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A simple characterization is given of finitely generated subspaces of L 2 (IRd) 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.

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A complete characterization is given of closed shift-invariant subspaces of) , Lt( ' whllE-r6ve-a-specified-approx-imation order-. vWh-en such a space is prii-l-(i.e:.generated by a single function), then this characterization is in terms of the Fourier transform of the generator. As a special case, we obtain the classical Strang-Fix conditions, but without requiring the generating function to decay at infinity. The approximation order of a general closed shift-invariant space is shown to be already realized by a specifiable principal subspace.,-AMS (MOS) Subject Classifications: 41A25, 41A63; 41A30, 41A15, 42B99, 46E30

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We provide a map | e which associates each finite set | in complex s-space with a polynomial space ~r e from which interpolation to arbitrary data given at the points in O is possible and uniquely so. Among all polynomial spaces Q from which interpolation at O is uniquely possible, our 9 re is of smallest degree. It is also D-and scale-invariant. Our map is monotone, thus providing a Newton form for the resulting interpolant. Our map is also continuous within reason, allowing us to interpret certain cases of coalescence as Hermite interpolation. In fact, our map can be extended to the ease where, with each 0 e 6, there is associated a polynomial space Po, and, for given smooth f, a polynomial q ~ Q is sought for whichWe obtain ~'o as the "'sealed limit at the origin" (expo)~ of the exponential space expo with frequencies O, and base our results on a study of the map H ~ HS defined on subspaces H of the space of functions analytic at the origin. This study also allows us to determine the local approximation order from such H and provides an algorithm for the construction of H~ from any basis for H.