On universal functions and series
type: Item
instance of: scholarly article
title: On universal functions and series
zbMATH Open document ID: 0629.40001
DOI: 10.1007/BF01951004
published in: Acta Mathematica Hungarica
publication date: 1987-01-01T00:00:00Z
review text: In the first part of the paper the non-density of the difference quotients is proved in \(L^ p\) spaces when \(p\geq 1\). In the second part universal series are studied. Suppose that \(\phi_ i\) is a complete orthonormal system in \(L^ 2\). The series \(\sum^{+\infty}_{1}c_ i\phi_ i\) is called universal in \(L^ p\) if its partial sums are dense in \(L^ p\). The effect of Toeplitz summation on the universality is investigated. Roughly speaking: if \(p\geq 1\) then there are no universal series in the Toeplitz sense while for \(0\leq p<1\) ``typical'' series are universal. Finally, as a by-product, it is proved that if \(\phi_ n\) is a complete orthonormal system in \(L^ 2[0,1]\) then \(\sum^{+\infty}_{n=1}\| \phi_ n\|^ 2_ 1=+\infty\).
Mathematics Subject Classification ID: 40A05
Mathematics Subject Classification ID: 26B05
zbMATH DE Number: 4023925
zbMATH Keywords: difference quotients
zbMATH Keywords: \(L^ p\) spaces
zbMATH Keywords: universal series
zbMATH Keywords: Toeplitz summation
author: Zoltán Buczolich
MaRDI profile type: Publication
cites work: Q3278441
cites work: On universal functions
cites work: Differentiation of real functions
cites work: Q3804957
cites work: Q3361372
cites work: Sur les nombres dérivés
cites work: Q5768777
full work available at URL: https://doi.org/10.1007/bf01951004
OpenAlex ID: W2007624043
description: scientific article; zbMATH DE number 4023925