Newton basis for multivariate Birkhoff interpolation

On Multivariate Birkhoff Rational Interpolation

A new non-polynomial solution to multivariate Hermite-Birkhoff interpolation

A new solution to the multivariate Hermite-Birkhoff interpolation problem is presented. The classical approach to this problem consists in constructing the minimum degree polynomial, which coincides with the prescribed function and derivative values at the sample points. Here the interpolant is represented as a truncated Multipoint Taylor (MT) series. A MT series can be regarded as an extension...

Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis

The purpose of this paper is twofold. Firstly, we provide explicit and compact formulas for computing both Caputo and (modified) Riemann-Liouville (RL) fractional pseudospectral differentiation matrices (F-PSDMs) of any order at general Jacobi-Gauss-Lobatto (JGL) points. We show that in the Caputo case, it suffices to compute F-PSDM of order μ ∈ (0, 1) to compute that of any order k + μ with in...

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( √...

Polya Conditions for Multivariate Birkhoff Interpolation: from General to Rectangular Sets of Nodes

Polya conditions are certain algebraic inequalities that regular Birkhoff interpolation schemes must satisfy, and they are useful in deciding if a given scheme is regular or not. Here we review the classical Polya condition and then we show how it can be strengthened in the case of rectangular nodes.