Abstract
a(α)ϕ( Mx � α), where ϕ is the unknown function defined on the s-dimensional Euclidean space R s ,a is a finitely supported nonnegative sequence on Z s , and M is an s× s dilation matrix with m := |detM|. We characterize the existence of L2-solution of refinement equation in terms of spectral radius of a certain finite matrix or transition operator associated with refinement mask a and dilation matrix M. For s =1 and M =2 , the sufficient and necessary conditions are obtained to characterize the existence of continuous solution of this refinement equation.
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