On constants in some inequalities for intermediate derivatives on a finite interval
type: Item
instance of: scholarly article
title: On constants in some inequalities for intermediate derivatives on a finite interval
zbMATH Open document ID: 1053.41018
author: Semyon Rafalson
published in: Journal of Approximation Theory
publication date: 2004-08-10T00:00:00Z
review text: Denoting by \(L_q\) the space of functions \(f\) measurable on \(I=[-1,1]\) and integrable to power \(q\) and by \(AC\) the set of all functions absolutely continuous on \(I\), for \(n\in\mathbb N\), \(q\in[1, +\infty)\), the author set \[ W_{n,q}=\left\{f; f^{(n-1)}\in AC, f^{(n)}\in L_q\right\}. \] The problem of accuracy of constants \(A, B\) in the inequalities \[ \begin{aligned} & \| f^{(m)}\| _q\leq A\| f\| _p+B\| f^{(m+k+1)}\| _r,\\ & m\in\mathbb{N}, k\in W,\; p,q,r\in[1,+\infty], f\in W_{n+k+1,r} \end{aligned} \] is considered. The author used the following notations \[ \begin{aligned} & A_{m+k+1, m} (p,q,r)=\inf\limits_A A\\ & B_{m+k+1,m} (p,q,r)=\inf\limits_{B}\\ & B:A=A_{m+k+1,m}(p,q,r) \end{aligned} \] Explicit expressions for \(A_{m+k+1,m}(2,\infty,r)\) \((m\in\mathbb N,\; k\in W, 1\leq r\leq\infty)\) and estimates for \(A_{m+k+1,m}(p,q,r)\) are established.
reviewed by: Dan Bărbosu
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 2086007
zbMATH Keywords: measurable function
zbMATH Keywords: absolutely continuous function
zbMATH Keywords: orthogonal polynomials
MaRDI profile type: Publication
full work available at URL: j.jat.2004.03.003
OpenAlex ID: W1969226265
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DOI: 10.1016/J.JAT.2004.03.003
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description: scientific article; zbMATH DE number 2086007