Sets of uniqueness on the $2$-torus

Abstract

${H^{(J)}}$-sets are defined on the 2-torus and the following results are established: (1) ${H^{(J)}}$-sets are sets of uniqueness both for Abel summability and circular convergence of double trigonometric series; (2) a countable union of closed sets of uniqueness of type (A) (i.e., Abel summability) is also a set of uniqueness of type (A).