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SIA matrices and non-negative stationary subdivision
(2012) Li, Xianjun; Jetter, Kurt
This dissertation is concerned with SIA matrices and non-negative stationary subdivision, and is organized as follows: After an introducing chapter where some basic notation is given we describe, in Chapter 3, how non-negative subdivision is connected to a corresponding non-homogenous Markov process. The family of matrices A, built from the mask of the subdivision scheme, is introduced. Among other results, Lemma 3.1 and Lemma 3.2 relate the coefficients of the iterated masks to matrix products from the family A, and in the limiting case the values of the basic limit function are found from the entries in an infinite product of matrices. Chapter 4 and Chapter 5 are the core of this dissertation. In Chapter 4, we first review some spectral and graph properties of row-stochastic matrices and, in particular, of SIA matrices. We point to the notion of scrambling power, introduced by Hajnal [16], and of the related coefficient of ergodicity. We also consider the directed graph of such matrices, and we improve upon a condition given by Ren and Beard in [30]. Then we study finite families of SIA matrices, the properties of their indicator matrices and the connectivity of their directed graphs. We consider this chapter to be an important contribution to the theory of non-negative subdivision, since it explains the background in order to apply the convergence result of Anthonisse and Tijms [2], which we reprove in Section 4.6, to rank one convergence of infinite products of row stochastic matrices. It does not use the notion of joint spectral radius but the (equivalent) coefficient of ergodicity. Properties equivalent to SIA are listed in Lemma 4.7 and in the subsequent Lemma 4.8; they connect the SIA property to equivalent conditions (scrambling property, positive column property) as they appear in the existing literature dealing with convergence of non-negative subdivision. The fifth chapter of the dissertation contains the full proof of the characterization of uniform convergence for non-negative subdivision, for the univariate and bivariate case, the latter one being a representative for multivariate aspects. It uses the pointwise definition of the limit function at dyadic points - refering to the dyadic expansion of real vectors from the unit cube - using the Anthonisse-Tijms pointwise convergence result, and employs the proper extension of the Micchelli-Prautzsch compatibility condition to the multivariate case, taking care of the ambiguity of representation of dyadic points. As a consequence, the Hölder exponent of the basic limit function can be expressed in terms of the coefficient of ergodicity of the family A. Our convergence theorems, in Theorem 5.1 and Theorem 5.8, include the existing characterizations of uniform convergence for non-negative univariate and bivariate subdivision from the literature, except for the GCD condition, which seems to be a condition applicable to univariate subdivision only. Chapter 5 also reports on some further attempts where we have tried to extend conditions from univariate subdivision, which are sufficient for convergence, to the bivariate case. We could find a bivariate analogue of Melkman's univariate string condition, which we call - in the bivariate case - a rectangular string condition. The chapter concludes with stating the fact that uniform convergence of non-negative stationary subdivision is a property of the support of the mask alone, modulo some apparent necessary conditions such as the sum rules. A typical application of this support property characterizes uniform convergence in the case where the mask is a convex combination of other masks. The dissertation ends with two short chapters on tensor product and box spline subdivision, and an appendix where some definitions and useful lemmas and theorems about matrix and graph theory are stated without proofs.