Coconvex Approximation of Functions with More than One Inflection Point
Assume that f ∈ C[−1, 1] belongs to C[−1, 1] and changes its convexity at s > 1 different points y i, \(\overline {1,s} \) , from (−1, 1). For n ∈ N, n ≥ 2, we construct an algebraic polynomial P n of order ≤ n that changes its convexity at the same points y i as f and is such that $$|f(...
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| Date: | 2004 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2004
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3759 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509892083712000 |
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| author | Dzyubenko, H. A. Zalizko, V. D. Дзюбенко, Г. А. Залізко, В. Д. |
| author_facet | Dzyubenko, H. A. Zalizko, V. D. Дзюбенко, Г. А. Залізко, В. Д. |
| author_sort | Dzyubenko, H. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:07:47Z |
| description | Assume that f ∈ C[−1, 1] belongs to C[−1, 1] and changes its convexity at s > 1 different points y i, \(\overline {1,s} \) , from (−1, 1). For n ∈ N, n ≥ 2, we construct an algebraic polynomial P n of order ≤ n that changes its convexity at the same points y i as f and is such that $$|f(x) - P_n (x)|\;\; \leqslant \;\;C(Y)\omega _3 \left( {f;\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right),\;\;\;\;\;x\;\; \in \;\;[ - 1,\;1],$$ where ω3(f; t) is the third modulus of continuity of the function f and C(Y) is a constant that depends only on \(\mathop {\min }\limits_{i = 0,...,s} \left| {y_i - y_{i + 1} } \right|,\;\;y_0 = 1,\;\;y_{s + 1} = - 1\) , y 0 = 1, y s + 1 = −1. |
| first_indexed | 2026-03-24T02:48:19Z |
| format | Article |
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| id | umjimathkievua-article-3759 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:48:19Z |
| publishDate | 2004 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b9/a2414b7624ad707ba1f6a01cf8a035b9.pdf |
| spelling | umjimathkievua-article-37592020-03-18T20:07:47Z Coconvex Approximation of Functions with More than One Inflection Point Коопукле наближення функцій, які мають більше однієї точки перегину Dzyubenko, H. A. Zalizko, V. D. Дзюбенко, Г. А. Залізко, В. Д. Assume that f ∈ C[−1, 1] belongs to C[−1, 1] and changes its convexity at s > 1 different points y i, \(\overline {1,s} \) , from (−1, 1). For n ∈ N, n ≥ 2, we construct an algebraic polynomial P n of order ≤ n that changes its convexity at the same points y i as f and is such that $$|f(x) - P_n (x)|\;\; \leqslant \;\;C(Y)\omega _3 \left( {f;\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right),\;\;\;\;\;x\;\; \in \;\;[ - 1,\;1],$$ where ω3(f; t) is the third modulus of continuity of the function f and C(Y) is a constant that depends only on \(\mathop {\min }\limits_{i = 0,...,s} \left| {y_i - y_{i + 1} } \right|,\;\;y_0 = 1,\;\;y_{s + 1} = - 1\) , y 0 = 1, y s + 1 = −1. Нехай $f \in C[−1, 1]$, змінює свою опуклість в $s > 1$ різних точках $y_i = 1, \;i = \overline {1,s}$, з $(-1,1)$. Для $n ∈ N, n ≥ 2$, побудовано алгебраїчний многочлен $P_n$ степеня $≤ n$, який змінює опуклість в тих самих точках $y_i$, щой $f$, і такий, що $$|f(x) - P_n (x)|\;\; \leqslant \;\;C(Y)\omega _3 \left( {f;\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right),\;\;\;\;\;x\;\; \in \;\;[ - 1,\;1],$$ де $ω_3(f; t)$ —третій модуль неперервності функції $f, C(Y)$ — стала, що залежить тільки від $\mathop {\min }\limits_{i = 0,...,s} \left| {y_i - y_{i + 1} } \right|,\;\;y_0 = 1,\;\;y_{s + 1} = - 1$ Institute of Mathematics, NAS of Ukraine 2004-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3759 Ukrains’kyi Matematychnyi Zhurnal; Vol. 56 No. 3 (2004); 352-365 Український математичний журнал; Том 56 № 3 (2004); 352-365 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3759/4240 https://umj.imath.kiev.ua/index.php/umj/article/view/3759/4241 Copyright (c) 2004 Dzyubenko H. A.; Zalizko V. D. |
| spellingShingle | Dzyubenko, H. A. Zalizko, V. D. Дзюбенко, Г. А. Залізко, В. Д. Coconvex Approximation of Functions with More than One Inflection Point |
| title | Coconvex Approximation of Functions with More than One Inflection Point |
| title_alt | Коопукле наближення функцій, які мають більше однієї точки перегину |
| title_full | Coconvex Approximation of Functions with More than One Inflection Point |
| title_fullStr | Coconvex Approximation of Functions with More than One Inflection Point |
| title_full_unstemmed | Coconvex Approximation of Functions with More than One Inflection Point |
| title_short | Coconvex Approximation of Functions with More than One Inflection Point |
| title_sort | coconvex approximation of functions with more than one inflection point |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3759 |
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