Strengthening of the Kolmogorov Comparison Theorem and Kolmogorov Inequality and Their Applications
We obtain a strengthened version of the Kolmogorov comparison theorem. In particular, this enables us to obtain a strengthened Kolmogorov inequality for functions x ∈ L ∞ x (r), namely, $$\left\| {x^{(k)} } \right\|_{L_\infty (R)} \leqslant \frac{{\left\| {\phi _{r - k} } \right\|_\infty }}{{\l...
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| Datum: | 2002 |
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| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2002
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/4173 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510308581244928 |
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| author | Kofanov, V. A. Кофанов, В. А. Кофанов, В. А. |
| author_facet | Kofanov, V. A. Кофанов, В. А. Кофанов, В. А. |
| author_sort | Kofanov, V. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:23:51Z |
| description | We obtain a strengthened version of the Kolmogorov comparison theorem. In particular, this enables us to obtain a strengthened Kolmogorov inequality for functions x ∈ L ∞ x (r), namely, $$\left\| {x^{(k)} } \right\|_{L_\infty (R)} \leqslant \frac{{\left\| {\phi _{r - k} } \right\|_\infty }}{{\left\| {\phi _r } \right\|_\infty ^{1 - k/r} }}M(x)^{1 - k/r} \left\| {x^{(r)} } \right\|_{L_\infty (R)}^{k/r} ,$$ where $$M(x): = \frac{1}{2}\mathop {\sup }\limits_{\alpha ,\beta } \left\{ {\left| {x(\beta ) - x(\alpha )} \right|:x'(t) \ne 0{\text{ }}\forall t \in (\alpha ,\beta )} \right\}{\text{,}}$$ k, r ∈ N, k < r, and ϕ r is a perfect Euler spline of order r. Using this inequality, we strengthen the Bernstein inequality for trigonometric polynomials and the Tikhomirov inequality for splines. Some other applications of this inequality are also given. |
| first_indexed | 2026-03-24T02:54:56Z |
| format | Article |
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| id | umjimathkievua-article-4173 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | rus English |
| last_indexed | 2026-03-24T02:54:56Z |
| publishDate | 2002 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/43/2c91f3c47fe8a34e2982d71f994fab43.pdf |
| spelling | umjimathkievua-article-41732020-03-18T20:23:51Z Strengthening of the Kolmogorov Comparison Theorem and Kolmogorov Inequality and Their Applications Усиленно теореми сравнения и неравенства Колмогорова и их приложения Kofanov, V. A. Кофанов, В. А. Кофанов, В. А. We obtain a strengthened version of the Kolmogorov comparison theorem. In particular, this enables us to obtain a strengthened Kolmogorov inequality for functions x ∈ L ∞ x (r), namely, $$\left\| {x^{(k)} } \right\|_{L_\infty (R)} \leqslant \frac{{\left\| {\phi _{r - k} } \right\|_\infty }}{{\left\| {\phi _r } \right\|_\infty ^{1 - k/r} }}M(x)^{1 - k/r} \left\| {x^{(r)} } \right\|_{L_\infty (R)}^{k/r} ,$$ where $$M(x): = \frac{1}{2}\mathop {\sup }\limits_{\alpha ,\beta } \left\{ {\left| {x(\beta ) - x(\alpha )} \right|:x'(t) \ne 0{\text{ }}\forall t \in (\alpha ,\beta )} \right\}{\text{,}}$$ k, r ∈ N, k < r, and ϕ r is a perfect Euler spline of order r. Using this inequality, we strengthen the Bernstein inequality for trigonometric polynomials and the Tikhomirov inequality for splines. Some other applications of this inequality are also given. Одержано посилений варіант теореми порівпяння Колмогорова. Це дозволило, зокрема, отримати підсилепу нерівність Колмогорова $$\left\| {x^{(k)} } \right\|_{L_\infty (R)} \leqslant \frac{{\left\| {\phi _{r - k} } \right\|_\infty }}{{\left\| {\phi _r } \right\|_\infty ^{1 - k/r} }}M(x)^{1 - k/r} \left\| {x^{(r)} } \right\|_{L_\infty (R)}^{k/r} ,$$ для функцій $x ∈ L_{∞}^x(r)$, де $$M(x): = \frac{1}{2}\mathop {\sup }\limits_{\alpha ,\beta } \left\{ {\left| {x(\beta ) - x(\alpha )} \right|:x'(t) \ne 0{\text{ }}\forall t \in (\alpha ,\beta )} \right\}{\text{,}}$$ $k,\; r ∈ N,\; k < r, ϕ_r$ — ідеальний сплайн Ейлера порядку $r$ за допомогою якої підсиленї нерівністі Бернштейпа для тригонометричних поліномів і нерівність Тихомирова для сплайнів. Наведено інші застосування цієї нерівності. Institute of Mathematics, NAS of Ukraine 2002-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4173 Ukrains’kyi Matematychnyi Zhurnal; Vol. 54 No. 10 (2002); 1348-1356 Український математичний журнал; Том 54 № 10 (2002); 1348-1356 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4173/5055 https://umj.imath.kiev.ua/index.php/umj/article/view/4173/5056 Copyright (c) 2002 Kofanov V. A. |
| spellingShingle | Kofanov, V. A. Кофанов, В. А. Кофанов, В. А. Strengthening of the Kolmogorov Comparison Theorem and Kolmogorov Inequality and Their Applications |
| title | Strengthening of the Kolmogorov Comparison Theorem and Kolmogorov Inequality and Their Applications |
| title_alt | Усиленно теореми сравнения и неравенства Колмогорова и их приложения
|
| title_full | Strengthening of the Kolmogorov Comparison Theorem and Kolmogorov Inequality and Their Applications |
| title_fullStr | Strengthening of the Kolmogorov Comparison Theorem and Kolmogorov Inequality and Their Applications |
| title_full_unstemmed | Strengthening of the Kolmogorov Comparison Theorem and Kolmogorov Inequality and Their Applications |
| title_short | Strengthening of the Kolmogorov Comparison Theorem and Kolmogorov Inequality and Their Applications |
| title_sort | strengthening of the kolmogorov comparison theorem and kolmogorov inequality and their applications |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4173 |
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