Stability of Nonlinear Subdivision and Multiscale Transforms

Abstract

Extending upon the work of Cohen, Dyn, and Matei (Appl. Comput. Harmon. Anal. 15:89–116, 2003) and of Amat and Liandrat (Appl. Comput. Harmon. Anal. 18:198–206, 2005), we present a new general sufficient condition for the Lipschitz stability of nonlinear subdivision schemes and multiscale transforms in the univariate case. It covers the special cases (weighted essentially nonoscillatory scheme, piecewise polynomial harmonic transform) considered so far but also implies the stability in some new cases (median interpolating transform, power-p schemes, etc.). Although the investigation concentrates on multiscale transforms $$\bigl\{v^0,d^1,\ldots,d^J\bigr\}\longmapsto v^J,\quad J\ge1,$$ in l ∞(ℤ) given by a stationary recursion of the form $$v^{j}=Sv^{j-1}+d^{j},\quad j\ge1,$$ involving a nonlinear subdivision operator S acting on l ∞(ℤ), the approach is extendable to other nonlinear multiscale transforms and norms, as well.