Geodesic
The generalized Bari theorem for the~Walsh system
Sbornik. Mathematics, Tome 77 (1994) no. 1, pp. 139-147

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For Walsh series in the Paley arrangement the author proves a generalized Bari theorem on the union of sets of uniqueness, from which it follows in particular that the union of two $\mathcal U$-sets, one of which is simultaneously an $F_\sigma$-set and a $G_\delta$-set, is a $\mathcal U$-set, and the union of two disjoint $\mathcal U$-sets of type $G_\delta$ is again a $\mathcal U$-set. It is shown that the last two assertions hold for sets of uniqueness of those classes of series for which the principle of localization of the kernel holds. 
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     title = {The generalized {Bari} theorem for {the~Walsh} system},
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N. N. Kholshchevnikova. The generalized Bari theorem for the~Walsh system. Sbornik. Mathematics, Tome 77 (1994) no. 1, pp. 139-147. http://geodesic.mathdoc.fr/item/SM_1994_77_1_a8/