An inequality between the norms of a function and its derivatives in integral metrics
type: Item
instance of: scholarly article
title: An inequality between the norms of a function and its derivatives in integral metrics
zbMATH Open document ID: 0535.26007
DOI: 10.1007/BF01141198
author: S. Z. Rafal'son
published in: Mathematical Notes
publication date: 1983-01-01T00:00:00Z
review text: Let \(L_{2,r}\) be the function space with the norm \[ \| f\|_{2,r}=\{\int^{+1}_{-1}(1-x^ 2)^ r| f(x)|^ 2dx\}^{1/2} \] and \(c_ k(f)=\int^{+1}_{-1}f(x)X_ k(x)dx\), where \(X_ k\) is a Legendre polynomial. Theorem. Let \(r\in {\mathbb{N}}\), \(s\in {\mathbb{N}}\cup \{0\}\) and \(0\leq s\leq r-1.\) Suppose that \(f\in L_ 2\), \(f^{(r-1)}\in W_{\infty}\!^{r-1}(-1+\epsilon,1-\epsilon)\) for all \(\epsilon>0\) and \(f^{(r)}\in L_{2,r}.\) If also \(c_ k(f)=0\) for \(k=s,...,r-1\), then the following exact inequality is valid \[ \| f^{(s)}\|_{2,s}\leq \frac{\Gamma^{1/2}(r+s+1)}{\Gamma^{1/2}(r- s+1)\Gamma^{s/2r}(2r+1)}\| f\|_ 2\!^{1-s/r}\| f^{(r)}\|_{2,r}\!\!\!^{s/r}; \] here \(\Gamma\) is the \(\Gamma\)- function.
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 42C10
zbMATH DE Number: 3849531
zbMATH Keywords: Kolmogorov-type inequality
zbMATH Keywords: Legendre polynomials
zbMATH Keywords: Fourier Legendre coefficients
reviewed by: Yuri A. Brudnyi
MaRDI profile type: Publication
cites work: Q5531363
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cites work: Functions of exponential type
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description: scientific article; zbMATH DE number 3849531