Menshov representation spectra

Details Menshov representation spectra Menshov representation spectra

2005-10-28

arxiv.org/abs/math/0510616v1

J. Anal. Math. 84 (2001), 361-393

Mathematics

Classical Analysis and ODEs

25 pages. Part of GK's PhD thesis

Scientific paper

A Menshov spectrum is a subset of the integers that is sufficient for representing every measurable function as an almost-everywhere converging trigonometric (non-Fourier) sum. In this language the celebrated "Menshov representation theorem" states that Z is a Menshov spectrum. In this paper we construct 1) Menshov spectra that are almost exponentially sparse 2) that are almost squares. Then we show that the positive integers are not a Menshov spectrum but are a Menshov spectrum in measure, and repeat 1) and 2) in the analytic settings.

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