Compactly Supported Distributional Solutions of Nonstationary Nonhomogeneous Refinement Equations

Abstract

Let A be a matrix with the absolute values of all eigenvalues strictly larger than one, and let Zo be a subset of Z such that n ∈ Z0 implies n + 1 ∈ Z0. Denote the space of all compactly supported distributions by D′, and the usual convolution between two compactly supported distributions f and g by f* g. For any bounded sequence Gn and Hn, n ∈ Z0, in D′, define the corresponding nonstationary nonhomogeneous refinement equation Φn = Hn * Φn+1 (A ·) + Gn for all n ∈ Z0, (*) where Φ n,n ∈ Z0, is in a bounded set of D′. The nonstationary nonhomogeneous refinement equation (*) arises in the construction of wavelets on bounded domain, multiwavelets, and of biorthogonal wavelets on nonuniform meshes. In this paper, we study the existence problem of compactly supported distributional solutions Φn,n ∈ Zo, of the equation (*). In fact, we reduce the existence problem to finding a bounded solution F̃n of the linear equations F̃n - SnF̃n+1 = G̃n for all n ∈ Z0, where the matrices Sn and the vectors G̃n, n ∈ Zo, can be constructed explicitly from Hn and Gn respectively. The results above are still new even for stationary nonhomogeneous refinement equations.