Thomas Sauer

On multivariate Lagrange interpolation

Lagrange interpolation by polynomials in several variables is studied through a finite difference approach. We establish an interpolation formula analogous to that of Newton and a remainder formula, both of them in terms of finite differences. We prove that the finite difference admits an integral representation involving simplex spline functions. In particular, this provides a remainder formula for Lagrange interpolation of degree n of a function f, which is a sum of integrals of certain (n + 1)st directional derivatives of f multiplied by simplex spline functions. We also provide two algorithms for the computation of Lagrange interpolants which use only addition, scalar multiplication, and point evaluation of polynomials.

On the history of multivariate polynomial interpolation

Multivariate polynomial interpolation is a basic and fundamental subject in Approximation Theory and Numerical Analysis, which has received and continues receiving not deep but constant attention. In this short survey, we review its development in the first 75 years of this century, including a pioneering paper by Kronecker in the 19th century.

Polynomial interpolation in several variables

This is a survey of the main results on multivariate polynomial interpolation in the last twenty-five years, a period of time when the subject experienced its most rapid development. The problem is considered from two different points of view: the construction of data points which allow unique interpolation for given interpolation spaces as well as the converse. In addition, one section is devoted to error formulas and another to connections with computer algebra. An extensive list of references is also included.

H-bases for polynomial interpolation and system solving

The H-basis concept allows, similarly to the Gröbner basis concept, a reformulation of nonlinear problems in terms of linear algebra. We exhibit parallels of the two concepts, show properties of H-bases, discuss their construction and uniqueness questions, and prove that n polynomials in n variables are, under mild conditions, already H-bases. We apply H-bases to the solution of polynomial systems by the eigenmethod and to multivariate interpolation.

ON VECTOR SUBDIVISION

Abstract. In this paper we give a complete characterization of the convergence of stationary vector subdivision schemes and the regularity of the associated limit function. These results extend and complete our earlier work on vector subdivision and its use in the construction of multiwavelets.

Gröbner Bases and Applications: Polynomial interpolation of Minimal Degree and Gröbner Bases

This paper investigates polynomial interpolation with respect to a finite set of appropriate linear functionals and the close relations to the Grobner basis of the associated finite dimensional ideal.

Gröbner bases, H–bases and interpolation

The paper is concerned with a construction for H-bases of polynomial ideals without relying on term orders. The main ingredient is a homogeneous reduction algorithm which orthogonalizes leading terms instead of completely canceling them. This allows for an extension of Buchbergers algorithm to construct these H-bases algorithmically. In addition, the close connection of this approach to minimal degree interpolation, and in particular to the least interpolation scheme due to de Boor and Ron, is pointed out.

On multivariate Hermite interpolation

We study the problem of Hermite interpolation by polynomials in several variables. A very general definition of Hermite interpolation is adopted which consists of interpolation of consecutive chains of directional derivatives. We discuss the structure and some aspects of poisedness of the Hermite interpolation problem; using the notion of blockwise structure which we introduced in [10], we establish an interpolation formula analogous to that of Newton in one variable and use it to derive an integral remainder formula for a regular Hermite interpolation problem. For Hermite interpolation of degreen of a functionf, the remainder formula is a sum of integrals of certain (n + 1)st directional derivatives off multiplied by simplex spline functions.

Computational aspects of multivariate polynomial interpolation

The paper is concerned with the practical implementation of two methods to compute the solution of polynomial interpolation problems. In addition to a description of the implementation, practical results and several improvements will be discussed, focusing on speed and robustness of the algorithms under consideration.

Multivariate Bernstein polynomials and convexity

Abstract It is well-known that in two or more variables Bernstein polynomials do not preserve convexity. Here we present two variations, one stronger than the classical notion, the other one weaker, which are preserved and do coincide with classical convexity in the univariate case. Moreover, it will be shown that even the weaker notion is sufficient for the monotonicity of successive Bernstein polynomials, strengthening the well-known result that monotonicity holds for classically convex functions.