Spectral factorization of 2-block Toeplitz matrices and refinement equations

Abstract:

Pairs of 2-block Toeplitz $(N\times N)$-matrices $(T_s)_{ij} = p_{2i - j +s -1}$, $s = 0, 1$, $i,j \in \lbrace 1, \ldots , N\rbrace$, are considered for arbitrary sequences of complex coefficients $p_0,\ldots ,p_N$. A complete spectral resolution of the matrices $T_0$, $T_1$ in the system of their common invariant subspaces is obtained. A criterion of nondegeneracy and of irreducibility of these matrices is derived, and their kernels, root subspaces, and all common invariant subspaces are found explicitly. The results are applied to the study of refinement functional equations and also subdivision and cascade approximation algorithms. In particular, the well-known formula for the exponent of regularity of a refinable function is simplified. A factorization theorem that represents solutions of refinement equations by certain convolutions is obtained, along with a characterization of the manifold of smooth refinable functions. The problem of continuity of solutions of the refinement equations with respect to their coefficients is solved. A criterion of convergence of the corresponding cascade algorithms is obtained, and the rate of convergence is computed.