On the density order of the principal shift-invariant subspaces of L2(Rd)L2(Rd)

We give necessary and sufficient conditions on the Fourier transform of a generator function of a principal shift-invariant subspace of L2(Rd)L2(Rd) providing density order αα. 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 L2(Rd)L2(Rd), 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 L2(Rd)L2(Rd) when it is dilated by a diagonalizable expansive linear map AA. 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 AA is dense in L2(Rd)L2(Rd).