Refinable shift invariant spaces in R-d
Let phi : R-d --> C be a compactly supported function which satisfies a refinement equation of the form phi(x) = Sigma(k is an element of Lambda) ck phi (Ax - k), c(k) is an element of C, where Gamma subset of R-d is a lattice, Lambda is a finite subset of Gamma, and A is a dilation matrix. We prove, under the hypothesis of linear independence of the Gamma-translates of phi, that there exists a correspondence between the vectors of the Jordan basis of a finite submatrix of L = [c(Ai-j)](i,j is an element of Gamma) and a finite-dimensional subspace H in the shift-invariant space generated by W. 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 phi has accuracy kappa, this basis can be chosen to contain a basis for all the multivaxiate polynomials of degree less than kappa. These latter functions are associated to eigenvalues that are powers of the eigenvalues of A(-1). Furthermore we show that the dimension of H coincides with the local dimension of phi, and hence, every function in the shift-invariant space generated by phi can be written locally as a linear combination of translates of the homogeneous functions.
shift-invariant spaces accuracy homogeneous functions refinable functions