A lacunary version of Mergelian’s approximation theorem Original Research Article

Let KK be a compact plane set having connected complement. Then Mergelian’s theorem states that the linear span of the monomials znzn, i.e. the polynomials, are dense in the Banach space A(K)A(K) of all functions continuous on KK and holomorphic in the interior of KK endowed with the sup-norm. We consider the question under which conditions the linear span of znzn, with nn running through a sequence of nonnegative integers having upper density one, is dense in A(K)A(K) or appropriate subspaces.