On certain properties of orthogonal systems (Q1333655)

The author presents four essential theorems. The first one reads as follows: The terms of trigonometrical series can be rearranged in a way such that the obtained system \(\{e^{i\sigma(k)x}\}\) will possess the following property: for any \(\varepsilon> 0\) there exists a measurable set \(E\subset [0, 2\pi]\) of measure \(| E|> 2\pi- \varepsilon\) such that for any continuous function \(f(x)\) on \(E\) one can find a function \(g(x)\in L_{[0, 2\pi]}\) (coinciding with \(f(x)\) on \(E\)) such that its Fourier series with respect to the system \(\{e^{i\sigma(k)x}\}\) converges uniformly on \(E\). The further theorems state analogous results for complete uniformly bounded orthonormed systems and for only complete orthonormal systems.