The Bojanov-Naidenov problem for trigonometric polynomials and periodic splines
For given $n, r \in \mathbb{N}$; $p, A > 0$ and any fixed interval $[a,b] \subset \mathbb{R}$ we solve the extremal problem $\int\limits_a^b |x(t)|^q dt \rightarrow \sup$, $q \geqslant p$, over sets of trigonometric polynomials $T$ of order $\leqslant n$ and $2\pi$-periodic splines $s$ of order $...
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| Format: | Article |
| Language: | English |
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Oles Honchar Dnipro National University
2019-07-01
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| Series: | Researches in Mathematics |
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| Online Access: | https://vestnmath.dnu.dp.ua/index.php/rim/article/view/108 |
| _version_ | 1811293226803396608 |
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| author | E.V. Asadova V.A. Kofanov |
| author_facet | E.V. Asadova V.A. Kofanov |
| author_sort | E.V. Asadova |
| collection | DOAJ |
| description | For given $n, r \in \mathbb{N}$; $p, A > 0$ and any fixed interval $[a,b] \subset \mathbb{R}$ we solve the extremal problem $\int\limits_a^b |x(t)|^q dt \rightarrow \sup$, $q \geqslant p$, over sets of trigonometric polynomials $T$ of order $\leqslant n$ and $2\pi$-periodic splines $s$ of order $r$ and minimal defect with knots at the points $k\pi / n$, $k \in \mathbb{Z}$, such that $\| T \| _{p, \delta} \leqslant A \| \sin n (\cdot) \|_{p, \delta} \leqslant A \| \varphi_{n,r} \|_{p, \delta}$, $\delta \in (0, \pi / n]$, where $\| x \|_{p, \delta} := \sup \{ \| x \|_{L_p[a,b]} \colon a, b \in \mathbb{R}, 0 < b - a < \delta\}$ and $\varphi_{n, r}$ is the $(2\pi / n)$-periodic spline of Euler of order $r$. In particular, we solve the same problem for the intermediate derivatives $x^{(k)}$, $k = 1, ..., r-1$, with $q \geqslant 1$. |
| first_indexed | 2024-04-13T04:57:19Z |
| format | Article |
| id | doaj.art-ad4b9f13c73e4b0d886b9561d7c254e7 |
| institution | Directory Open Access Journal |
| issn | 2664-4991 2664-5009 |
| language | English |
| last_indexed | 2024-04-13T04:57:19Z |
| publishDate | 2019-07-01 |
| publisher | Oles Honchar Dnipro National University |
| record_format | Article |
| series | Researches in Mathematics |
| spelling | doaj.art-ad4b9f13c73e4b0d886b9561d7c254e72022-12-22T03:01:26ZengOles Honchar Dnipro National UniversityResearches in Mathematics2664-49912664-50092019-07-0127131310.15421/241901The Bojanov-Naidenov problem for trigonometric polynomials and periodic splinesE.V. Asadova0V.A. Kofanov1Oles Honchar Dnipro National UniversityOles Honchar Dnipro National UniversityFor given $n, r \in \mathbb{N}$; $p, A > 0$ and any fixed interval $[a,b] \subset \mathbb{R}$ we solve the extremal problem $\int\limits_a^b |x(t)|^q dt \rightarrow \sup$, $q \geqslant p$, over sets of trigonometric polynomials $T$ of order $\leqslant n$ and $2\pi$-periodic splines $s$ of order $r$ and minimal defect with knots at the points $k\pi / n$, $k \in \mathbb{Z}$, such that $\| T \| _{p, \delta} \leqslant A \| \sin n (\cdot) \|_{p, \delta} \leqslant A \| \varphi_{n,r} \|_{p, \delta}$, $\delta \in (0, \pi / n]$, where $\| x \|_{p, \delta} := \sup \{ \| x \|_{L_p[a,b]} \colon a, b \in \mathbb{R}, 0 < b - a < \delta\}$ and $\varphi_{n, r}$ is the $(2\pi / n)$-periodic spline of Euler of order $r$. In particular, we solve the same problem for the intermediate derivatives $x^{(k)}$, $k = 1, ..., r-1$, with $q \geqslant 1$.https://vestnmath.dnu.dp.ua/index.php/rim/article/view/108Bojanov-Naidenov problempolynomialsplinerearrangement |
| spellingShingle | E.V. Asadova V.A. Kofanov The Bojanov-Naidenov problem for trigonometric polynomials and periodic splines Researches in Mathematics Bojanov-Naidenov problem polynomial spline rearrangement |
| title | The Bojanov-Naidenov problem for trigonometric polynomials and periodic splines |
| title_full | The Bojanov-Naidenov problem for trigonometric polynomials and periodic splines |
| title_fullStr | The Bojanov-Naidenov problem for trigonometric polynomials and periodic splines |
| title_full_unstemmed | The Bojanov-Naidenov problem for trigonometric polynomials and periodic splines |
| title_short | The Bojanov-Naidenov problem for trigonometric polynomials and periodic splines |
| title_sort | bojanov naidenov problem for trigonometric polynomials and periodic splines |
| topic | Bojanov-Naidenov problem polynomial spline rearrangement |
| url | https://vestnmath.dnu.dp.ua/index.php/rim/article/view/108 |
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