Subdivision difference scheme ($\Delta S_M=S_B\Delta ,B(z)=M(z)/(1+z+\cdots +z^{a-1})$)

For backward differences $\Delta P_i=P_i-P_{i-1}$, a scalar stationary arity-$a$ subdivision mask has symbol $M(z)={\sum }_i{M}_i{z}^i$. The identity $(1-z)M(z)=B(z)(1-z^a)$ gives the displayed intertwining whenever the quotient is a finite Laurent polynomial. Iterating supplies higher-difference masks. The coefficients $a^{r\ell }{\Delta }^r{P}_i^{\ell }$ are candidate $r$th-derivative controls. A convergent derivative scheme, proved by contraction of its first differences, establishes corresponding smoothness of the original limit under the standard consistent parameterization.

Finite local subdivision matrices for the scaled difference scheme track derivative variations in nested parameter intervals. A product-norm bound below one proves uniform contraction for every interval and therefore gives a lower bound on derivative continuity. A product eigenvalue of modulus at least one is an obstruction when its mode is reachable from the data and visible in the limit: generic data excite it, so the corresponding derivative differences fail to vanish. Harmless polynomial or unobservable modes must be removed before applying such an upper bound. Borderline smoothness needs direct analysis, not rounding a numerical exponent. Stability and convergence of the underlying scheme must be checked rather than inferred from mask width.