Convergence of univariate non-stationary subdivision schemes via asymptotical similarity
For researchers in subdivision schemes and geometric modeling, this provides a weaker condition to verify convergence, making analysis easier for a broad class of non-stationary schemes.
The paper introduces a new notion of asymptotical similarity for non-stationary subdivision schemes, which is weaker than asymptotical equivalence, and proves that for schemes reproducing constants, this condition still guarantees convergence. This relaxes existing convergence conditions and applies to many important non-stationary schemes.
A new equivalence notion between non-stationary subdivision schemes, termed asymptotical similarity, which is weaker than asymptotical equivalence, is introduced and studied. It is known that asymptotical 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 asymptotical equivalence can be relaxed to asymptotical similarity. This result applies to a wide class of non-stationary schemes of importance in theory and applications.