The almost everywhere convergence of Fourier series according to complete orthonormal systems (Q1316843)

The following theorem is proved. The terms of any bounded orthonormal system \(\{\varphi_ n(x)\}\) complete in \(L^ 2 (0,1)\) can be permuted so that the system \(\{\varphi_{\sigma (k)} (x)\}\) obtained possesses the following property: for any \(\varepsilon>0\) there exists a measurable set \({\mathcal G} \in [0,1]\) with a measure \(| {\mathcal G} | > 1 - \varepsilon\) such that for each function \(f(x) \in L(0,1)\) one can find a function \(g(x) \in L(0,1)\) coinciding with \(f(x)\) on \({\mathcal G}\) such that its Fourier series according to the system \(\{\varphi_{\sigma(k)} (x)\}\) converges almost everywhere on \([0,1]\).