Refinable shift invariant spaces in R^d
| dc.creator | Cabrelli, Carlos | |
| dc.creator | Heineken, Sigrid | |
| dc.creator | Molter, Ursula | |
| dc.date | 2005-11-16 | |
| dc.date.accessioned | 2026-07-07T06:51:21Z | |
| dc.date.available | 2026-07-07T06:51:21Z | |
| dc.description | 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. | |
| dc.description | 21 pages, 3 figures | |
| dc.identifier | https://arxiv.org/abs/math/0511421 | |
| dc.identifier | http://arxiv.org/abs/math/0511421 | |
| dc.identifier | International Journal of Wavelets, Multiresolution and Information Processing, vol. 3 (2005), no. 3, 321-345 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/104956 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.title | Refinable shift invariant spaces in R^d | |
| dc.type | text |