Subsequence convergence in subdivision

We study the phenomenon that regularly spaced subsequences of the control points in subdivision may converge to scalar multiples of the same limit function, even though subdivision itself is divergent. We present different sets of easily checkable sufficient conditions for this phenomenon (which we term subsequence convergence) to occur, study the basic properties of subsequence convergence, show how certain results from subdivision carry over to this case, show an application for decorative effects, and use our results to build nested sets of refinement masks, which provide some insight into the structure of the set of refinable functions. All our results are formulated for a general integer dilation factor.