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Constructive Approximation

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

On Convergent Interpolatory Subdivision Schemes in Riemannian Geometry

  • Johannes WallnerEmail author
Article

Abstract

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.

Keywords

Interpolatory subdivision Subdivision in Riemannian geometry Convergence Interpolatory wavelets Hölder continuity 

Mathematics Subject Classification

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

Notes

Acknowledgments

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.

References

  1. 1.
    Cavaretta, A.S., Dahmen, W., Micchelli, C.A.: Stationary Subdivision, Memoirs of the AMS, vol. 93. American Mathematical Society (1991)Google Scholar
  2. 2.
    Choi, S.W., Lee, B.-G., Lee, Y.J., Yoon, J.: Stationary subdivision schemes reproducing polynomials. Comput. Aided Geom. Des. 23, 351–360 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Deslauriers, G., Dubuc, S.: Symmetric iterative interpolation processes. Constr. Approx. 5, 49–68 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    do Carmo, M.P.: Riemannian Geometry. Birkhäuser, Basel (1992)CrossRefzbMATHGoogle Scholar
  5. 5.
    Donoho, D.L.: Interpolating Wavelet Transforms, Tech. report, Statistics Dep., Stanford (1992)Google Scholar
  6. 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
  7. 7.
    Dubuc, S.: Interpolation through an iterative scheme. J. Math. Anal. Appl. 114, 185–204 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Dyn, N., Gregory, J., Levin, D.: A four-point interpolatory subdivision scheme for curve design. Comput. Aided Geom. Des. 4, 257–268 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Dyn, N., Hormann, K.: Geometric conditions for tangent continuity of interpolatory planar subdivision curves. Comput. Aided Geom. Des. 29, 332–347 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Ebner, O.: Convergence of iterative schemes in metric spaces. Proc. Am. Math. Soc 141, 677–686 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Ebner, O.: Stochastic aspects of refinement schemes on metric spaces. SIAM J. Numer. Anal. 52, 717–734 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Grohs, P.: A general proximity analysis of nonlinear subdivision schemes. SIAM J. Math. Anal. 42(2), 729–750 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Grohs, P., Wallner, J.: Interpolatory wavelets for manifold-valued data. Appl. Comput. Harmon. Anal. 27, 325–333 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Grohs, P., Wallner, J.: Definability and stability of multiscale decompositions for manifold-valued data. J. Frankl. Inst. 349, 1648–1664 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Hechler, J., Mößner, B., Reif, U.: \(C^1\)-continuity of the generalized four-point scheme. Linear Algebra Appl. 430, 3019–3029 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Klingenberg, W.: Riemannian Geometry, 2nd edn. de Gruyter, Berlin (1995)CrossRefzbMATHGoogle Scholar
  17. 17.
    Ur Rahman, I., Drori, I., Stodden, V.C., Donoho, D.L., Schröder, P.: Multiscale representations for manifold-valued data. Multiscale Model. Simul. 4, 1201–1232 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Wallner, J., Dyn, N.: Convergence and \(C^1\) analysis of subdivision schemes on manifolds by proximity. Comput. Aided Geom. Des. 22(7), 593–622 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Wallner, J., Nava Yazdani, E., Weinmann, A.: Convergence and smoothness analysis of subdivision rules in Riemannian and symmetric spaces. Adv. Comput. Math. 34, 201–218 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Xie, G., Yu, T.P.-Y.: Smoothness equivalence properties of general manifold-valued data subdivision schemes. Multiscale Model. Simul. 7(3), 1073–1100 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Xie, G., Yu, T.P.-Y.: Smoothness equivalence properties of interpolatory Lie group subdivision schemes. IMA J. Numer. Anal. 30, 731–750 (2010)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institut für GeometrieTechnische Universität GrazGrazAustria

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