Support of a stationary subdivision scheme ($[j_{\min }/(a-1),j_{\max }/(a-1)]$)

Under $P_i^{\ell +1}={\sum }_j{M}_{i-aj}{P}_j^{\ell }$, 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.