On a generalization of the Bessel inequality
type: Item
instance of: scholarly article
title: On a generalization of the Bessel inequality
zbMATH Open document ID: 0813.42015
DOI: 10.1007/BF01262190
author: A. S. Krantsberg
published in: Mathematical Notes
publication date: 1994-04-12T00:00:00Z
review text: Let \(\{\varphi_ k(x)\}^ n_{i= 1}\) and \(\{\psi_ i(x)\}^ s_{i= 1}\) be orthonormal systems on the interval \([0,1]\) \((s\leq n)\) and \(\{a_ i\}^ s_{i= 1}\) a sequence of numbers. We put \[ f_ r(x)= \sum^ r_{i= 1} a_ i \psi_ i(x),\quad r= 1,2,\dots, s,\quad\text{and } (f,\varphi)= \int^ 1_ 0 f(x) \varphi(x) dx. \] Let \(r= r(k)\) be a monotone increasing function taking natural values \((r\leq s)\), and we denote by \(\widetilde s\) the smallest number of the form \(2^ l\) which is larger than \(s\). The following inequality holds: \[ \sum^ n_{k= 1} (f_ r, \varphi_ k)^ 2\leq (\log_ 2 \widetilde s+ 1)^ 2 \sum^ s_{i= 1} a^ 2_ i. \] {}.
Mathematics Subject Classification ID: 42C15
zbMATH DE Number: 525683
zbMATH Keywords: Bessel inequality
zbMATH Keywords: Menshov-Rademacher lemma
zbMATH Keywords: orthonormal systems
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf01262190
OpenAlex ID: W1990231356
reviewed by: Károly Tandori
description: scientific article; zbMATH DE number 525683