Refinable shift invariant spaces in R^d

Abstract

Let $\phi: \R^d \longrightarrow \C$ 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 $\Gamma\subset\R^d$ 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 $L=[c_{Ai-j}]_{i,j\in\Gamma}$ and a finite dimensional subspace $\mathcal H$ in the shift invariant space generated by $\phi$. 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 $\phi$ 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 $\phi$, 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.