On the bounded divergence by rearranged orthonormal systems
The following two theorems are proved: Theorem 1. Let \(\{\varphi_n: n= 1,2,\dots\}\) be either the trigonometric system or a basis in the space \(C[0, 1]\). Then there exist a continuous function \(f\) and a permutation \(\tau\) of the positive integers such that the rearranged expansion \[ \sum^\infty_{n= 1} a_{\tau(n)} \varphi_{\tau(n)}(t),\quad a_n:= \int^1_0 f(t) \varphi^*_n(t) dt,\tag{*} \] diverges a.e., where \(\{\varphi^*_n\}\) is the biorthogonal system of \(\{\varphi_n\}\), and the partial sums of the series (*) are uniformly bounded. Theorem 2. Let \(\{\varphi_n\}\) be a complete orthonormal system on \([0,1]\). Then there exist a continuous function \(f\) and a permutation \(\tau\) of the positive integers such that the rearranged Fourier series (*) of \(f\) diverges a.e. (this time \(\varphi^*_n\equiv \varphi_n\)), and the maximal function of the partial sums of the series (*) belongs to the space \(L^2[0, 1]\).
scientific article; zbMATH DE number 795392