Convergence of univariate non-stationary subdivision schemes via asymptotical similarity

A new equivalence notion between non-stationary subdivision schemes, termed asymptotic similarity, which is weaker than asymptotic equivalence, is introduced and studied. It is known that asymptotic equivalence between a non-stationary subdivision scheme and a convergent stationary scheme guarantees the convergence of the non-stationary scheme. We show that for non-stationary schemes reproducing constants, the condition of asymptotic equivalence can be relaxed to asymptotic similarity. This result applies to a wide class of non-stationary schemes.