Uniqueness theorems for one-dimensional and double Franklin series

The paper contains two main results. First we describe one-dimensional Franklin series converging everywhere except possibly on a finite set to an everywhere-finite integrable function. Second we establish a class of subsets of $[0, 1]^2$ with the following property. If a double Franklin series converges everywhere except on this set to an everywhere-finite integrable function, then it is the Fourier–Franklin series of this function. In particular, all countable sets are in this class.