## Real Analysis Exchange

### Essential Divergence in Measure of Multiple Orthogonal Fourier Series

#### 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

https://projecteuclid.org/euclid.rae/1404230142

Mathematical Reviews number (MathSciNet)
MR1006530

Zentralblatt MATH identifier
1296.13021

#### 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