Convergence and preservation of cyclicity

Abstract

We prove that the set of cyclic (respectively, non-cyclic) functions in Dirichlet type spaces Dα is not closed in the topology induced by the norm. We also show that some additional conditions on a convergent sequence of cyclic functions {fn} force cyclicity of the limit f. We show counterexamples satisfying all but one of these conditions. Then we find precise estimates for the distance between the corresponding optimal polynomial approximants of each degree d and control its blow up as the choice of f moves within Dα.