Partition of unity and density: A counterexample

By providing a counterexample we show that there exists a shift-invariant spaceS generated by a piecewise linear function such that the union of the corresponding scaled spacesS ^h (h>0) is dense inC ₀(R ²) butS does not contain a stable and locally supported partition of unity. This settles a question raised by C. de Boor and R. DeVore a decade ago.