Book Review: Birkhoff interpolation

On Multivariate Birkhoff Rational 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( √...

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 ...

Book Review

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Innovative Methodologies for Secure Multimedia Communications Using Birkhoff Polynomial Interpolation

Security of multimedia communications is regarded by both academy and industry research as one of the most important and urgent problems of the last decades. In order to improve existing solutions, we propose a framework based on Birkhoff polynomial interpolation. Such new mathematical tool based on this theory improves the classical secret sharing scheme based on Lagrange interpolation widely ...