Refinable Shift Invariant Spaces in R^n

CABRELLI, CARLOS A.; HEINEKEN, SIGRID; MOLTER, URSULA M.

International Journal of Wavelets, Multiresolution and Information Processing

World Scientific

Año: 2005 vol. 3 p. 321 - 345

0219-6913

Abstract. Let φ : Rn −→ C be a compactly supported function which satis- fies a refinement equation of the form φ(x) = Xk∈Λ ck φ(Ax − k), ck ∈ C, where Γ ⊂ R n 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.