Abstract: Given a nested sequence of triangulations $\triangle_0,\triangle_1,\ldots,\triangle_n,\ldots$ of a polygonal domain $\Omega$, we construct for any $r\ge1$, $d\ge 4r+1$, locally stable bases for some spaces ${\cSw}_d^r(\triangle_0)\subset{\cSw}_d^r(\triangle_1)\subset\cdots \subset{\cSw}_d^r(\triangle_n)\subset\cdots$ of bivariate polynomial splines of smoothness $r$ and degree $d$. In particular, the bases are stable and locally linearly independent simultaneously.

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