Constructive Approximation

, Volume 40, Issue 3, pp 473–486 | Cite as

On Convergent Interpolatory Subdivision Schemes in Riemannian Geometry

  • Johannes WallnerEmail author


We show the convergence (for all input data) of refinement rules in Riemannian manifolds which are analogous to the linear four-point scheme and similar univariate interpolatory schemes, and which are generalized to the Riemannian setting by the so-called log/exp analogy. For this purpose, we use a lemma on the Hölder regularity of limits of contractive refinement schemes in metric spaces. In combination with earlier results on smoothness of limits, we settle the question of existence of interpolatory refinement rules intrinsic to Riemannian geometry which have \(C^r\) limits for all input data, for \(r \le 3\). We further establish well-definedness of the reconstruction procedure of “interpolatory” multiscale transforms intrinsic to Riemannian geometry.


Interpolatory subdivision Subdivision in Riemannian geometry Convergence Interpolatory wavelets Hölder continuity 

Mathematics Subject Classification

41A25 (primary) 65T60 42C40 (secondary) 



This research has been performed within the framework of the Austrian Science Foundation’s DK+ program Discrete Mathematics (FWF Grant No. W1230). The author thanks the anonymous reviewers for their comments.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institut für GeometrieTechnische Universität GrazGrazAustria

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