THE GENERALIZED BARI THEOREM FOR THE WALSH SYSTEM

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 U-sets, one of which is simultaneously an F(sigma)-set and a G(delta)-set, is a U-set, and the union of two disjoint U-sets of type G(delta) is again a 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.