Space Of Bivariate Polynomials Research Articles

Bivariate Lagrange interpolation at the checkerboard nodes

In this paper, we derive an explicit formula for the bivariate Lagrange basis polynomials of a general set of checkerboard nodes. This formula generalizes existing results of bivariate Lagrange basis polynomials at the Padua nodes, Chebyshev nodes, Morrow-Patterson nodes, and Geronimus nodes. We also construct a subspace spanned by linearly independent bivariate vanishing polynomials that vanish at the checkerboard nodes and prove the uniqueness of the set of bivariate Lagrange basis polynomials in the quotient space defined as the space of bivariate polynomials with a certain degree over the subspace of bivariate vanishing polynomials.

Bivariate ideal projectors and their perturbations

In this paper we present a complete description of ideal projectors from the space of bivariate polynomials $\mathbb{F}[x,y]$ onto its subspace $\mathbb{F }_{<n}[x,y]$ of polynomials of degree less than n. Several applications are given. In particular, we study small perturbations of ideal projectors as well as the limits of Lagrange projectors. The latter results verify one particular case of a conjecture of Carl de Boor.