On DPT representations of solutions to the Helmholtz equation in a convex N-gon

The conditions of linear independence of systems of functions which arise in the framework of the domain-product technique (DPT) to expand the sought-for function within a convex N - gon when solving boundary-value problems for the 2D Helmholtz equation are discussed. It is established that linear dependence appears for at most countable set of values of the spectral parameter. Outside this set and resonant sets of auxiliary domains the expansions unambiguously represent the solution provided it exists and is unique.