The Structure of Finitely Generated Shift-Invariant Spaces in L2( [formula omitted]d)

A simple characterization is given of finitely generated subspaces of L 2( R d ) which are invariant under translation by any (multi)integer, and is used to give conditions under which such a space has a particularly nice generating set, namely a basis, and, more than that, a basis with desirable properties, such as stability, orthogonality, or linear independence. The last property makes sense only for "local" spaces, i.e., shift-invariant spaces generated by finitely many compactly supported functions, and special attention is paid to such spaces. As an application, we prove that the approximation order provided by a given local space is already provided by the shift-invariant space generated by just one function, with this function constructible as a finite linear combination of the finite generating set for the whole space, hence compactly supported. This settles a question of some 20 years′ standing.