= Solution
Fix the indexing convention
$$
P_i^{\ell+1}=\sum_jM_{i-aj}P_j^\ell,\qquad t_i^\ell=i/a^\ell,
$$
where the <subdivision arity> $a>1$ is an <integer>. A change of the initial control $P_0^0$ can influence only indices $i=j_1$ after one step, where $M_{j_1}\ne0$. After $\ell$ steps the possible descendant indices are
$$
i=a^{\ell-1}j_1+a^{\ell-2}j_2+\cdots+j_\ell.
$$
Let $j_{\min},j_{\max}$ be the smallest and largest active <subdivision mask> indices. Dividing by $a^\ell$ gives the enclosure
$$
\frac{j_{\min}}{a-1}(1-a^{-\ell})\le t_i^\ell\le\frac{j_{\max}}{a-1}(1-a^{-\ell}).
$$
Therefore the <support of a stationary subdivision scheme>, in initial control-grid units, is enclosed by
$$
\boxed{\left[\frac{j_{\min}}{a-1},\frac{j_{\max}}{a-1}\right],\qquad\text{width }\frac{j_{\max}-j_{\min}}{a-1}.}
$$
For the stated symmetric <subdivision mask> with active extremes $-w,w$, the width is $2w/(a-1)$. For a nonzero compactly supported basic limit function $\phi$, this is an exact endpoint calculation: its refinement equation is $\phi(x)=\sum_jM_j\phi(ax-j)$. If $\beta$ is its right support endpoint, only the largest active <subdivision mask> index can contribute sufficiently close to the rightmost point of this sum; all smaller indices have argument beyond $\beta$. Hence $\beta=(\beta+j_{\max})/a$, giving $\beta=j_{\max}/(a-1)$. The left endpoint follows in the same way. This argument also excludes cancellation at an extreme for signed <subdivision masks>. The support for initial control $P_k$ is translated by $k$. If the nominal <subdivision mask> was padded by zero coefficients, use its actual extremes; and a limiting <subdivision curve> has to exist before its support is meaningful. This calculation describes the entire possible influence region, rather than the width of a single refinement stencil on its finer grid.
Back to article page
