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How smooth is the smoothest function in a given refinable space? (English) Zbl 0901.46024
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 $$g\in V$$.

##### MSC:
 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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