Abstract
A criterion for almost everywhere convergence of Franklin series on a set is proved. The criterion is similar to the ones obtained by Y.S. Chow for martingales and F.G. Arutyunyan for Haar series.
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F. G. Arutyunyan, On series with respect to the Haar system, Akad. Nauk Armjan. SSR Dokl., 42 (1966), 134–140 (in Russian).
Y. S. Chow, Convergence theorems of martingales, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 1 (1963), 340–346.
Ph. Franklin, A set of continuous orthogonal functions, Math. Ann., 100 (1928), 522–529.
G. G. Gevorkyan, On series by the Franklin system, Anal. Math., 16 (1990) 87-114 (in Russian).
G. G. Gevorkyan, Uniqueness theorem for series in the Franklin system, Mat. Zametki, 98 (2015), 786–789 (in Russian); translation in Math. Notes, 98 (2015), 847–851
G. G. Gevorkyan, On the uniqueness of series in the Franklin system, Matem. Sbornik, 207 (2016), 30–53 (in Russian); translation in Sb. Math., 207 (2016), 1650–1673
G. G. Gevorkyan, Uniqueness theorems of Franklin series, converging to integrable functions, Matem. Sbornik, 209 (2018), 25–46 (in Russian); translation in Sb. Math., 209 (2018).
R. Gundy, Martingale theory and pointwise convergence of certain orthogonal series, Trans. Amer. Math. Soc., 124 (1966), 228–248.
S. V. Konyagin, Limits of indeterminacy of trigonometric series, Mat. Zametki, 44 (1988), 770–784 (in Russian); translated in Math. Notes, 44 (1988), 910–920.
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This research was supported by SCS RA grant 18T-1A074.
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Gevorkyan, G.G. On a “Martingale Property” of Franklin Series. Anal Math 45, 803–815 (2019). https://doi.org/10.1007/s10476-019-0008-z
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DOI: https://doi.org/10.1007/s10476-019-0008-z