On L p-error of bivariate polynomial interpolation (original) (raw)

On Lp-error of bivariate polynomial interpolation on the square

Journal of Approximation Theory, 2018

We obtain estimates of the Lp-error of the bivariate polynomial interpolation on the Lissajous-Chebyshev node points for wide classes of functions including non-smooth functions of bounded variation in the sense of Hardy-Krause. The results show that Lperrors of polynomial interpolation on the Lissajous-Chebyshev nodes have almost the same behavior as the polynomial interpolation in the case of the tensor product Chebyshev grid.

A unifying theory for multivariate polynomial interpolation on general Lissajous-Chebyshev nodes

arXiv: Numerical Analysis, 2017

The goal of this article is to provide a general multivariate framework that synthesizes well-known non-tensorial polnomial interpolation schemes on the Padua points, the Morrow-Patterson-Xu points and the Lissajous node points into a single unified theory. The interpolation nodes of these schemes are special cases of the general Lissajous-Chebyshev points studied in this article. We will characterize these Lissajous-Chebyshev points in terms of Lissajous curves and Chebyshev varieties and derive a general discrete orthogonality structure related to these points. This discrete orthogonality is used as the key for the proof of the uniqueness of the polynomial interpolation and the derivation of a quadrature rule on these node sets. Finally, we give an efficient scheme to compute the polynomial interpolants.

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.

Bivariate Lagrange Interpolation at the Chebyshev Nodes

2010

We discuss Lagrange interpolation on two sets of nodes in two dimensions where the coordinates of the nodes are Chebyshev points having either the same or opposite parity. We use a formula of Xu for Lagrange polynomials to obtain a general interpolation theorem for bivariate polynomials at either set of Chebyshev nodes. An extra term must be added to the interpolation formula to handle all polynomials with the same degree as the Lagrange polynomials. We express this term as a specifically determined linear combination of canonical polynomials that vanish on the set of Chebyshev nodes being considered. As an application we deduce in an elementary way known minimal and near minimal cubature formulas applying to both the even and the odd Chebyshev nodes. Finally, we restrict to triangular subsets of the Chebyshev nodes to show unisolvence and deduce a Lagrange interpolation formula for bivariate symmetric and skew-symmetric polynomials. This result leads to another proof of the interpolation formula.

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 Chung and Yao's geometric characterization for bivariate polynomial interpolation

2002

Interpolation problems on sets of points in the plane satisfying Chung and Yao's geometric characterization give rise to La- grange interpolation formulae in the space of polynomials of degree not greater than n. A conjecture on these sets states that there exists a line containing n + 1 nodes. It has only been proved for degree • 4. In this

Some Recent Advances in Multivariate Polynomial Interpolation

AIP Conference Proceedings, 2007

The generalization of Lagrange and Newton univariate interpolation formulae is one of the topics of multivariate polynomial interpolation. Two classes of geometric configurations of points in the plane, suitable for the use of those formulas, were given by Chung and Yao in 1978 for the Lagrange formula, and by Gasca and Maeztu in 1982 for the Newton formula. The latter authors conjectured that every configuration of the first class belongs to the second class and proved that the converse is not true. In 1990 J. R. Busch proved the conjecture for polynomials of degree not greater than 4, showing the difficulty of extending his reasoning to higher degree. In this paper we prove the same result using different arguments with similar difficulties, in the hope that these arguments could shed more light to the problem. Una conjetura sobre interpolación polinómica en varias variables Resumen. La generalización de las fórmulas de interpolación de Lagrange y Newton a varias variables es uno de los temas habituales de estudio en interpolación polinómica. Dos clases de configuraciones geométricas particularmente interesantes en el plano fueron obtenidas por Chung y Yao en 1978 para la fórmula de Lagrange y por Gasca y Maeztu en 1982 para la de Newton. Estosúltimos autores conjeturaron que toda configuración de la primera clase es de la segunda, y probaron que el recíproco no es cierto. En 1990 J. R. Busch probó la conjetura para polinomios de grado no mayor que 4, viendo la dificultad de extender su razonamiento a grado superior. En este trabajo damos otra demostración del mismo resultado con otros argumentos que muestran similar dificultad pero ofrecen alguna esperanza de generalización.

On the filtered polynomial interpolation at Chebyshev nodes

arXiv (Cornell University), 2020

The paper deals with a special filtered approximation method, which originates interpolation polynomials at Chebyshev zeros by using de la Vallée Poussin filters. These polynomials can be an useful device for many theoretical and applicative problems since they combine the advantages of the classical Lagrange interpolation, with the uniform convergence in spaces of locally continuous functions equipped with suitable, Jacobi-weighted, uniform norms. The uniform boundedness of the related Lebesgue constants, which equals to the uniform convergence and is missing from Lagrange interpolation, has been already proved in literature under different, but only sufficient, assumptions. Here, we state the necessary and sufficient conditions to get it. These conditions are easy to check since they are simple inequalities on the exponents of the Jacobi weight defining the norm. Moreover, they are necessary and sufficient to get filtered interpolating polynomials with a near best approximation error, which tends to zero as the number n of nodes tends to infinity. In addition, the convergence rate is comparable with the error of best polynomial approximation of degree n, hence the approximation order improves with the smoothness of the sought function. Several numerical experiments are given in order to test the theoretical results, to make a comparison with the Lagrange interpolation at the same nodes and to show how the Gibbs phenomenon can be strongly reduced.

Uniform weighted approximation on the square by polynomial interpolation at Chebyshev nodes

Applied Mathematics and Computation, 2020

The paper deals with de la Vallée Poussin type interpolation on the square at tensor product Chebyshev zeros of the first kind. The approximation is studied in the space of locally continuous functions with possible algebraic singularities on the boundary, equipped with weighted uniform norms. In particular, simple necessary and sufficient conditions are proved for the uniform boundedness of the related Lebesgue constants. Error estimates in some Sobolev-type spaces are also given. Pros and cons of such a kind of filtered interpolation are analyzed in comparison with the Lagrange polynomials interpolating at the same Chebyshev grid or at the equal number of Padua nodes. The advantages in reducing the Gibbs phenomenon are shown by means of some numerical experiments.