Copositive pointwise approximation
We prove that if a functionf ∈C (1) (I),I: = [−1, 1], changes its signs times (s ∈ ℕ) within the intervalI, then, for everyn > C, whereC is a constant which depends only on the set of points at which the function changes its sign, andk ∈ ℕ, there exists an algebraic polynomialP n =P n (x)...
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| Дата: | 1996 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1996
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/5330 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860511553330085888 |
|---|---|
| author | Dzyubenko, H. A. Дзюбенко, Г. А. Дзюбенко, Г. А. |
| author_facet | Dzyubenko, H. A. Дзюбенко, Г. А. Дзюбенко, Г. А. |
| author_sort | Dzyubenko, H. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:30:31Z |
| description | We prove that if a functionf ∈C (1) (I),I: = [−1, 1], changes its signs times (s ∈ ℕ) within the intervalI, then, for everyn > C, whereC is a constant which depends only on the set of points at which the function changes its sign, andk ∈ ℕ, there exists an algebraic polynomialP n =P n (x) of degree ≤n which locally inherits the sign off(x) and satisfies the inequality $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c\left( {s,k} \right)\left( {\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right)\omega _k \left( {f'; \frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in I$$ , where ω k (f′;t) is thekth modulus of continuity of the functionf’. It is also shown that iff ∈C (I) andf(x) ≥ 0,x ∈I then, for anyn ≥k − 1, there exists a polynomialP n =P n (x) of degree ≤n such thatP n (x) ≥ 0,x ∈I, and |f(x) −P n (x)| ≤c(k)ω k (f;n −2 +n −1 √1 −x 2),x ∈I. |
| first_indexed | 2026-03-24T03:14:43Z |
| format | Article |
| fulltext |
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| id | umjimathkievua-article-5330 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:14:43Z |
| publishDate | 1996 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/5f/df6f0562702ff31c754b595cf8a6d95f.pdf |
| spelling | umjimathkievua-article-53302020-03-18T21:30:31Z Copositive pointwise approximation Коположительное поточечное приближение Dzyubenko, H. A. Дзюбенко, Г. А. Дзюбенко, Г. А. We prove that if a functionf ∈C (1) (I),I: = [−1, 1], changes its signs times (s ∈ ℕ) within the intervalI, then, for everyn > C, whereC is a constant which depends only on the set of points at which the function changes its sign, andk ∈ ℕ, there exists an algebraic polynomialP n =P n (x) of degree ≤n which locally inherits the sign off(x) and satisfies the inequality $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c\left( {s,k} \right)\left( {\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right)\omega _k \left( {f'; \frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in I$$ , where ω k (f′;t) is thekth modulus of continuity of the functionf’. It is also shown that iff ∈C (I) andf(x) ≥ 0,x ∈I then, for anyn ≥k − 1, there exists a polynomialP n =P n (x) of degree ≤n such thatP n (x) ≥ 0,x ∈I, and |f(x) −P n (x)| ≤c(k)ω k (f;n −2 +n −1 √1 −x 2),x ∈I. Доведено, що коли функція $f ∈ C^{(1)}\; (I), I: = [−1, 1]$, змінює знак $s$ разів на $f\; (s ∈ ℕ)$, тоді для кожного $n > C$, де стала $С$ залежить тільки від множини точок зміни знаку функції і $k ∈ ℕ$, існує алгебраїчний многочлен $P_n =P_n (x) $ степеня $ ≤n $, який локально успадковує знак $(x)$ і $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c\left( {s,k} \right)\left( {\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right)\omega _k \left( {f'; \frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in I$$ де $ω k (f′;t)$ — $k$-й модуль неперервності функції $f$. Також показано, що коли $f ∈ C (I)$ і $f(x) ≥ 0,x ∈I$, тоді для кожного $n ≥ k − 1$ існує многочлен $P_n =P_n(x)$ степеня $≤n$ такий, що $P_n (x) ≥ 0,x ∈ I$, і $|f(x) −P n (x)| ≤c(k)ω k (f;n −2 +n −1 √1 −x 2),x ∈ I.$ Institute of Mathematics, NAS of Ukraine 1996-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/5330 Ukrains’kyi Matematychnyi Zhurnal; Vol. 48 No. 3 (1996); 326-334 Український математичний журнал; Том 48 № 3 (1996); 326-334 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/5330/7351 https://umj.imath.kiev.ua/index.php/umj/article/view/5330/7352 Copyright (c) 1996 Dzyubenko H. A. |
| spellingShingle | Dzyubenko, H. A. Дзюбенко, Г. А. Дзюбенко, Г. А. Copositive pointwise approximation |
| title | Copositive pointwise approximation |
| title_alt | Коположительное поточечное приближение |
| title_full | Copositive pointwise approximation |
| title_fullStr | Copositive pointwise approximation |
| title_full_unstemmed | Copositive pointwise approximation |
| title_short | Copositive pointwise approximation |
| title_sort | copositive pointwise approximation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/5330 |
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