On One Counterexample in Convex Approximation
We prove the existence of a function fcontinuous and convex on [−1, 1] and such that, for any sequence {p n} n= 1 ∞of algebraic polynomials p nof degree ≤ nconvex on [−1, 1], the following relation is true: \(\begin{array}{*{20}c} {\lim \sup } \\ {n \to \infty } \\ \end{array} \begin{array}{*{20}c...
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| Date: | 2000 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2000
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/4576 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510720816316416 |
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| author | Yushchenko, L. P. Ющенко, Л. П. |
| author_facet | Yushchenko, L. P. Ющенко, Л. П. |
| author_sort | Yushchenko, L. P. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:31:34Z |
| description | We prove the existence of a function fcontinuous and convex on [−1, 1] and such that, for any sequence {p n} n= 1 ∞of algebraic polynomials p nof degree ≤ nconvex on [−1, 1], the following relation is true: \(\begin{array}{*{20}c} {\lim \sup } \\ {n \to \infty } \\ \end{array} \begin{array}{*{20}c} {\max } \\{x \in [ - 1,1]} \\ \end{array} \frac{{|f(x) - p_n (x)|}}{{\omega _4 (\rho _n (x),f)}} = \infty \) , where ω4(t, f) is the fourth modulus of continuity of the function fand \(\rho _n \left( x \right): = \frac{1}{{n^2 }} + \frac{1}{n}\sqrt {1 - x^2 } \) . We generalize this result to q-convex functions. |
| first_indexed | 2026-03-24T03:01:29Z |
| format | Article |
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| id | umjimathkievua-article-4576 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:01:29Z |
| publishDate | 2000 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/88/4cdf61d2856dd8bdb7e90b7dce2eb788.pdf |
| spelling | umjimathkievua-article-45762020-03-18T20:31:34Z On One Counterexample in Convex Approximation Один контрприклад у опуклому наближенні Yushchenko, L. P. Ющенко, Л. П. We prove the existence of a function fcontinuous and convex on [−1, 1] and such that, for any sequence {p n} n= 1 ∞of algebraic polynomials p nof degree ≤ nconvex on [−1, 1], the following relation is true: \(\begin{array}{*{20}c} {\lim \sup } \\ {n \to \infty } \\ \end{array} \begin{array}{*{20}c} {\max } \\{x \in [ - 1,1]} \\ \end{array} \frac{{|f(x) - p_n (x)|}}{{\omega _4 (\rho _n (x),f)}} = \infty \) , where ω4(t, f) is the fourth modulus of continuity of the function fand \(\rho _n \left( x \right): = \frac{1}{{n^2 }} + \frac{1}{n}\sqrt {1 - x^2 } \) . We generalize this result to q-convex functions. Доведено існування неперервної і опуклої на $[-1, 1]$ функції $f$ такої, що для будь-якої послідовності ${p_n}_{n = 1}^{∞}$ опуклих на $[-1,1]$ алгебраїчних многочленів ${p_n}$ степеня Institute of Mathematics, NAS of Ukraine 2000-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4576 Ukrains’kyi Matematychnyi Zhurnal; Vol. 52 No. 12 (2000); 1715-1721 Український математичний журнал; Том 52 № 12 (2000); 1715-1721 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4576/5856 https://umj.imath.kiev.ua/index.php/umj/article/view/4576/5857 Copyright (c) 2000 Yushchenko L. P. |
| spellingShingle | Yushchenko, L. P. Ющенко, Л. П. On One Counterexample in Convex Approximation |
| title | On One Counterexample in Convex Approximation |
| title_alt | Один контрприклад у опуклому наближенні |
| title_full | On One Counterexample in Convex Approximation |
| title_fullStr | On One Counterexample in Convex Approximation |
| title_full_unstemmed | On One Counterexample in Convex Approximation |
| title_short | On One Counterexample in Convex Approximation |
| title_sort | on one counterexample in convex approximation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4576 |
| work_keys_str_mv | AT yushchenkolp ononecounterexampleinconvexapproximation AT ûŝenkolp ononecounterexampleinconvexapproximation AT yushchenkolp odinkontrprikladuopuklomunabliženní AT ûŝenkolp odinkontrprikladuopuklomunabliženní |