Discrete Polynomial Blending

Abstract: 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.