Compactly Supported Distributional Solutions of Nonstationary Nonhomogeneous Refinement Equations

Let A be a matrix with the absolute values of all eigenvalues strictly larger than one, and let Z 0 be a subset of Z such than nZ 0 implies n * 1 Z 0. 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 G n and H n , nZ 0, in D, define the corresponding nonstationary nonhomogeneous refinement equation n=Hnn*1(A)*GnforallnZ0, ((*)) where n , nZ 0, 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 n , nZ 0, of the equation (*). In fact, we reduce the existence problem to finding a bounded solution Fn of the linear equations \ifmmode\expandafter\else\expandafter\\fiFnSn\ifmmode\expan dafter\else\expandafter\\fiFn*1=\ifmmode\expandafter\else\expandafter\\f iGnforallnZ0, where the matrices S n and the vectors Gn, nZ 0, can be constructed explicitly from H n and G n respectively. The results above are still new even for stationary nonhomogeneous refinement equations.