Constructive Approximation

, Volume 31, Issue 3, pp 359–393 | Cite as

Stability of Nonlinear Subdivision and Multiscale Transforms

  • S. Harizanov
  • P. Oswald


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 (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 (ℤ), the approach is extendable to other nonlinear multiscale transforms and norms, as well.


Nonlinear subdivision and multiscale transforms Lipschitz stability Finite differences Derived subdivision schemes Spectral radius conditions 

Mathematics Subject Classification (2000)

65D15 65D17 65T50 26A16 


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Jacobs University, SESBremenGermany

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