Geodesic
$A$-integrable martingale sequences and Walsh series
Izvestiya. Mathematics , Tome 65 (2001) no. 3, pp. 607-615.

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A sufficient condition for a Walsh series converging to an $A$-integrable function $f$ to be the $A$-Fourier's series of $f$ is stated in terms of uniform $A$-integrability of a martingale subsequence of partial sums of the Walsh series. Moreover, the existence is proved of a Walsh series that converges almost everywhere to an $A$-integrable function and is not the $A$-Fourier series of its sum. 
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V. A. Skvortsov. $A$-integrable martingale sequences and Walsh series. Izvestiya. Mathematics , Tome 65 (2001) no. 3, pp. 607-615. http://geodesic.mathdoc.fr/item/IM2_2001_65_3_a9/

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