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**The structure of finitely generated shift-invariant spaces in \(L_ 2(\mathbb{R}^ d)\).**
*(English)*
Zbl 0806.46030

A simple characterization is given of finitely generated subspaces of \(L_ 2(\mathbb{R}^ d)\) 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, it is proved 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.

### MSC:

46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |

47B38 | Linear operators on function spaces (general) |