On a discrete equation in a quarter-plane and a related boundary-value problem

Discrete equations of the convolution type in a quarter-plane are considered. We prove that each such equation is equivalent to an analog of the two-dimensional periodic Riemann problem on the torus. We describe sufficient conditions for the unique solvability of such a periodic Riemann problem and, as a consequence, conditions for the unique solvability of a discrete equation in terms of the symbol of the convolution operator.