**Abstract:** Let $\Delta$ be a triangulation of some polygonal domain
$\Omega\subset\RR^2$ and let $S_q^r(\Delta)$ denote the space of all bivariate
polynomial splines of smoothness $r$ and degree $q$ with respect to $\Delta$.
We develop the first Hermite type interpolation scheme for $S_q^r(\Delta)$,
$q\ge 3r+2$, whose approximation error is bounded above by $Kh^{q+1}$,
where $h$ is the maximal diameter of the triangles in $\Delta$, and the
constant $K$ only depends on the smallest angle of the triangulation and
is independent of near-degenerate edges and near-singular vertices. Moreover,
the fundamental functions of our scheme are minimally supported and form
a locally linearly independent basis for a superspline subspace of $S_q^r(\Delta)$.
This shows that the optimal approximation order can be achieved by using
minimally supported splines. Our method of proof is completely different
from the quasi-interpolation techniques for the study of the approximation
power of bivariate splines developed in [7,18].

**Preprint version available:** pdf

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