Lacunary subsets of orthonormal sets
type: Item
instance of: scholarly article
title: Lacunary subsets of orthonormal sets
zbMATH Open document ID: 0676.42011
DOI: 10.1007/BF01909193
author: I. Agaev
published in: Analysis Mathematica
publication date: 1985-01-01T00:00:00Z
review text: The author studies the density of certain subsets in sets of functions \(\Phi =\{\phi_ k(x)\}^{\infty}_{k=1}\), which are orthonormal on [0,1]. One such type of subset is that of a lacunary set. These are defined as follows. An orthonormal set (ONS) of functions \(\Phi =\{\phi_ k(x)\}^{\infty}_{k=1}\), \(x\in [0,1]\), is said to be lacunary of order p \((p>2)\), or an \(S_ p\)-set, if for each polynomial P(x) with respect to this set the following estimate holds: (1) \(\| P(x)\|_ p\equiv P(x)\|_{L_ p}\leq C_{\Phi}\| P(x)\|_ 2.\) Using earlier work by Banach and Kashin, the author establishes several results about the density of lacunary subsets (and similar subsets) in orthonormal sets. One such result is the following theorem: Let \(p>2\) be an even integer, \(0<\delta \leq p-2\), \(\Phi =\{\phi_ n(x)\}^ N_{n=1}\) an ONS, \(x\in [0,1]\), with \(\| \phi_ n\|_{\tilde p}\leq M\), \(n=1,2,...,N\), where \(\tilde p=p+\delta\), \(0<M<\infty\). Then a subset \(\Phi '=\{\phi_{nk}\}^{N'}_{k=1}\) can be extracted from the set \(\Phi\), with \(N'\geq N^{\alpha (\delta)}\), \(\alpha (\delta)=2\delta /(p(p-2+\delta))\), such that for each polynomial \(P(x)=\sum^{N'}_{k- 1}a_ k\phi_{n_ k}(x)\) the following estimate holds: \[ (\sum^{N'}_{k=1}a^ 2_ k)^{1/2}\leq \| P\|_ p\leq c_{p,M,\delta}(\sum^{N'}_{k=1}a^ 2_ k)^{1/2}. \] The author also shows (based on work by Gaposhkin) that the estimates in this theorem are sharp.
Mathematics Subject Classification ID: 42A55
zbMATH DE Number: 4107391
zbMATH Keywords: density of lacunary subsets
zbMATH Keywords: orthonormal sets
MaRDI profile type: Publication
cites work: Sur la convergence et la sommation des séries de fonctions orthogonales
cites work: Q3265794
cites work: Q5835464
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description: scientific article; zbMATH DE number 4107391