## Subsequence convergence in subdivision

HTML articles powered by AMS MathViewer

- by Deter de Wet PDF
- Math. Comp.
**80**(2011), 973-994 Request permission

## Abstract:

We study the phenomenon that regularly spaced subsequences of the control points in subdivision may converge to scalar multiples of the same limit function, even though subdivision itself is divergent. We present different sets of easily checkable sufficient conditions for this phenomenon (which we term subsequence convergence) to occur, study the basic properties of subsequence convergence, show how certain results from subdivision carry over to this case, show an application for decorative effects, and use our results to build nested sets of refinement masks, which provide some insight into the structure of the set of refinable functions. All our results are formulated for a general integer dilation factor.## References

- Lothar Berg and Gerlind Plonka,
*Compactly supported solutions of two-scale difference equations*, Proceedings of the Sixth Conference of the International Linear Algebra Society (Chemnitz, 1996), 1998, pp. 49–75. MR**1628382**, DOI 10.1016/S0024-3795(97)10012-X - Lothar Berg and Gerlind Plonka,
*Some notes on two-scale difference equations*, Functional equations and inequalities, Math. Appl., vol. 518, Kluwer Acad. Publ., Dordrecht, 2000, pp. 7–29. MR**1792070** - 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 - W. Dahmen and C. A. Micchelli,
*Biorthogonal wavelet expansions*, Constr. Approx.**13**(1997), no. 3, 293–328. MR**1451708**, DOI 10.1007/s003659900045 - Ingrid Daubechies and Jeffrey C. Lagarias,
*Two-scale difference equations. I. Existence and global regularity of solutions*, SIAM J. Math. Anal.**22**(1991), no. 5, 1388–1410. MR**1112515**, DOI 10.1137/0522089 - W.d.V. de Wet,
*On the analysis of refinable functions with respect to mask factorisation, regularity and corresponding subdivision convergence.*, Ph.D. thesis, University of Stellenbosch, 2007. - Vera Latour, Jürgen Müller, and Werner Nickel,
*Stationary subdivision for general scaling matrices*, Math. Z.**227**(1998), no. 4, 645–661. MR**1621951**, DOI 10.1007/PL00004397 - Wayne Lawton, S. L. Lee, and Zuowei Shen,
*Characterization of compactly supported refinable splines*, Adv. Comput. Math.**3**(1995), no. 1-2, 137–145. MR**1314906**, DOI 10.1007/BF03028364 - Charles A. Micchelli,
*Mathematical aspects of geometric modeling*, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 65, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. MR**1308048**, DOI 10.1137/1.9781611970067 - Marian Neamtu,
*Convergence of subdivision versus solvability of refinement equations*, East J. Approx.**5**(1999), no. 2, 183–210. MR**1705396** - Thomas Sauer,
*Differentiability of multivariate refinable functions and factorization*, Adv. Comput. Math.**25**(2006), no. 1-3, 211–235. MR**2231702**, DOI 10.1007/s10444-004-7635-y

## Additional Information

**Deter de Wet**- Affiliation: Department of Mathematical Sciences, Mathematics Division, Private Bag X1, Matieland 7602, South Africa
- Received by editor(s): April 16, 2008
- Received by editor(s) in revised form: April 29, 2009
- Published electronically: October 18, 2010
- © Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**80**(2011), 973-994 - MSC (2010): Primary 65D10, 65D17, 41A99
- DOI: https://doi.org/10.1090/S0025-5718-2010-02380-4
- MathSciNet review: 2772104