## Real Analysis Exchange

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

### 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

**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