Matrix analysis
Ten lectures on wavelets
An introduction to wavelets
Some elementary properties of multiresolution analyses of L2(Rn)
Wavelets: a tutorial in theory and applications
Approximation properties of multivariate wavelets
Mathematics of Computation
Approximation with scaled shift-invariant spaces by means of quasi-projection operators
Journal of Approximation Theory
Multiresolution analysis. Haar bases, and self-similar tilings of Rn
IEEE Transactions on Information Theory - Part 2
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We give necessary and sufficient conditions on the Fourier transform of a generator function of a principal shift-invariant subspace of L^2(R^d) providing density order @a. 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(R^d), 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(R^d) 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(R^d).