Continuous and bounded harmonic functions. Exact and approximate methods

The problem of representation of functions harmonic on an open square is considered for the case when these functions satisfy one of the following conditions: a) they have continuous extensions to the closed square and b) they are bounded in the open square. A full description of these classes of harmonic functions is obtained in terms of the properties of boundary values, double-layer potential densities, and the intrinsic properties of harmonic functions in an open square.