Simultaneous approximation, interpolation and Birkhoff systems

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( √...

On the Simultaneous Approximation of a Function and Its Derivatives by Sums of Birkhoff Type*

be a given linear differential system of the nth order subject to the following hypotheses: (i) the functions P2 , • • • , Pn are continuous and have continuous derivatives of all orders on (0, 1) ; (ii) the boundary conditions, consisting of n linearly independent linear equations involving w(0), u{\), (fe=0, 1, • • • , n — 1), are regular ;f (iii) X = 0 is not a characteristic value, so that ...

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 ...

Multilevel Interpolation and Approximation

Interpolation by translates of a given radial basis function (RBF) has become a well-recognized means of tting functions sampled at scattered sites in R d. A major drawback of these methods is their inability to interpolate very large data sets in a numerically stable way while maintaining a good t. To circumvent this problem, a multilevel interpolation (ML) method for scattered data was presen...