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

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.

There are no numerical continuity bounds determined by $w$ and $a$ alone: the subdivision mask coefficients matter. First check constant reproduction, ${\sum }_j{M}_{ϵ-aj}=1$ for every residue $ϵ$, and convergence. For the usual stationary scalar scheme, two useful calculations are a uniform contraction test and a spectral obstruction test. Higher smoothness is tested with derived difference subdivision masks.

Write $M(z)={\sum }_j{M}_j{z}^j$ and $s_a(z)=1+z+\cdots +z^{a-1}$. Using backward differences $\Delta P_i=P_i-P_{i-1}$, the generating-function identityproves the subdivision difference scheme relation $\Delta S_M=S_{M/s_a}\Delta$. Iterating gives ${\Delta }^r{S}_M=S_{M/s_a^r}{\Delta }^r$. To test a candidate $C^r$ limit, suppose the requisite sum rules hold, so $M/s_a^{r+1}$ is a finite Laurent polynomial. The scaled derivative controls $q_i^{\ell }=a^{r\ell }{\Delta }^r{P}_i^{\ell }$ are refined by the subdivision maskwhose residue sums are one. Their first differences are refined byThe powers of $a$ are essential because each refinement shrinks parameter intervals by $a$.

A lower continuity bound comes from proving uniform contraction. The infinity norm of one difference-refinement step is bounded byIf $q<1$, adjacent derivative controls contract geometrically at every location. More generally, form the $p$-step subdivision maskand test ${\max }_{ϵ\mathrm{mod}{a}^p}{\sum }_j|(B_r^{[p]}{)}_{ϵ-a^{p}j}|<1$. This proves convergence of the constant-preserving derivative scheme to a continuous limit. One justification is to compare consecutive piecewise-linear interpolants: their difference is bounded by a constant times the maximum adjacent-control difference, and the geometrically decreasing bound is summable. To identify the limit as $P^{(r)}$, use fixed-degree Cardinal B-spline interpolants of the original refined controls. Their $r$th derivatives are sums of the scaled finite differences; convergence of both functions and derivatives gives $P\in C^r$. Testing successive $r$ supplies a guaranteed lower bound.

An upper continuity bound comes from an observable noncontracting shape mode. Build finite local subdivision matrices for $B_r$ on a refinement-invariant control neighborhood. For a digit $ϵ$ their entries are $(A_{ϵ}{)}_{ij}=(B_r{)}_{ϵ+i-aj}$, since the new index is $ak+ϵ+i$. Restrict to the realizable difference space and discard modes that do not affect the limit. A word ${ϵ}_1,\dots ,{ϵ}_p$ corresponds to a nested parameter location. If its product has an observable eigenvalue $\lambda$ with $|\lambda |>1$, repetition of this word makes some scaled derivative differences grow rather than tend to zero. For a stable limit representation, $C^r$ regularity would force these differences to vanish; hence general-position data cannot have $C^r$ continuity. A modulus-one mode requires examination of whether it actually persists, rather than an automatic rounding rule.

The two calculations can be organized through the joint spectral radius $\hat{\rho }$ of these restricted matrices:These norm and spectral bounds for subdivision regularity make the test computable from the subdivision mask. The right bound certifies all-word contraction when it is less than one; the left bound detects a repeated-word obstruction when it exceeds one. Longer products improve the tests. A single matrix's spectrum is not a substitute for controlling arbitrary products. The product estimates also appear in Charina's analysis of subdivision regularity; the strict contraction and observable-mode arguments above explain their respective roles.

A norm estimate controls the worst possible refined differences for every choice of the initial controls. Submultiplicativity turns a length-$p$ contraction into a bound for arbitrary long products, so it proves a certain number of continuous derivatives. Its absolute values can lose cancellations; failure of the estimate is inconclusive. It therefore gives a lower bound on smoothness.

Conversely, an observable eigenmode of a repeated local product exhibits a particular sequence of shrinking neighborhoods with inadequate decay. General-position controls have a nonzero coefficient in that mode, so greater derivative continuity is impossible. Special symmetric or polynomial data may suppress the bad mode; this is why the general-position qualification matters. The obstruction gives an upper bound on smoothness, while failure to find an obstruction proves nothing. A lower bound on spectral growth becomes an upper bound on smoothness because large surviving differences correspond to rough behavior.

For example, the binary Cardinal cubic B-spline subdivision mask has symbol $M(z)=(1+z{)}^4/8$, up to an irrelevant index shift. At derivative order $r=2$, the first-difference subdivision mask is $B_2=(1+z)/2$, whose maximum parity sum is $1/2$: the limit is at least $C^2$. At $r=3$, $B_3=1$, and the surviving difference mode does not decay. For generic controls the third derivative is piecewise constant with nonzero jumps, so the subdivision curve is not $C^3$. Here the lower and upper tests determine the exact integer continuity, but they need not coincide for an arbitrary subdivision mask. Stability, sum rules and borderline modes must be handled explicitly before interpreting any numerical estimate as a continuity level.