On Birkhoff interpolation: Free Birkhoff nodes

Birkhoff Interpolation with Rectangular Sets of Nodes

Although it is important both in theory as well as in applications, a theory of Birkhoff interpolation with main emphasis on the shape of the set of nodes is still missing. Here we explain the problem, and then we concentrate on one of the simplest shapes (“rectangular”) in the case of bivariate uniforme Birkhoff interpolation schemes. The ultimate goal is to obtain a geometrical understanding ...

On Multivariate Birkhoff Rational Interpolation

Uniform Birkhoff Interpolation with Rectangular Sets of Nodes

In this paper we initiate the study of Birkhoff interpolation schemes with main emphasis on the shape of the set of nodes. We concentrate here on the simplest shapes (“rectangular”). The ultimate goal is to obtain a geometrical understanding of the solvability. We present several regularity criteria, a few conjectures, with geometric interpretations, and we illustrate with many examples.

Lower Bounds by Birkhoff Interpolation

In this paper we give lower bounds for the representation of real univariate polynomials as sums of powers of degree 1 polynomials. We present two families of polynomials of degree d such that the number of powers that are required in such a representation must be at least of order d. This is clearly optimal up to a constant factor. Previous lower bounds for this problem were only of order Ω( √...

Interpolation approximations based on Gauss-Lobatto-Legendre-Birkhoff quadrature

We derive in this paper the asymptotic estimates of the nodes and weights of the Gauss-Lobatto-Legendre-Birkhoff (GLLB) quadrature formula, and obtain optimal error estimates for the associated GLLB interpolation in Jacobi weighted Sobolev spaces. We also present a useroriented implementation of the pseudospectral methods based on the GLLB quadrature nodes for Neumann problems. This approach al...