Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 66 / 3 / i

Fix the indexing conventionwhere 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 $M_{j_1}\ne 0$. After $\ell$ steps the possible descendant indices areLet $j_{\min },j_{\max }$ be the smallest and largest active subdivision mask indices. Dividing by $a^{\ell }$ gives the enclosureTherefore the support of a stationary subdivision scheme, in initial control-grid units, is enclosed byFor the stated symmetric subdivision mask with active extremes $-w,w$, the width is $2w/(a-1)$. For a nonzero compactly supported basic limit function $\varphi$, this is an exact endpoint calculation: its refinement equation is $\varphi (x)={\sum }_j{M}_j\varphi (ax-j)$. 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.