Refinable shift invariant spaces in R^d

Abstract

Let $ϕ: \R^d \longrightarrow \C$ be a compactly supported function which satisfies a refinement equation of the form $ϕ(x) = \sum_{k\inΛ} c_k ϕ(Ax - k),\quad c_k\in\C$, where $Γ\subset\R^d$ 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 $L=[c_{Ai-j}]_{i,j\inΓ}$ and a finite dimensional subspace $\mathcal H$ in the shift invariant space generated by $ϕ$. We provide a basis of $\mathcal H$ 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 $\mathcal H$ 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.