A Generalization of the Remainder in Multivariate Polynomial Interpolation (original) (raw)

On Error Formulas for Multivariate Polynomial Interpolation

2007

In this paper we prove that the existence of an error for- mula of a form suggested in (2) leads to some very specific restrictions on an ideal basis that can be used in such formulas. As an applica- tion, we provide a negative answer to one version of the question posed by Carl de Boor (cf. (2)) regarding the existence of certain minimal error formulas for multivariate interpolation.

On Error Formulas for Multivariate Interpolation

In this paper we prove that the existence of an error formula of a form suggested in [2] leads to some very specific restrictions on an ideal basis that can be used in such formulas. As an application, we provide a negative answer to one version of the question posed by Carl de Boor (cf.

On multivariate polynomial interpolation

Constructive Approximation, 1990

We provide a map 7 ! which associates each nite set of points in C s with a polynomial space from which interpolation to arbitrary data given at the points in is possible and uniquely so. Among all polynomial spaces Q from which interpolation at is uniquely possible, our is of smallest degree. It is also Dand scale-invariant. Our map is monotone, thus providing a Newton form for the resulting interpolant. Our map is also continuous within reason, allowing us to interpret certain cases of coalescence as Hermite interpolation. In fact, our map can be extended to the case where, with each 2 , there is associated a polynomial space P , and, for given smooth f, a polynomial q 2 Q is sought for which p(D)(f ? q)() = 0; all p 2 P ; 2 : We obtain as the \scaled limit at the origin" (exp) # of the exponential space exp with frequencies , and base our results on a study of the map H 7 ! H # de ned on subspaces H of the space of functions analytic at the origin. This study also allows us to determine the local approximation order from such H and provides an algorithm for the construction of H # from any basis for H.

On non-existence of certain error formulas for ideal interpolation

Journal of Approximation Theory, 2010

In this paper we prove that the existence of an error formula of a form suggested in [2] leads to some very specific restrictions on an ideal basis that can be used in such formulas. As an application, we provide a negative answer to one version of the question posed by Carl de Boor (cf.

Topics in Polynomial Interpolation Theory

This thesis studies two aspects of polynomial interpolation theory. The first part sets forth explicit formulas for the coefficients of polynomial interpolants to implicit functions. A formula for the higher-order derivatives of implicit functions appears as a limiting case of these formulas. The second part delves into certain configurations of points in space -generalized principal lattices -that are well suited as interpolation points. Applying the theory of real algebraic curves then allows the construction of many new examples of such configurations.

On the history of multivariate polynomial interpolation

Journal of Computational and Applied Mathematics, 2000

Analysis, which has received and continues receiving not deep but constant attention. In this short survey, we review its development in the ÿrst 75 years of this century, including a pioneering paper by Kronecker in the 19th century.

On the Trivariate Polynomial Interpolation

2012

Abstract: This paper is concerned with the formulae for computing the coefficients of the trivariate polynomial interpolation (TPI) passing through ) 1 )( 1 )( 1 ( + + + r n m distinct points in the solid rectangular region. The TPI is formulated as a matrix equation using Kronecker product and Khatri-Rao product of the matrices and the coefficients of the TPI are computed using the generalized inverse of a matrix. In addition, the closed formulae of the coefficients of the bivariate and univariate polynomial interpolations are obtained by the use of the inverse of the Vandermonde matrix. It is seen that the trivariate polynomial interpolation can be investigated as the matrix equation and the coefficients of the TPI can be computed directly from the solution of the matrix equation. Also, it is shown that the bivariate polynomial interpolation (BPI) is the special case of the TPI when 0 = r . Numerical examples are represented.