An improved Proximity =⇒ Smoothness theorem

In this note, we point out that the proximity condition proposed in [8] can be weakened. We reprove the “Proximity ⇒ Smoothness” theorem under the so-called weak proximity condition. This improvement is embarrassingly trivial; and appears to be just an artifact of the authors overlooking a detail during the writing of [8]. But this raises a question: the unnecessarily strong proximity condition proposed in [8] was used in a number of subsequent papers and, according to this note the authors of these subsequent papers had worked unnecessarily hard to establish the strong proximity condition. If the original proximity condition is truly unnecessary, why would it persist in all these subsequent papers? Coupling the observation in this note and the main result in our recent work [1], we conclude that, under a natural compatibility condition, the strong and weak proximity conditions are equivalent. This equivalence is attributable to an algebraic structure of subdivision schemes. The goal of this note is prove Theorem 0.7 below; it weakens the proximity condition in [8, Theorem 2.4] to a weaker proximity condition. Throughout this section, we let Z := {0} ∪ N. For any sequence x = (xi)i in an Euclidian space, let |x|∞ := sup i ‖xi‖2. For any M ⊆ R and δ > 0, let XM := { x : Z→M ∣∣ |∆x|∞ <∞} and XM,δ := {x : Z→M ∣∣ |∆x|∞ < δ} . For j ∈ N, let Γj := { γ = (γ1, · · · , γj) ∣∣∣ γi ∈ Z, j ∑