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- W2009979968 abstract "Let φ : ℝ d → ℂ be a compactly supported function which satisfies a refinement equation of the form [Formula: see text] where Γ ⊂ ℝ d is a lattice, Λ is a finite subset of Γ, and A is a dilation matrix. We prove, under the hypothesis of linear independence of the Γ-translates of φ, that there exists a correspondence between the vectors of the Jordan basis of a finite submatrix of L = [c Ai-j ] i,j∈Γ and a finite-dimensional subspace [Formula: see text] in the shift-invariant space generated by φ. We provide a basis of [Formula: see text] and show that its elements satisfy a property of homogeneity associated to the eigenvalues of L. If the function φ has accuracy κ, this basis can be chosen to contain a basis for all the multivariate polynomials of degree less than κ. These latter functions are associated to eigenvalues that are powers of the eigenvalues of A -1 . Furthermore we show that the dimension of [Formula: see text] coincides with the local dimension of φ, and hence, every function in the shift-invariant space generated by φ can be written locally as a linear combination of translates of the homogeneous functions." @default.
- W2009979968 created "2016-06-24" @default.
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- W2009979968 date "2005-09-01" @default.
- W2009979968 modified "2023-09-23" @default.
- W2009979968 title "REFINABLE SHIFT INVARIANT SPACES IN ℝd" @default.
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- W2009979968 doi "https://doi.org/10.1142/s0219691305000889" @default.
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