Abstract
We give necessary and sufficient conditions for the existence and uniqueness of compactly supported distribution solutionsf=(f1,...,fr)T of nonhomogeneous refinement equations of the form $$f(x) = h(x) + \sum\limits_{\alpha \in A} {c_\alpha f(2x - \alpha )(x \in R^s )} $$ , where h=(h1,...,hr)Tis a compactly supported vector-valued multivariate distribution, A⊂Z+s has compact support, and the coefficientsc α are real-valued r×r matrices. In particular, we find a finite dimensional matrix B, constructed from the coefficientsc α of the equation (I−B)q=p, where the vectorp depends on h. Our proofs proceed in the time domain and allow us to represent each solution regardless of the spectral radius of P(0):=2−s∑c α , which has been a difficulty in previous investigations of this nature.
Published Version
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