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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.

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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