How smooth is the smoothest function in a given refinable space? (English) Zbl 0901.46024

Cohen, Albert; Daubechies, Ingrid; Ron, Amos

Appl. Comput. Harmon. Anal. 3, No. 1, 87-89 (1996).

A closed subspace \(V\) of \(L_2:= L_2(\mathbb{R}^d)\) is called principal shift-invariant (abbrev. PSI) if it is the smallest space that contains all shifts (i.e. integer translates) of some function \(\Phi\in L_2\). A PSI space is refinable in the sense that, for some integer \(N>1\), the space \(\{f(\cdot/N): f\in V\}\) is a subspace of \(V\). It provides approximation order \(k\) of \(\text{dist} (f,V_j)= O(N^{-jk})\) for every sufficiently smooth function \(f\). Here \(V_j:= V(N^j)\). The result of this paper is: the last condition does not imply the smoothness of the “smoothest” nonzero function \(g\in V\).

Reviewer: A.Smajdor (Katowice)

MSC:

46E30

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

Keywords:

smoothness; refinable spaces; approximation order; principal shift-invariant