Refinable shift invariant spaces in Rᵈ

الأصل · EN

Let ϕ: ᵈ be a compactly supported function which satisfies a refinement equation of the form ϕ(x) = ∑ₖ∈Λ cₖ ϕ(Ax - k), cₖ∈, where Γ⊂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=[cAi-j]ᵢ,ⱼ∈ᵧ and a finite dimensional subspace H in the shift invariant space generated by ϕ. We provide a basis of H 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. Further we show that the dimension of H 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.