On Convergent Interpolatory Subdivision Schemes in Riemannian Geometry
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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.
KeywordsInterpolatory subdivision Subdivision in Riemannian geometry Convergence Interpolatory wavelets Hölder continuity
Mathematics Subject Classification41A25 (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.
- 1.Cavaretta, A.S., Dahmen, W., Micchelli, C.A.: Stationary Subdivision, Memoirs of the AMS, vol. 93. American Mathematical Society (1991)Google Scholar
- 5.Donoho, D.L.: Interpolating Wavelet Transforms, Tech. report, Statistics Dep., Stanford (1992)Google Scholar
- 6.Donoho, D.L.: Wavelet-type representation of Lie-valued data. In: 2001, Talk at the IMI “Approximation and Computation” meeting, Charleston (SC) (2001)Google Scholar