Convergence of refinement schemes on metric spaces
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Abstract:
We analyze the convergence of iterative refinement processes on metric spaces, imposing the principle of contractivity to obtain convergence criteria. As a major result, we show that on Hadamard spaces a wide natural class of contractible barycentric subdivision schemes converges.References
- Werner Ballmann, Lectures on spaces of nonpositive curvature, DMV Seminar, vol. 25, Birkhäuser Verlag, Basel, 1995. With an appendix by Misha Brin. MR 1377265, DOI 10.1007/978-3-0348-9240-7
- Alfred S. Cavaretta, Wolfgang Dahmen, and Charles A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 93 (1991), no. 453, vi+186. MR 1079033, DOI 10.1090/memo/0453
- Jonathan Dahl, Steiner problems in optimal transport, Trans. Amer. Math. Soc. 363 (2011), no. 4, 1805–1819. MR 2746666, DOI 10.1090/S0002-9947-2010-05065-2
- G. Derfel, N. Dyn, and D. Levin, Generalized refinement equations and subdivision processes, J. Approx. Theory 80 (1995), no. 2, 272–297. MR 1315413, DOI 10.1006/jath.1995.1019
- Philipp Grohs, A general proximity analysis of nonlinear subdivision schemes, SIAM J. Math. Anal. 42 (2010), no. 2, 729–750. MR 2607928, DOI 10.1137/09075963X
- Serge Lang, Fundamentals of differential geometry, Graduate Texts in Mathematics, vol. 191, Springer-Verlag, New York, 1999. MR 1666820, DOI 10.1007/978-1-4612-0541-8
- Charles A. Micchelli, Interpolatory subdivision schemes and wavelets, J. Approx. Theory 86 (1996), no. 1, 41–71. MR 1397613, DOI 10.1006/jath.1996.0054
- Karl-Theodor Sturm, Probability measures on metric spaces of nonpositive curvature, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002) Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp. 357–390. MR 2039961, DOI 10.1090/conm/338/06080
- L. N. Vasershtein, Markov processes over denumerable products of spaces describing large system of automata, Problems Inform. Transmission 5 (1969), no. 3, 47–52. MR 314115, DOI 10.1016/s0016-0032(33)90010-1
- Johannes Wallner, Esfandiar Nava Yazdani, and Andreas Weinmann, Convergence and smoothness analysis of subdivision rules in Riemannian and symmetric spaces, Adv. Comput. Math. 34 (2011), no. 2, 201–218. MR 2762953, DOI 10.1007/s10444-010-9150-7
- Xinlong Zhou, On multivariate subdivision schemes with nonnegative finite masks, Proc. Amer. Math. Soc. 134 (2006), no. 3, 859–869. MR 2180904, DOI 10.1090/S0002-9939-05-08118-9
Additional Information
- Oliver Ebner
- Affiliation: Institute of Geometry, TU Graz, Kopernikusgasse 24/IV, A-8010 Graz, Austria
- Email: o.ebner@tugraz.at
- Received by editor(s): March 25, 2011
- Received by editor(s) in revised form: June 30, 2011
- Published electronically: June 7, 2012
- Additional Notes: The author was supported by the Austrian science fund, grant W1230.
- Communicated by: Walter Van Assche
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 677-686
- MSC (2010): Primary 53C23, 65D17
- DOI: https://doi.org/10.1090/S0002-9939-2012-11331-0
- MathSciNet review: 2996972