Discrete Polynomial Blending

TL;DR

This paper introduces a novel discrete polynomial blending method that combines Boolean Sum techniques with Bernstein basis duals to create a quasi-interpolant with Bernstein-like geometric properties and tensor product approximation rates.

Contribution

It develops a new quasi-interpolant for discrete polynomial blending using Boolean Sum methods and Bernstein basis duals, enhancing geometric and approximation properties.

Findings

The blended element exhibits Bernstein-like geometric properties.

Achieves approximation rates comparable to tensor product polynomial approximation.

Provides a new framework for discrete curve 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.