Refinable shift invariant spaces in R^d

Let $\varphi :{\mathbb{R}}^d⟶\text{C}$ be a compactly supported function which satisfies a refinement equation of the form $\phi(x) = \sum_{k\in\Lambda} c_k \phi(Ax - k),\quad c_k\in\C$,where$\Gamma\subset\R^d$isalattice,$\Lambda$isafinitesubsetof$\Gamma$,and$A$ is a dilation matrix. We prove, under the hypothesis of linear independence of the $\mathrm{\Gamma }$-translates of $\varphi$, that there exists a correspondence between the vectors of the Jordan basis of a finite submatrix of $L=[c_{Ai-j}{]}_{i,j\in \mathrm{\Gamma }}$ and a finite dimensional subspace $\mathcal{H}$ in the shift invariant space generated by $\varphi$. 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 $\varphi$ has accuracy $\kappa$, this basis can be chosen to contain a basis for all the multivariate polynomials of degree less than $\kappa$. 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 $\varphi$, and hence, every function in the shift invariant space generated by $\varphi$ can be written locally as a linear combination of translates of the homogeneous functions.