Divergence in measure of rearranged multiple orthogonal Fourier series (Q1047488)

The paper deals with the convergence in measure of Fourier series with respect to rearranged multiple complete orthonormal systems. The main result is the following. Let \(I:= [0,1)\) , \(\{\varphi_n\}\) \((n= 1,2,\dots)\) a complete orthonormal system of a.e. bounded functions on \(I\) and \(\varphi: [0,+\infty)\to \mathbb{R}^+\) a continuous nondecreasing function with \(x\varphi(x)\) convex and \(x\varphi(x)= o(\ln x)\) for \(x\to+\infty\). Then there exist a rearrangement \(\{\varphi_{h(n)}\}\) \((n= 1,2,\dots)\) of \(\{\varphi_n\}\) and a function \(f\in L^1(I^2)\) such that the sequence of square partial sums of the Fourier series of \(f\) with respect to the double system \(\{\varphi_{h(n)}, \varphi_{h(m)}\}\) \((n,m= 1,2,\dots)\) on \(I^2\) is essentially bounded in measure on \(I^2\).