A closed subspace V of L 2 := L 2 (R d ) is called PSI (principal shift-invariant) if it is the smallest space that contains all the shifts (i.e., integer translates) of some function Ο β L 2 . Ideally, each function f in such PSI V can be written uniquely as a convergent series
In this case one says that the shifts of Ο form a Riesz basis or that they are L 2 -stable; this is, in particular, the case when these shifts form an orthonormal set.
We are interested here in PSI spaces which are refinable in the sense that, for some integer N > 1, the space
The role of refinable PSI spaces in the construction of wavelets from multiresolution analysis, as well as in the study of subdivision algorithms is wellknown, well-understood, and well-documented (cf. e.g., [6,4]). The two properties of a refinable PSI space that we compare here are:
(s) the smoothness of the "smoothest" nonzero function g β V.
(ao) the approximation orders provided by V. This latter notion refers to the decay of the error when approximating smooth functions from dilations of V; roughly speaking, V provides approximation order k if dist(f, V j ) = O(N -jk