Uniform multipole approximation for scattered fields with boundary continuity

Let the obstacles be planar scatterers, and let the scattered field be defined in the exterior of the obstacles and on their boundaries. Assume that the scattered field is analytic in the exterior and continuous on the boundaries. Uniform multipole approximation conjecture. The scattered field can be uniformly approximated by sums of multipoles whenever it is analytic in the exterior of the obstacles and continuous on their boundaries. This conjecture is motivated by the analogy with Runge's and Mergelyan's theorems and by Vekua theory, which connects approximation of analytic functions with approximation by solutions of the Helmholtz equation.