Polynomial approximation of polyharmonic functions on a complement of a John domain

We prove a Jackson–Mergelyan type theorem on the uniform polynomial approximation of continuous polyharmonic functions on a set “without cusps on the boundary that point inside of the set”. We apply this theorem to derive a harmonic counterpart of a result by Mezhevich and Shirokov on the analytic polynomial approximation of continuous functions on a set consisting of two parallel segments.