Math @ Duke

Publications [#287132] of Ingrid Daubechies
Papers Published
 Daubechies, I; Lagarias, JC, Sets of matrices all infinite products of which converge,
Linear Algebra and Its Applications, vol. 161 no. C
(January, 1992),
pp. 227263, ISSN 00243795 [doi]
(last updated on 2019/08/21)
Abstract: An infinite product ∏∞i=1Mi of matrices converges (on the right) if limi→∞ M1 ... Mi exists. A set ∑={Ai:i≥1}of n x n matrices is called an RCP set (right convergent product set) if all infinite products with each element drawn from ∑ converge. Such sets of matrices arise in constructing selfsimilar objects like von Koch's snowflake curve, in various interpolation schemes, in constructing wavelets of compact support, and in studying nonhomogeneous Markov chains. This paper gives necessary conditions and also some sufficient conditions for a set ∑ to be an RCP set. These are conditions on the eigenvalues and left eigenspaces of matrices in ∑ and finite products of these matrices. Necessary and sufficient conditions are given for a finite set ∑ to be an RCP set having a limit function M∑(d)=π∞i=1Adi , where d=(d1,...,dn,...), which is a continuous function on the space of all sequences d with the sequence topology. Finite RCP sets of columnstochastic matrices are completely characterized. Some results are given on the problem of algorithmically deciding if a given set ∑ is an RCP set. © 1992.


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