Constructive Approximation

, Volume 32, Issue 3, pp 569–596 | Cite as

Stability of Manifold-Valued Subdivision Schemes and Multiscale Transformations

  • Philipp GrohsEmail author


Linear subdivision schemes can be adapted in various ways so as to operate in nonlinear geometries such as Lie groups or Riemannian manifolds. It is well known that along with a linear subdivision scheme a multiscale transformation is defined. Such transformations can also be defined in a nonlinear setting. We show the stability of such nonlinear multiscale transforms. To do this we introduce a new kind of proximity condition which bounds the difference of the differential of a nonlinear subdivision scheme and a linear one. It turns out that—unlike the generic nonlinear case and modulo some minor technical assumptions—in the manifold-valued setting, convergence implies stability of the nonlinear subdivision scheme and associated nonlinear multiscale transformations.


Lipschitz Interpolatory wavelet transform Manifold-valued data Nonlinear subdivision 

Mathematics Subject Classification (2000)

41AXX 41A25 53B 22E 


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Institute of GeometryTU GrazGrazAustria

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