= Support of a stationary subdivision scheme
{title2=$[j_{\min}/(a-1),j_{\max}/(a-1)]$}
Under $P_i^{\ell+1}=\sum_jM_{i-aj}P_j^\ell$, the descendants of control index zero lie at $a^{\ell-1}j_1+\cdots+j_\ell$, where each $j_k$ is in the mask support. Dividing by the refined parameter scale $a^\ell$ gives a limiting influence enclosure with the displayed endpoints. For a mask supported on $[-w,w]$ its width is $2w/(a-1)$ when the endpoints are active and a nonzero compactly supported basic limit exists. To prove exactness, the largest active mask term is the only surviving term near the right endpoint of the refinement equation, forcing $\beta=(\beta+j_{\max})/a$; the analogous left-end argument completes the result. Mask padding should be removed, and a nonconvergent rule has no limit support.
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