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Spectral radii and eigenvalues of subdivision operators

Author: Di-Rong Chen
Journal: Proc. Amer. Math. Soc. 132 (2004), 1113-1123
MSC (2000): Primary 42C15, 47B06
Published electronically: October 9, 2003
MathSciNet review: 2045428
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Abstract: This paper discusses the spectra of matrix subdivision operators. We establish some formulas for spectral radii of subdivision operators on various invariant subspaces in $\ell _{p}$. A formula for the spectral radius of a subdivision operator, in terms of the moduli of eigenvalues, is derived under a mild condition. The results are even new in the scalar case. In this case, we show that the subdivision operator has no eigenvector in $\ell _{p}$ if the corresponding subdivision scheme converges for some $p\in [1, \infty )$.

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  • 1. A. S. Cavaretta, W. Dahmen, and C. A. Micchelli, Stationary subdivision, Memoirs Amer. Math. Soc. 93 (1991), 1-186. MR 92h:65017
  • 2. D. R. Chen, Algebraic properties of subdivision operators with matrix mask and their applications, J. Approx. Theory 97 (1999), 294-310. MR 2000a:41025
  • 3. D. R. Chen, Construction of smooth refinable function vectors by cascade algorithms, SIAM J. Numer. Anal. 40 (2002), 1354-1368.
  • 4. I. Daubechies and J. Lagarias, Two-scale difference equations I. Existence and global regularity of solutions, SIAM J. Math. Anal. 22 (1991), 1388-1410. MR 92d:39001
  • 5. T. N. T. Goodman, C. A. Micchelli, and J. D. Ward, Spectral radius formulas for subdivision operators, Recent Advances in Wavelet Analysis (L. L. Schumaker and G. Webb, eds.), Academic Press, Boston, MA, 1994, pp. 335-360. MR 94m:47076
  • 6. Mark C. Ho, Spectra of slant Toeplitz operators with continuous symbols, Michigan Math. J. 44 (1997), 157-164. MR 98c:47034
  • 7. R. Q. Jia, Subdivision schemes in $L_{p}$spaces, Adv. Comput. Math. 3 (1995), 309-341. MR 96d:65028
  • 8. R. Q. Jia, S. D. Riemenschneider, and D. X. Zhou, Approximation by multiple refinable functions, Canadian J. Math. 49 (1997), 944-962. MR 99f:39036
  • 9. K. S. Lau and J. Wang, Characterization of $L^{p}$-solutions for two-scale dilation equations, SIAM J. Math. Anal. 26 (1995), 1018-1046. MR 96f:39004
  • 10. C. A. Micchelli and T. Sauer, Regularity of multiwavelets, Adv. Comput. Math. 7 (1997), 455-545. MR 99d:42067
  • 11. C. A. Micchelli and T. Sauer, On vector subdivision, Math. Z. 229 (1998), 621-674. MR 2000d:42016
  • 12. G.-C. Rota and G. Strang, A note on the joint spectral radius, Indag. Math. 22 (1960), 379-381. MR 26:5434
  • 13. Y. Wang, Two-scale dilation equations and the mean spectral radius, Random and Computational Dynamics 4 (1996), 49-72. MR 96j:42023
  • 14. D. X. Zhou, The $p$-norm joint spectral radius for even integers, Methods and Applications of Analysis 5 (1998), 39-54. MR 99e:42054
  • 15. D. X. Zhou, Spectra of subdivision operators, Proc. Amer. Math. Soc. 129 (2001), 191-202. MR 2001h:47049

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Additional Information

Di-Rong Chen
Affiliation: Department of Applied Mathematics, Beijing University of Aeronautics, Astronautics, Beijing 100083, China; Department of Mathematics, Hubei Institute for Nationalities, Enshi 445000, Hubei, China

Keywords: Subdivision operator, spectral radius, joint spectral radius
Received by editor(s): February 21, 2001
Received by editor(s) in revised form: December 12, 2002
Published electronically: October 9, 2003
Additional Notes: Supported in part by NSF of China under Grant 10171007 and City University of Hong Kong under Grant 7001442
Communicated by: David R. Larson
Article copyright: © Copyright 2003 American Mathematical Society