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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Gespeichert in:

Autor*in:	E.V. Asadova [verfasserIn] V.A. Kofanov [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch ; Ukrainisch

Erschienen:	2019

Schlagwörter:	Bojanov-Naidenov problem polynomial spline rearrangement

Übergeordnetes Werk:	In: Researches in Mathematics - Oles Honchar Dnipro National University, 2019, 27(2019), 1, Seite 3-13
Übergeordnetes Werk:	volume:27 ; year:2019 ; number:1 ; pages:3-13

Links:	Link aufrufen Link aufrufen Link aufrufen Journal toc Journal toc

DOI / URN:	10.15421/241901

Katalog-ID:	DOAJ024882739

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520			\|a 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 $q \geqslant 1$.
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callnumber-first	Q - Science
author	E.V. Asadova
spellingShingle	E.V. Asadova misc QA1-939 misc Bojanov-Naidenov problem misc polynomial misc spline misc rearrangement misc Mathematics The Bojanov-Naidenov problem for trigonometric polynomials and periodic splines
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topic_title	QA1-939 The Bojanov-Naidenov problem for trigonometric polynomials and periodic splines Bojanov-Naidenov problem polynomial spline rearrangement
topic	misc QA1-939 misc Bojanov-Naidenov problem misc polynomial misc spline misc rearrangement misc Mathematics
topic_unstemmed	misc QA1-939 misc Bojanov-Naidenov problem misc polynomial misc spline misc rearrangement misc Mathematics
topic_browse	misc QA1-939 misc Bojanov-Naidenov problem misc polynomial misc spline misc rearrangement misc Mathematics
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title	The Bojanov-Naidenov problem for trigonometric polynomials and periodic splines
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title_full	The Bojanov-Naidenov problem for trigonometric polynomials and periodic splines
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title_sort	bojanov-naidenov problem for trigonometric polynomials and periodic splines
callnumber	QA1-939
title_auth	The Bojanov-Naidenov problem for trigonometric polynomials and periodic splines
abstract	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 $q \geqslant 1$.
abstractGer	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 $q \geqslant 1$.
abstract_unstemmed	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 $q \geqslant 1$.
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title_short	The Bojanov-Naidenov problem for trigonometric polynomials and periodic splines
url	https://doi.org/10.15421/241901 https://doaj.org/article/ad4b9f13c73e4b0d886b9561d7c254e7 https://vestnmath.dnu.dp.ua/index.php/rim/article/view/108 https://doaj.org/toc/2664-4991 https://doaj.org/toc/2664-5009
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up_date	2024-07-04T00:45:16.548Z
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The Bojanov-Naidenov problem for trigonometric polynomials and periodic splines

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