Expansions with Poisson kernels and related topics

Let P(r,theta) be the two dimensional Poisson kernel in the unit disk D. In this paper it is proved that there exists a special sequence a_k of points of D which is non tangentially dense for the boundary bD and such that any function f (theta) on bD can be expanded in series of P(|a_k|, (theta)- arg(a_k)) with coefficients depending continuously on f in various classes of functions. The result is used to solve a Cauchy type problem for Delta u=m, where m is a measure supported on the set {a_k}.