On the density order of the principal shift-invariant subspaces of $L^2\left({\mathbb{R}}^d\right)$

Abstract

We give necessary and sufficient conditions on the Fourier transform of a generator function of a principal shift-invariant subspace of $L^2\left({\mathbb{R}}^d\right)$ providing density order $\alpha$. Our starting point is a paper by de Boor, DeVore, and Ron [C. de Boor, R.A. DeVore, A. Ron, Approximation from shift-invariant subspaces of $L^2\left({\mathbb{R}}^d\right)$, Trans. Amer. Math. Soc. 341 (2) (1994) 787–806] and the new conditions that we present here involve the classical notion of the point of approximate continuity. In addition, we study properties of the approximation of a shift-invariant subspace of $L^2\left({\mathbb{R}}^d\right)$ when it is dilated by a diagonalizable expansive linear map $A$. Indeed, we present a necessary and sufficient condition on a principal shift-invariant subspace such that its union with itself dilated by integer powers of $A$ is dense in $L^2\left({\mathbb{R}}^d\right)$.