On Kolmogorov-Type Inequalities Taking into Account the Number of Changes in the Sign of Derivatives
For 2π-periodic functions \(x \in L_\infty ^r \) and arbitrary q ∈ [1, ∞] and p ∈ (0, ∞], we obtain the new exact Kolmogorov-type inequality \(|| x^(k) ||_q \leqslant (\frac{v(x^(k))}{2})^{1/q} \frac{|| \phi_{r-k} ||_q}{||| \phi_r |||_p^\alpha} ||| x |||_p^\alpha || x^(r) ||_\infty^{1- \alpha}...
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| Date: | 2003 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
2003
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3919 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510052671029248 |
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| author | Kofanov, V. A. Кофанов, В. А. Кофанов, В. А. |
| author_facet | Kofanov, V. A. Кофанов, В. А. Кофанов, В. А. |
| author_sort | Kofanov, V. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:15:31Z |
| description | For 2π-periodic functions \(x \in L_\infty ^r \) and arbitrary q ∈ [1, ∞] and p ∈ (0, ∞], we obtain the new exact Kolmogorov-type inequality \(|| x^(k) ||_q \leqslant (\frac{v(x^(k))}{2})^{1/q} \frac{|| \phi_{r-k} ||_q}{||| \phi_r |||_p^\alpha} ||| x |||_p^\alpha || x^(r) ||_\infty^{1- \alpha}, k, r \in N, k < r,\) which takes into account the number of changes in the sign of the derivatives ν(x (k)) over the period. Here, α = (r − k + 1/q)/(r + 1/p), ϕ r is the Euler perfect spline of degree r, \(\begin{gathered} \left\| {\left| x \right|} \right\|_p : = {\text{sup}}_{a,b \in {\text{R}}} \{ E_0 (x)_{L_p [a,b]} :x'(t) \ne 0{\text{ }}\forall t \in (a,b)\} , \\ {\text{ }} \\ {\text{ }}E_0 (x)_{L_p [a,b]} : = {\text{ inf}}_{c \in {\text{R}}} \left\| {x - c} \right\|_{L_p [a,b]} , \\ \\ \left\| x \right\|_{L_p [a,b]} : = \left\{ {\int\limits_a^b {\left| {x(t)} \right|^p dt} } \right\}^{1/p} {\text{ for }}0 < p < \infty , \\ \end{gathered} \) and \(\left\| x \right\|_{L_p [a,b]} : = {\text{ sup vrai}}_{t \in \left[ {a,b} \right]} \left| {x(t)} \right|\) . The inequality indicated turns into the equality for functions of the form x(t) = aϕ r (nt + b), a, b ∈ R, n ∈ N. We also obtain an analog of this inequality in the case where k = 0 and q = ∞ and prove new exact Bernstein-type inequalities for trigonometric polynomials and splines. |
| first_indexed | 2026-03-24T02:50:52Z |
| format | Article |
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| id | umjimathkievua-article-3919 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:50:52Z |
| publishDate | 2003 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2a/a54887296e1414b490552ad69b571e2a.pdf |
| spelling | umjimathkievua-article-39192020-03-18T20:15:31Z On Kolmogorov-Type Inequalities Taking into Account the Number of Changes in the Sign of Derivatives О некоторых неравенствах типа Колмогорова, учитывающих число перемен знака производных Kofanov, V. A. Кофанов, В. А. Кофанов, В. А. For 2π-periodic functions \(x \in L_\infty ^r \) and arbitrary q ∈ [1, ∞] and p ∈ (0, ∞], we obtain the new exact Kolmogorov-type inequality \(|| x^(k) ||_q \leqslant (\frac{v(x^(k))}{2})^{1/q} \frac{|| \phi_{r-k} ||_q}{||| \phi_r |||_p^\alpha} ||| x |||_p^\alpha || x^(r) ||_\infty^{1- \alpha}, k, r \in N, k < r,\) which takes into account the number of changes in the sign of the derivatives ν(x (k)) over the period. Here, α = (r − k + 1/q)/(r + 1/p), ϕ r is the Euler perfect spline of degree r, \(\begin{gathered} \left\| {\left| x \right|} \right\|_p : = {\text{sup}}_{a,b \in {\text{R}}} \{ E_0 (x)_{L_p [a,b]} :x'(t) \ne 0{\text{ }}\forall t \in (a,b)\} , \\ {\text{ }} \\ {\text{ }}E_0 (x)_{L_p [a,b]} : = {\text{ inf}}_{c \in {\text{R}}} \left\| {x - c} \right\|_{L_p [a,b]} , \\ \\ \left\| x \right\|_{L_p [a,b]} : = \left\{ {\int\limits_a^b {\left| {x(t)} \right|^p dt} } \right\}^{1/p} {\text{ for }}0 < p < \infty , \\ \end{gathered} \) and \(\left\| x \right\|_{L_p [a,b]} : = {\text{ sup vrai}}_{t \in \left[ {a,b} \right]} \left| {x(t)} \right|\) . The inequality indicated turns into the equality for functions of the form x(t) = aϕ r (nt + b), a, b ∈ R, n ∈ N. We also obtain an analog of this inequality in the case where k = 0 and q = ∞ and prove new exact Bernstein-type inequalities for trigonometric polynomials and splines. Одержано нову точну нерівність типу Колмогорова $$|| x^(k) ||_q \leqslant (\frac{v(x^(k))}{2})^{1/q} \frac{|| \phi_{r-k} ||_q}{||| \phi_r |||_p^\alpha} ||| x |||_p^\alpha || x^(r) ||_\infty^{1- \alpha}, k, r \in N, k < r,$$ у якій враховано число змін знаку похідних $ν(x^{(k)})$ на періоді, для $2\pi$-періодичних функцій $x \in L_{\infty} ^r$ і для довільних $q ∈ [1, ∞]$, $p ∈ (0, ∞]$, де $α = (r − k + 1/q)/(r + 1/p)$, $ϕ_r$— ідеальний сплайн Ейлера порядку $r$, $$\begin{gathered} \left\| {\left| x \right|} \right\|_p : = {\text{sup}}_{a,b \in {\text{R}}} \{ E_0 (x)_{L_p [a,b]} :x'(t) \ne 0{\text{ }}\forall t \in (a,b)\} , \hfill \\ {\text{ }} \hfill \\ {\text{ }}E_0 (x)_{L_p [a,b]} : = {\text{ inf}}_{c \in {\text{R}}} \left\| {x - c} \right\|_{L_p [a,b]} , $$ $∥x∥_{L_p[a,b]}:= \sup \text{vrai}_{t∈[a,b]}|x(t)|. Ця нерівність перетворюється в рівність для функцій вигляду $x(t) = aϕ_r(nt + b), a, b ∈ R, n ∈ N$. Одержано також аналог даної нерівності у випадку $k = 0, q = ∞$ і доведено нові точні нерівності типу Бернштейна для тригонометричних поліномів та сплайнів. Institute of Mathematics, NAS of Ukraine 2003-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3919 Ukrains’kyi Matematychnyi Zhurnal; Vol. 55 No. 4 (2003); 456-469 Український математичний журнал; Том 55 № 4 (2003); 456-469 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3919/4552 https://umj.imath.kiev.ua/index.php/umj/article/view/3919/4553 Copyright (c) 2003 Kofanov V. A. |
| spellingShingle | Kofanov, V. A. Кофанов, В. А. Кофанов, В. А. On Kolmogorov-Type Inequalities Taking into Account the Number of Changes in the Sign of Derivatives |
| title | On Kolmogorov-Type Inequalities Taking into Account the Number of Changes in the Sign of Derivatives |
| title_alt | О некоторых неравенствах типа Колмогорова, учитывающих число перемен знака производных |
| title_full | On Kolmogorov-Type Inequalities Taking into Account the Number of Changes in the Sign of Derivatives |
| title_fullStr | On Kolmogorov-Type Inequalities Taking into Account the Number of Changes in the Sign of Derivatives |
| title_full_unstemmed | On Kolmogorov-Type Inequalities Taking into Account the Number of Changes in the Sign of Derivatives |
| title_short | On Kolmogorov-Type Inequalities Taking into Account the Number of Changes in the Sign of Derivatives |
| title_sort | on kolmogorov-type inequalities taking into account the number of changes in the sign of derivatives |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3919 |
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