Bivariate Lagrange interpolation at the checkerboard nodes

TL;DR

This paper derives a general explicit formula for bivariate Lagrange basis polynomials at checkerboard nodes, extending previous results for various special node sets, and proves their uniqueness in a specific polynomial quotient space.

Contribution

It introduces a unified explicit formula for bivariate Lagrange basis polynomials at checkerboard nodes and establishes their uniqueness within a polynomial quotient space.

Findings

Explicit formula for bivariate Lagrange basis polynomials

Generalization of known basis polynomials at special nodes

Proof of uniqueness in a polynomial quotient space

Abstract

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 by the subspace of bivariate vanishing polynomials.