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multivariate B-splines %

The {\bf simplex spline} was the first multivariate B-spline to be studied. A bivariate, quadratic simplex spline appeared in a letter %Schoenberg66c of Schoenberg to P. Davis. A formal definition was first given in the B-spline survey %Boor76h, as the function $$%
M_\gs\Rd\to\RR: x\mapsto \vol_{n-d}(\gs\cap \inv{P}(x))/\vol_n(\gs), $$%
with $\gs$ a simplex in $\Rn$, and $P:\Rn\to\Rd:x\mapsto (x_1,\ldots,x_d,0,\ldots,0)$. %

Micchelli %Micchelli80b, see also, %Micchelli79a used instead the much more effective definition of $M_\gs$ as the {\it distribution} $$%
M_\gs :f\mapsto \int_\gs f\circ P/\vol_n(\gs), $$%
and used it very effectively to established recurrence relations and other properties of $M_\gs$. %

Following this lead, de Boor and H\"ollig %BoorHollig82a defined the distribution $$%
M_B :f\mapsto \int_B f\circ P $$%
for an arbitrary polyhedral body in $\Rn$ and an arbitrary affine map $P:\Rn\to\Rd$. See % ``A formula for $M_B$''(qv) % for a precise definition of $\int_B$ as well as a careful derivation of a formula for $M_B$ as a function on $\flat(PB)$. \bye