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Multivariate nonhomogeneous refinement equations. (English) Zbl 0945.42017
Summary: We give necessary and sufficient conditions for the existence and uniqueness of compactly supported distribution solutions $$f= (f_1,\dots, f_r)^T$$ of nonhomogeneous refinement equations of the form $f(x)= h(x)+ \sum_{\alpha\in A} c_\alpha f(2x- \alpha)\quad (x\in \mathbb{R}^s),$ where $$h= (h_1,\dots, h_r)^T$$ is a compactly supported vector-valued multivariate distribution, $$A\subset \mathbb{Z}^s_+$$ has compact support, and the coefficients $$c_\alpha$$ are real-valued $$r\times r$$ matrices. In particular, we find a finite-dimensional matrix $$B$$, constructed from the coefficients $$c_\alpha$$, such that there are precisely as many linearly independent compactly supported distribution solutions to the refinement equation as there are linearly independent vector solutions $${\mathbf q}$$ of the equation $$(I- B){\mathbf q}={\mathbf p}$$, where the vector $${\mathbf p}$$ depends on $$h$$. Our proofs proceed in the time domain and allow us to represent each solution regardless of the spectral radius of $$P(0):= 2^{-s} \sum c_\alpha$$, which has been a difficulty in previous investigations of this nature.

##### MSC:
 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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##### References:
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