  In this paper we study "discrete polynomial blending," a term used to define
a certain discretized version of curve blending whereby one approximates from
the "sum of tensor product polynomial spaces" over certain grids. Our strategy
is to combine the theory of Boolean Sum methods with dual bases connected to
the Bernstein basis to construct a new quasi-interpolant for discrete blending.
Our blended element has geometric properties similar to that of the
Bernstein-B\'ezier tensor product surface patch, and rates of approximation
that are comparable with those obtained in tensor product polynomial
approximation.
