How smooth is the smoothest function in a given refinable space? (Q1917553)

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 \(g\in V\).