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The spectral radius of matrix continuous refinement operators

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Abstract

A simple analytic formula for the spectral radius of matrix continuous refinement operators is established. On the space \(L_2^m({{\mathbb R}}^s)\), m ≥ 1 and s ≥ 1, their spectral radius is equal to the maximal eigenvalue in magnitude of a number matrix, obtained from the dilation matrix M and the matrix function c defining the corresponding refinement operator. A similar representation is valid for the continuous refinement operators considered on spaces L p for p ∈ [1, ∞ ), p ≠ 2. However, additional restrictions on the kernel c are imposed in this case.

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Authors and Affiliations

  1. Department of Mathematics, Universiti Brunei Darussalam, Bandar Seri Begawan, BE1410, Brunei

    Victor Didenko & Wee Ping Yeo

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  1. Victor Didenko
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  2. Wee Ping Yeo
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Correspondence to Victor Didenko.

Additional information

Communicated by Tim Goodman.

This research was supported in part by the Universiti Brunei Darussalam, Grant PNC2/2/RG/1(72)

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Didenko, V., Yeo, W.P. The spectral radius of matrix continuous refinement operators. Adv Comput Math 33, 113–127 (2010). https://doi.org/10.1007/s10444-009-9124-9

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Mathematics Subject Classifications (2000)