Compactly supported distributional solutions of nonstationary nonhomogeneous refinement equations

Let A be a matrix with the absolute values of all eigenvalues strictly larg er than one, and let Z(0) be a subset of Z such that n is an element of Z(0 ) implies n + 1 is an element of Z(0). Denote the space of all compactly su pported distributions by D', and the usual convolution between two compactl y supported distributions f and g by f * g. For any bounded sequence G(n) a nd H-n, n is an element of Z(0), in D', define the corresponding nonstation ary nonhomsgeneous refinement equation Phi (n) = H-n* Phi (n+1)(A.) + G(n) for all n is an element of Z(0), (*) where Phi (n), n is an element of Z(0), is in a bounded set of D'. The nons tationary nonhmogeneous 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 comp actly supported distributional solutions Phi (n), n is an element of Z(0), of the equation (*). In fact, we reduce the existence problem to finding a bounded solution (F) over tilde (n) of the linear equations (F) over tilde (n) - S-n(F) over tilde (n+1) = (G) over tilde (n) for all n is an element of Z(0), where the matrices S-n and the vectors (G) over tilde (n), n is an element of Z(0), can be constructed explicitly from H-n and G(n) respectively. The results above are still new even for stationary nonhomogeneous refinement e quations.