Inequalities of Different Metrics for Differentiable Periodic Functions, Polynomials, and Splines
We obtain new inequalities of different metrics for differentiable periodic functions. In particular, for p, q ∈ (0, ∞], q > p, and s ∈ [p, q], we prove that functions \(x \in L_\infty ^{{\text{ }}r}\) satisfy the unimprovable inequality $$|| (x-c_{s+1} (x))_{\pm} ||_q \leqslant \frac{|...
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| Datum: | 2001 |
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| Sprache: | Ukrainisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2001
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/4283 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510414579695616 |
|---|---|
| author | Kofanov, V. A. Кофанов, В. О. |
| author_facet | Kofanov, V. A. Кофанов, В. О. |
| author_sort | Kofanov, V. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:25:53Z |
| description | We obtain new inequalities of different metrics for differentiable periodic functions. In particular, for p, q ∈ (0, ∞], q > p, and s ∈ [p, q], we prove that functions \(x \in L_\infty ^{{\text{ }}r}\) satisfy the unimprovable inequality $$|| (x-c_{s+1} (x))_{\pm} ||_q \leqslant \frac{|| (\phi_r)_{\pm} ||_q}{|| \phi_r ||_p^{\frac{r+1/q}{r+1/p}}} || x-c_{s+1}(x) ||_p^{\frac{r+1/q}{r+1/P}} || x^(r) ||_\infty^{\frac{1/p-1/q}{r+1/p}},$$ where ϕ r is the perfect Euler spline of order r and c s + 1(x) is the constant of the best approximation of the function x in the space L s + 1. By using the inequality indicated, we obtain a new Bernstein-type inequality for trigonometric polynomials τ whose degree does not exceed n, namely, $$|| (\tau^(k))_{\pm} ||_q \leqslant n^{k+1/p-1/q} \frac{|| (\cos(\cdot))_{\pm} ||_q}{|| \cos(\cdot) ||_p} || \tau ||_p,$$ where k ∈ N, p ∈ (0, 1], and q ∈ [1, ∞]. We also consider other applications of the inequality indicated. |
| first_indexed | 2026-03-24T02:56:37Z |
| format | Article |
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| id | umjimathkievua-article-4283 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:56:37Z |
| publishDate | 2001 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/61/e01e0ff1a9032bf41ebf2996dfa52361.pdf |
| spelling | umjimathkievua-article-42832020-03-18T20:25:53Z Inequalities of Different Metrics for Differentiable Periodic Functions, Polynomials, and Splines Неравенства разных метрик для дифференцируемых периодических функций, полиномов и сплайнов Kofanov, V. A. Кофанов, В. О. We obtain new inequalities of different metrics for differentiable periodic functions. In particular, for p, q ∈ (0, ∞], q > p, and s ∈ [p, q], we prove that functions \(x \in L_\infty ^{{\text{ }}r}\) satisfy the unimprovable inequality $$|| (x-c_{s+1} (x))_{\pm} ||_q \leqslant \frac{|| (\phi_r)_{\pm} ||_q}{|| \phi_r ||_p^{\frac{r+1/q}{r+1/p}}} || x-c_{s+1}(x) ||_p^{\frac{r+1/q}{r+1/P}} || x^(r) ||_\infty^{\frac{1/p-1/q}{r+1/p}},$$ where ϕ r is the perfect Euler spline of order r and c s + 1(x) is the constant of the best approximation of the function x in the space L s + 1. By using the inequality indicated, we obtain a new Bernstein-type inequality for trigonometric polynomials τ whose degree does not exceed n, namely, $$|| (\tau^(k))_{\pm} ||_q \leqslant n^{k+1/p-1/q} \frac{|| (\cos(\cdot))_{\pm} ||_q}{|| \cos(\cdot) ||_p} || \tau ||_p,$$ where k ∈ N, p ∈ (0, 1], and q ∈ [1, ∞]. We also consider other applications of the inequality indicated. Одержано нові нерівності різних метрик для диференційовпих періодичних функцій, зокрема, доведено, що при $p, q ∈ (0, ∞], q > p$ і $s ∈ [p, q]$, для функцій $x \in L_\infty ^{{\text{ }}r}$ справедлива непокращувана нерівність $$|| (x-c_{s+1} (x))_{\pm} ||_q \leqslant \frac{|| (\phi_r)_{\pm} ||_q}{|| \phi_r ||_p^{\frac{r+1/q}{r+1/p}}} || x-c_{s+1}(x) ||_p^{\frac{r+1/q}{r+1/P}} || x^(r) ||_\infty^{\frac{1/p-1/q}{r+1/p}},$$ де $ϕ_r$ — ідеальний сплайн Ейлера порядку $r$, $c_{s + 1}(x)$— константа найкращого наближення функції $x$ у просторі $L_{s + 1}$. За допомогою наведеної нерівності одержано нову нерівність типу Бериштейна для тригонометричних поліномів $τ$ порядку, що не перевищує $n$: $$|| (\tau^(k))_{\pm} ||_q \leqslant n^{k+1/p-1/q} \frac{|| (\cos(\cdot))_{\pm} ||_q}{|| \cos(\cdot) ||_p} || \tau ||_p,$$ де $k ∈ N, p ∈ (0, 1], a q ∈ [1, ∞]$. Розглянуто інші застосування цієї нерівності. Institute of Mathematics, NAS of Ukraine 2001-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4283 Ukrains’kyi Matematychnyi Zhurnal; Vol. 53 No. 5 (2001); 597-609 Український математичний журнал; Том 53 № 5 (2001); 597-609 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4283/5270 https://umj.imath.kiev.ua/index.php/umj/article/view/4283/5271 Copyright (c) 2001 Kofanov V. A. |
| spellingShingle | Kofanov, V. A. Кофанов, В. О. Inequalities of Different Metrics for Differentiable Periodic Functions, Polynomials, and Splines |
| title | Inequalities of Different Metrics for Differentiable Periodic Functions, Polynomials, and Splines |
| title_alt | Неравенства разных метрик для дифференцируемых периодических
функций, полиномов и сплайнов |
| title_full | Inequalities of Different Metrics for Differentiable Periodic Functions, Polynomials, and Splines |
| title_fullStr | Inequalities of Different Metrics for Differentiable Periodic Functions, Polynomials, and Splines |
| title_full_unstemmed | Inequalities of Different Metrics for Differentiable Periodic Functions, Polynomials, and Splines |
| title_short | Inequalities of Different Metrics for Differentiable Periodic Functions, Polynomials, and Splines |
| title_sort | inequalities of different metrics for differentiable periodic functions, polynomials, and splines |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4283 |
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