Real Analysis Exchange

Essential Divergence in Measure of Multiple Orthogonal Fourier Series

Rostom Getsadze

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Abstract

In the present paper we prove the following theorem: \\ Let \(\{\vf _{m,n}(x,y)\}_{m,n=1}^{\infty} \) be an arbitrary uniformly bounded double orthonormal system on \(I^2:=[0,1]^2\) such that for some increasing sequence of positive integers \(\{N_n\}_{n=1}^\infty \) the Lebesgue functions \(L_{N_n,N_n}(x,y)\) of the system are bounded below a. e. by \( \ln^{1+\epsilon} N_n \), where \(\epsilon \) is a positive constant. Then there exists a function \(g \in L(I^2)\) such that the double Fourier series of \(g\) with respect to the system \(\{\vf _{m,n}(x,y)\}_{m,n=1}^{\infty} \) essentially diverges in measure by squares on \(I^2\). The condition is critical in the logarithmic scale in the class of all such systems %\footnote{ 2000 Mathematics Subject Classification : Primary 42B08; Secondary 40B05. % Key words and phrases: %Essential divergence in measure, orthogonal Fourier series, Lebesgue functions}.

Article information

Source
Real Anal. Exchange, Volume 39, Number 1 (2013), 91-100.

Dates
First available in Project Euclid: 1 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.rae/1404230142

Mathematical Reviews number (MathSciNet)
MR1006530

Zentralblatt MATH identifier
1296.13021

Subjects
Primary: 26B10: Implicit function theorems, Jacobians, transformations with several variables 42B08: Summability
Secondary: 40B05: Multiple sequences and series (should also be assigned at least one other classification number in this section)

Keywords
Measure Convergence Fourier Essential

Citation

Getsadze, Rostom. Essential Divergence in Measure of Multiple Orthogonal Fourier Series. Real Anal. Exchange 39 (2013), no. 1, 91--100. https://projecteuclid.org/euclid.rae/1404230142


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