Difference equations in discrete spaces

One considers a general difference equation with variable coefficients in discrete spaces. The conditions for unique solvability and Fredholmness for such equation are given using the theory of periodic Riemann boundary value problem. Key role in the studying takes the periodic analogue of the Hilbert transform, it permits to obtain explicit solution for particular cases. Also, this transform has very important properties related to a holomorphy. It leads to Fredholm properties for more general cases of difference equations.