How smooth is the smoothest function in a given re nable space ?

A closed subspace V of L 2 := L 2 (IR 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 2 L 2. Ideally, each function f in such PSI V can be written uniquely as a convergent series f = X 2ZZ d c()(?) with kck`2 kfk L 2. In this case one says that the shifts of 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 reenable in the sense that, for some integer N > 1, the space is a subspace of V. The role of reenable PSI spaces in the construction of wavelets from multiresolution analysis, as well as in the study of subdivision algorithms is well-known, well-understood and well-documented (cf. e.g., D2] and CDM]). The two properties of a reenable PSI space that we compare here are: (s) the smoothness of the \smoothest" non-zero function g 2 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) for every suuciently smooth function f. Here, V j := V (N j). One of the early discoveries in this area was the non-trivial observation that for a reenable PSI space V , (s) and (ao) are connected. For example, a result in M] shows that if decays rapidly and its shifts are L 2-stable, then V provides approximation order k as soon as lies in the Sobolev space W k?1 2. A closely related result appears in CDM]. More recently, the following is proved in R]: Result 1. Let V be an N-reenable PSI space. Then the following conditions are equivalent: (a) V provides approximation order k.