Bivariate Lagrange Interpolation at the Chebyshev Nodes

We discuss Lagrange interpolation on two sets of nodes in two dimensions where the coordinates of the nodes are Chebyshev points having either the same or opposite parity. We use a formula of Xu for Lagrange polynomials to obtain a general interpolation theorem for bivariate polynomials at either set of Chebyshev nodes. An extra term must be added to the interpolation formula to handle all polynomials with the same degree as the Lagrange polynomials. We express this term as a specifically determined linear combination of canonical polynomials that vanish on the set of Chebyshev nodes being considered. As an application we deduce in an elementary way known minimal and near minimal cubature formulas applying to both the even and the odd Chebyshev nodes. Finally, we restrict to triangular subsets of the Chebyshev nodes to show unisolvence and deduce a Lagrange interpolation formula for bivariate symmetric and skew-symmetric polynomials. This result leads to another proof of the interpolation formula.