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Journal of Operator Theory

Volume 58, Issue 1, Summer 2007  pp. 3-22.

Estimates of the spectral radius of refinement and subdivision operators with isotropic dilations

Authors Victor D. Didenko
Author institution: Department of Mathematics, Universiti Brunei Darussalam, Bandar Seri Begawan, BE1410 Brunei

Summary:  The paper presents lower bounds for the spectral radii of refinement and subdivision operators with continuous matrix symbols and with dilations from a class of isotropic matrices. This class contains the main dilation matrices used in wavelet analysis. After obtaining general formulas, two kinds of estimates for the spectral radii are established: namely, estimates using point values of the symbols, as well as other ones making use of integrals on special subsets of the torus $\mathbb{T}^s$. For some symbol classes the exact value of the spectral radius of the refinement operator is found.

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