The structure of finitely generated shift - invariant spaces in L

A simple characterization is given of finitely generated subspaces of L2(IR) which are invariant under translation by any (multi)integer, and 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. AMS (MOS) Subject Classifications: primary: 41A25, 41A63, 46C99; secondary: 41A30, 41A15, 42B99, 46E20.