Coconvex Pointwise Approximation
Assume that a function f ∈ C[−1, 1] changes its convexity at a finite collection Y := {y 1, ... y s} of s points y i ∈ (−1, 1). For each n > N(Y), we construct an algebraic polynomial P n of degree ≤ n that is coconvex with f, i.e., it changes its convexity at the same points y i as f and $...
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| Date: | 2002 |
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| Main Authors: | , , , , , |
| Format: | Article |
| Language: | English |
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Institute of Mathematics, NAS of Ukraine
2002
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/4159 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510294023864320 |
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| author | Dzyubenko, H. A. Gilewicz, J. Shevchuk, I. A. Дзюбенко, Г. А. Гілевич, Я. Я. Шевчук, І. О. |
| author_facet | Dzyubenko, H. A. Gilewicz, J. Shevchuk, I. A. Дзюбенко, Г. А. Гілевич, Я. Я. Шевчук, І. О. |
| author_sort | Dzyubenko, H. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:23:25Z |
| description | Assume that a function f ∈ C[−1, 1] changes its convexity at a finite collection Y := {y 1, ... y s} of s points y i ∈ (−1, 1). For each n > N(Y), we construct an algebraic polynomial P n of degree ≤ n that is coconvex with f, i.e., it changes its convexity at the same points y i as f and $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c{\omega }_{2} \left( {f,\frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in \left[ { - 1,1} \right],$$ where c is an absolute constant, ω2(f, t) is the second modulus of smoothness of f, and if s = 1, then N(Y) = 1. We also give some counterexamples showing that this estimate cannot be extended to the case of higher smoothness. |
| first_indexed | 2026-03-24T02:54:42Z |
| format | Article |
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| id | umjimathkievua-article-4159 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:54:42Z |
| publishDate | 2002 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/88/01436052555be09e0ca1a9e1198b0b88.pdf |
| spelling | umjimathkievua-article-41592020-03-18T20:23:25Z Coconvex Pointwise Approximation Коопукле поточкове наближення Dzyubenko, H. A. Gilewicz, J. Shevchuk, I. A. Дзюбенко, Г. А. Гілевич, Я. Я. Шевчук, І. О. Assume that a function f ∈ C[−1, 1] changes its convexity at a finite collection Y := {y 1, ... y s} of s points y i ∈ (−1, 1). For each n > N(Y), we construct an algebraic polynomial P n of degree ≤ n that is coconvex with f, i.e., it changes its convexity at the same points y i as f and $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c{\omega }_{2} \left( {f,\frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in \left[ { - 1,1} \right],$$ where c is an absolute constant, ω2(f, t) is the second modulus of smoothness of f, and if s = 1, then N(Y) = 1. We also give some counterexamples showing that this estimate cannot be extended to the case of higher smoothness. Нехай функція $f ∈ C[−1, 1]$ змінює свою опуклість у скінченному наборі $Y := \{y_1, ... y_s\}$ точок $y_i ∈ (−1, 1)$. Для кожного $n > N(Y)$ будується алгебраїчний многочлен $P_n$ степеня $≤ n$, який є коопуклим з $f$, тобто змінює свою опуклість в тих самих точках $y_i$, що й $f$, а $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c{\omega }_{2} \left( {f,\frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in \left[ { - 1,1} \right],$$ де $c$ — абсолютна стала, $ω_2(f, t)$—другий модуль неперервності $f$, і якщо $s = 1$, то $N(Y) = 1$. Наведено також контрприклади, що показують, зокрема, неможливість поширення цієї оцінки для більшої гладкості. Institute of Mathematics, NAS of Ukraine 2002-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4159 Ukrains’kyi Matematychnyi Zhurnal; Vol. 54 No. 9 (2002); 1200-1212 Український математичний журнал; Том 54 № 9 (2002); 1200-1212 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/4159/5027 https://umj.imath.kiev.ua/index.php/umj/article/view/4159/5028 Copyright (c) 2002 Dzyubenko H. A.; Gilewicz J.; Shevchuk I. A. |
| spellingShingle | Dzyubenko, H. A. Gilewicz, J. Shevchuk, I. A. Дзюбенко, Г. А. Гілевич, Я. Я. Шевчук, І. О. Coconvex Pointwise Approximation |
| title | Coconvex Pointwise Approximation |
| title_alt | Коопукле поточкове наближення |
| title_full | Coconvex Pointwise Approximation |
| title_fullStr | Coconvex Pointwise Approximation |
| title_full_unstemmed | Coconvex Pointwise Approximation |
| title_short | Coconvex Pointwise Approximation |
| title_sort | coconvex pointwise approximation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4159 |
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