Abstract: Stable locally supported bases are constructed for the spaces ${\cal S}_d^r(\triangle)$ of polynomial splines of degree $d\ge 3r+2$ and smoothness $r$ defined on triangulations $\triangle$, as well as for various superspline subspaces. In addition, we show that for $r\ge 1$, it is impossible to construct bases which are simultaneously stable and locally linearly independent.

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