Refinable Shift Invariant Spaces in R^d

SIGRID HEINEKEN

Samsun, Turquía

Conferencia; Sampling Theory and Applications (SampTa05); 2005

Let $varphi: R^d longrightarrow C$ be a compactly supportedfunction which satisfies a refinement equation of the formbegin{equation*}varphi(x) = sum_{kinLambda} c_k varphi(Ax - k),quadc_kinC,end{equation*}where $GammasubsetR^d$ is a lattice, $Lambda$ is a finitesubset of $Gamma$, and $A$ is a dilation matrix. We prove, underthe hypothesis of linear independence of the $Gamma$-translatesof $varphi$, that there exists a correspondence betweenthe vectors of the Jordan basis of afinite submatrix of $L=[c_{Ai-j}]_{i,jinGamma}$and a finite dimensional subspace$mathcal H$ in the shift invariant spacegenerated by $varphi$.We provide a basis of $mathcal H$ and show that its elementssatisfy a property of homogeneity associated to the eigenvalues of$L$.If the function $varphi$ has accuracy $kappa$, this basis can bechosen to containa basis for all the multivariate polynomials of degree less than$kappa$.These latter functions are associated to eigenvalues that are powersofthe 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.