Subdivision schemes for iterated function systems

Authors:
Charles A. Micchelli, Thomas Sauer and Yuesheng Xu

Journal:
Proc. Amer. Math. Soc. **129** (2001), 1861-1872

MSC (2000):
Primary 54C40, 14E20; Secondary 46E25, 20C20

Published electronically:
December 7, 2000

MathSciNet review:
1814120

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We identify iterated function systems and regular Borel measures such that the matrix subdivision process relative to a finite family converges if and only if satisfies certain spectral properties.

**1.**Zhongying Chen, Charles A. Micchelli, and Yuesheng Xu,*A construction of interpolating wavelets on invariant sets*, Math. Comp.**68**(1999), no. 228, 1569–1587. MR**1651746**, 10.1090/S0025-5718-99-01110-2**2.**John E. Hutchinson,*Fractals and self-similarity*, Indiana Univ. Math. J.**30**(1981), no. 5, 713–747. MR**625600**, 10.1512/iumj.1981.30.30055**3.**Rong Qing Jia,*Subdivision schemes in 𝐿_{𝑝} spaces*, Adv. Comput. Math.**3**(1995), no. 4, 309–341. MR**1339166**, 10.1007/BF03028366**4.**Charles A. Micchelli and Hartmut Prautzsch,*Uniform refinement of curves*, Linear Algebra Appl.**114/115**(1989), 841–870. MR**986909**, 10.1016/0024-3795(89)90495-3**5.**Charles A. Micchelli, Thomas Sauer, and Yuesheng Xu,*A construction of refinable sets for interpolating wavelets*, Results Math.**34**(1998), no. 3-4, 359–372. MR**1652740**, 10.1007/BF03322061**6.**Charles A. Micchelli and Yuesheng Xu,*Using the matrix refinement equation for the construction of wavelets on invariant sets*, Appl. Comput. Harmon. Anal.**1**(1994), no. 4, 391–401. MR**1310661**, 10.1006/acha.1994.1024**7.**Charles A. Micchelli and Yuesheng Xu,*Reconstruction and decomposition algorithms for biorthogonal multiwavelets*, Multidimens. Systems Signal Process.**8**(1997), no. 1-2, 31–69. MR**1433530**, 10.1023/A:1008264805830**8.**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****9.**K. R. Parthasarathy,*Probability measures on metric spaces*, Probability and Mathematical Statistics, No. 3, Academic Press, Inc., New York-London, 1967. MR**0226684**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
54C40,
14E20,
46E25,
20C20

Retrieve articles in all journals with MSC (2000): 54C40, 14E20, 46E25, 20C20

Additional Information

**Charles A. Micchelli**

Affiliation:
Department of Mathematics & Statistics, State University of New York, The University at Albany, Albany, New York 12222

Email:
cam@math.albany.edu,cam@watson.ibm.com

**Thomas Sauer**

Affiliation:
Mathematisches Institut, Universität Erlangen–Nürnberg, Bismarckstr. 1 $\frac12$, D–91054 Erlangen, Germany

Email:
sauer@mi.uni-erlangen.de

**Yuesheng Xu**

Affiliation:
Department of Mathematics, North Dakota State University, Fargo, North Dakota 58105 and Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China

Email:
xu@hypatia.math.ndsu.nodak.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-00-05966-9

Keywords:
Subdivision,
iterated function systems,
spectral radius

Received by editor(s):
September 27, 1999

Published electronically:
December 7, 2000

Additional Notes:
The first author was supported by National Science Foundation under grants DMS–9504780 and DMS–9973427, by the Alexander von Humboldt foundation and by the Wavelets Strategic Research Programme, National University of Singapore, under a grant from the National Science and Technology Board and the Ministry of Education, Singapore

The second author was supported by Deutsche Forschungsgemeinschaft with a Heisenberg fellowship, Sa 627/6

The third author was supported by National Science Foundation under grants DMS–9504780 and DMS–9973427, by the Alexander von Humboldt Foundation and by the “One Hundred Outstanding Young Chinese Scientists” program of the Chinese Academy of Sciences

Communicated by:
David R. Larson

Article copyright:
© Copyright 2000
American Mathematical Society