Landau-type extremal problem for the triple \(\| f\| _{\infty},\| f'\| _{p},\| f''\| _{\infty}\) on a finite interval.
type: Item
instance of: scholarly article
title: Landau-type extremal problem for the triple \(\| f\| _{\infty},\| f'\| _{p},\| f''\| _{\infty}\) on a finite interval.
zbMATH Open document ID: 1035.41008
DOI: 10.1016/S0021-9045(03)00081-9
author: Nikola Naidenov
published in: Journal of Approximation Theory
publication date: 2003-08-25T00:00:00Z
review text: Let \(f\) be a function defined on a given interval \(I\) (finite or infinite). The \(L_p\)-norms are defined by \[ \| f\| _p=\| f\| _{L_p(I)}= \begin{cases} \left(\int\limits_I\left| f\left(x\right)\right| ^p\,dx\right)^{1/p}, & 1\leq p<\infty\\ \mathop{\text{vrai\,sup}}\limits_{x\in I}| f(x)| , & p=\infty, \end{cases} \] as usual. The extremal problem \[ \| f\| _q\leq M_0,\quad \| f^{(n)}\| _r\leq M_n,\quad \| f^{(k)}\| _p\to\sup, \tag{1} \] for any fixed \(k\), \(1\leq k<n\), is known as a problem of Landau-Kolmogorov type. It is well investigated when \(I={\mathbb R}\), \(I={\mathbb R}_+\), or for classes of periodic functions. If the interval \(I\) is finite, the problem is much more difficult. In this case the extremal function depends essentially on the length \(| I| \). Thus, we have an additional parameter that plays an important role. The present article is devoted to the investigation of the problem (1) on a finite interval \(I\) with \(n=2\), \(q=r=\infty\), \(1\leq p<\infty\). In order to formulate the main result, the author defines \[ \Omega^2(I)=\{f\in W^2_\infty(I): \| f\| _\infty\leq 1,\;\| f''\| _\infty\leq 4\}. \] Bounds \(1\) and \(4\) appearing here are chosen according to the Chebyshev polynomial \(T_2(x)=2x^2-1\) and the interval \([-1,1]\). Let \(\Phi\) be the class of all functions \(\varphi\) that are positive, increasing and nonconcave on \((0,\infty)\). Then for every interval \(I\) we define \[ J_\varphi(g;I):= \int_I\varphi (| g(x)|)\,dx. \] Let us note that if \(\varphi(x)=x^p\), the functional \(J_\varphi(g;I)\) reduces to the \(L_p\)-norm of a function \(g\). A parabolic perfect spline with \(m\) knots is a zigzag spline on \([a,b]\) if it has \(m+1\) local extrema on \((a,b)\). The end points of the interval are considered as points of local extremum too. The main result of the article is the following theorem. Theorem. Assume that \(I=[a,b]\) with \(b-a>2\) and \(\varphi\in\Phi\). Then every extremal function of the problem \[ J_\…
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A15
zbMATH DE Number: 1970534
zbMATH Keywords: Landau type extremal problems
zbMATH Keywords: Kolmogorov type inequalities
zbMATH Keywords: extremal function
zbMATH Keywords: parabolic perfect spline
zbMATH Keywords: zigzag spline
reviewed by: Alexander Tovstolis
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1970534