Lower bounds by Birkhoff interpolation

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( √...

Lower complexity bounds for interpolation algorithms

We introduce and discuss a new computational model for Hermite–Lagrange interpolation by nonlinear classes of polynomial interpolants. We distinguish between an interpolation problem and an algorithm that solves it. Our model includes also coalescence phenomena and captures a large variety of known Lagrange-Hermite interpolation problems and algorithms. Like in traditional Hermite–Lagrange inte...

On Multivariate Birkhoff Rational Interpolation

Hermite Birkhoff interpolation of scattered data by combined Shepard operators

Methods approaching the problem of the Hermite Birkhoff interpolation of scattered data by combining Shepard operators with local interpolating polynomials are not new in literature [1–4]. In [3] combinations of Shepard operators with bivariate Hermite-Birkhoff local interpolating polynomials are introduced to increase the algebraic degree of precision (polynomial reproduction degree) of Shepar...

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 ...