Uniform Approximation by Polynomial Solutions of Elliptic Systems on Boundaries of Caratheodory Domains in R2

We study the problem on uniform approximation of functions on compact subsets of the complex plane by polynomial solutions of general second-order elliptic systems with constant coefficients. This problem is well-known for the systems corresponding to second order equations with constant complex coefficients and is rather poor studied in the general case. In particular case, when the compact set where the approximation is considered is the boundary of a Caratheodory domain in the plane, we establish some new sufficient approximability conditions. We also discuss new measure orthogonality conditions that appear in the problem under consideration.