Polynomial Approximation on Quasi-Uniform Grids

A simple generalization of univariate polynomial interpolation and approximation to the bivariate setting is by tensor products. Here, one can easily construct a polynomial interpolant (in Lagange or Newton form) or quasi-interpolant (using Bernstein polynomials and dual functionals) over uniform bivariate grids, and compute the error in approximation based on the error in univariate polynomial interpolation. In this talk we focus on interpolation and approximation over certain ``quasi-uniform'' grids.  These are grids much sparser than full tensor product grids.  In particular, we will construct a quasi-interpolant on these grids, and derive an error of approximation. In particular, we show that for certain quasi-uniform grids we can achieve the same rate of approximation as by interpolation on the full tensor product grid.  Hence, we have a class of ``serendipity'' elements analogous to those in the finite element literature.  The construction involves a technique called discrete blending based on Boolean sum methods for interpolation and approximation.  We conclude with a brief mention of the construction of a subdivision scheme on quasi-uniform grids.