t_0,..., t_s ]$ of order $s$ are nonnegative. Let $\Delta^s_{+} B_p := \Delta^s_{+} \bigcap B_p,\; 1 \leq p \leq \infty$, where $B_p$ is the unit
ball of the space $L_p$, and let $\Delta^s_{+} L_p := \Delta^s_{+} \bigcap L_p,\; 1 \leq q \leq \infty$. For every $s \geq 3$ and $1 \leq q \leq p \leq \infty$, exact orders of the shape-preserving Kolmogorov widths
$$d_n (\Delta^s_{+} B_p, \Delta^s_{+} L_p )_{L_p}^{\text{kol}} := \inf_{M^n \in \mathcal{M}^n} \sup_{x \in \Delta^s_{+} B_p} \inf_{y \in M^n \bigcap \Delta^s_{+} L_p} ||x - y||_{L_p},$$
are obtained, where $\mathcal{M}^n$ is the set of all affine linear manifolds $M^n$ in $L_q$ such that $\dim М^n \leq n$ and $M^n \bigcap \Delta^s_{+} L_p \neq \emptyset$. |
| first_indexed | 2026-03-24T02:49:07Z |
| format | Article |
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| id | umjimathkievua-article-3808 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:49:07Z |
| publishDate | 2004 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c4/8876c561d197f2b82ca239ecfdc883c4.pdf |
| spelling | umjimathkievua-article-38082020-03-18T20:11:04Z Shape-preserving kolmogorov widths of classes of s-monotone integrable functions Формосохраняющие поперечники типа Колмогорова классов s-монотонных интегрируемых функций Konovalov, V. N. Коновалов, В. Н. Коновалов, В. Н. Let $s ∈ ℕ$ and $Δ^s_{+}$ be a set of functions $x$ which are defined on a finite interval $I$ and are such that, for all collections of $s + 1$ pairwise different points $t_0,..., t_s \in I$, the corresponding divided differences $[x; t_0,..., t_s ]$ of order $s$ are nonnegative. Let $\Delta^s_{+} B_p := \Delta^s_{+} \bigcap B_p,\; 1 \leq p \leq \infty$, where $B_p$ is the unit ball of the space $L_p$, and let $\Delta^s_{+} L_p := \Delta^s_{+} \bigcap L_p,\; 1 \leq q \leq \infty$. For every $s \geq 3$ and $1 \leq q \leq p \leq \infty$, exact orders of the shape-preserving Kolmogorov widths $$d_n (\Delta^s_{+} B_p, \Delta^s_{+} L_p )_{L_p}^{\text{kol}} := \inf_{M^n \in \mathcal{M}^n} \sup_{x \in \Delta^s_{+} B_p} \inf_{y \in M^n \bigcap \Delta^s_{+} L_p} ||x - y||_{L_p},$$ are obtained, where $\mathcal{M}^n$ is the set of all affine linear manifolds $M^n$ in $L_q$ such that $\dim М^n \leq n$ and $M^n \bigcap \Delta^s_{+} L_p \neq \emptyset$. Нехай $s ∈ ℕ$ і $Δ^s_{+}$ — множина функцій $x$, визначених на скінченному інтервалі $I$ і таких, що для всіх наборів з $s + 1$ попарно різних точок $t_0,..., t_s \in I$ відповідні поділені різниці $[x; t_0,..., t_s ]$ порядку $s$ є невід'ємними. Нехай $\Delta^s_{+} B_p := \Delta^s_{+} \bigcap B_p,\; 1 \leq p \leq \infty$, де $B_p$ — одинична куля простору $L_p$, і $\Delta^s_{+} L_p := \Delta^s_{+} \bigcap L_p,\; 1 \leq q \leq \infty$. Для всіх $s \geq 3$ i $1 \leq q \leq p \leq \infty$ знайдено точні порядки формозберігаючих поперечників типу Колмогорова $$d_n (\Delta^s_{+} B_p, \Delta^s_{+} L_p )_{L_p}^{\text{kol}} := \inf_{M^n \in \mathcal{M}^n} \sup_{x \in \Delta^s_{+} B_p} \inf_{y \in M^n \bigcap \Delta^s_{+} L_p} ||x - y||_{L_p},$$ де $\mathcal{M}^n$ — сукупність усіх афінно лінійних многовидів $M^n$ з $L_q$ таких, що $\dim М^n \leq n$ і $M^n \bigcap \Delta^s_{+} L_p \neq \emptyset$. Institute of Mathematics, NAS of Ukraine 2004-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3808 Ukrains’kyi Matematychnyi Zhurnal; Vol. 56 No. 7 (2004); 901–926 Український математичний журнал; Том 56 № 7 (2004); 901–926 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3808/4336 https://umj.imath.kiev.ua/index.php/umj/article/view/3808/4337 Copyright (c) 2004 Konovalov V. N. |
| spellingShingle | Konovalov, V. N. Коновалов, В. Н. Коновалов, В. Н. Shape-preserving kolmogorov widths of classes of s-monotone integrable functions |
| title | Shape-preserving kolmogorov widths of classes of s-monotone integrable functions |
| title_alt | Формосохраняющие поперечники типа Колмогорова классов
s-монотонных интегрируемых функций |
| title_full | Shape-preserving kolmogorov widths of classes of s-monotone integrable functions |
| title_fullStr | Shape-preserving kolmogorov widths of classes of s-monotone integrable functions |
| title_full_unstemmed | Shape-preserving kolmogorov widths of classes of s-monotone integrable functions |
| title_short | Shape-preserving kolmogorov widths of classes of s-monotone integrable functions |
| title_sort | shape-preserving kolmogorov widths of classes of s-monotone integrable functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3808 |
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