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We consider sequences of linear operators $U_nf(x)$ with localizationproperty. It is proved that for any set $E$ of measure zero there exists a set$G$ for which $U_n\ZI_G(x)$ diverges at each point $x\in E$. This result is ageneralization of analogous theorems known for the Fourier sums operators withrespect to different orthogonal systems.

Divergence of general localized operators on the sets of measure zero