Some of the known Haar spaces are linear hulls of shifts of a single function G on ℂ{0}. We study N-dimensional and universal analytic Haar space generators for some closed sets F of ℂ (in the sense that an arbitrary finite number of shifts generates Haar spaces by forming linear hulls). The suitable function space for our investigation is Cο(F), the space of all complex valued, continuous functions f on F with the defining property lim z∈ F,z→∞f(z) = 0. In many cases we are able to characterize universal Haar space generators. We show, in addition, that in Cο(F) a best approximation by elements of finite dimensional spaces V is unique if and only if V is a Haar space.