Strong contraction of the representations of the three dimensional Lie algebras

For any Inonu-Wigner contraction of a three dimensional Lie algebra we construct the corresponding contractions of representations. Our method is quite canonical in the sense that in all cases we deal with realizations of the representations on some spaces of functions; we contract the differential operators on those spaces along with the representation spaces themselves by taking certain pointwise limit of functions. We call such contractions strong contractions. We show that this pointwise limit gives rise to a direct limit space. Many of these contractions are new and in other examples we give a different proof.