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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.
##### MSC:
 42C20 Other transformations of harmonic type