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On certain properties of orthogonal systems. (English. Russian original) Zbl 0814.42020
Russ. Math. 36, No. 10, 78-80 (1992); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1992, No. 10 (365) 80-82 (1992).
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.
42C20 Other transformations of harmonic type