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Spectral radii of refinement and subdivision operators


Author: Victor Didenko
Journal: Proc. Amer. Math. Soc. 133 (2005), 2335-2346
MSC (2000): Primary 42C40, 47B35, 47B33
DOI: https://doi.org/10.1090/S0002-9939-05-07899-8
Published electronically: March 17, 2005
MathSciNet review: 2138876
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Abstract: The spectral radii of refinement and subdivision operators considered on the space $L_2$ can be estimated by using norms of their symbols. In several cases, including those arising in wavelet analysis, the exact value of the spectral radius is found. For example, if $\mathbb{T} $ is the unit circle and if the symbol $a$ of a refinement operator satisfies the conditions $\vert a(z)\vert^2 + \vert a(-z)\vert^2 = 4 ,{z\in \mathbb{T} }$, and $a(1)=2,$ then the spectral radius of this operator is equal to $\sqrt{2}.$


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Additional Information

Victor Didenko
Affiliation: Department of Mathematics, Universiti Brunei Darussalam, Bandar Seri Begawan, BE1410 Brunei
Email: victor@fos.ubd.edu.bn

DOI: https://doi.org/10.1090/S0002-9939-05-07899-8
Keywords: Spectral radius, subdivision operator, refinement operator
Received by editor(s): September 16, 2002
Received by editor(s) in revised form: November 25, 2003, and December 10, 2003
Published electronically: March 17, 2005
Additional Notes: This research was supported in part by UBD Grant PNC2/2/RG/1(21).
Communicated by: David R. Larson
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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