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A conjecture on multivariate polynomial interpolation.

Jesús Miguel Carnicer, Mariano Gasca (2001)


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. Estos últimos autores conjeturaron que toda configuración de la primera clase es de la segunda, y probaron que el recíproco no es cierto....

A note on the rate of convergence for Chebyshev-Lobatto and Radau systems

Elías Berriochoa, Alicia Cachafeiro, Jaime Díaz, Eduardo Martínez (2016)

Open Mathematics

This paper is devoted to Hermite interpolation with Chebyshev-Lobatto and Chebyshev-Radau nodal points. The aim of this piece of work is to establish the rate of convergence for some types of smooth functions. Although the rate of convergence is similar to that of Lagrange interpolation, taking into account the asymptotic constants that we obtain, the use of this method is justified and it is very suitable when we dispose of the appropriate information.

A particular smooth interpolation that generates splines

Segeth, Karel (2017)

Programs and Algorithms of Numerical Mathematics

There are two grounds the spline theory stems from – the algebraic one (where splines are understood as piecewise smooth functions satisfying some continuity conditions) and the variational one (where splines are obtained via minimization of some quadratic functionals with constraints). We use the general variational approach called 𝑠𝑚𝑜𝑜𝑡ℎ𝑖𝑛𝑡𝑒𝑟𝑝𝑜𝑙𝑎𝑡𝑖𝑜𝑛 introduced by Talmi and Gilat and show that it covers not only the cubic spline and its 2D and 3D analogues but also the well known tension spline (called also spline...

A priori error estimates for Lagrange interpolation on triangles

Kenta Kobayashi, Takuya Tsuchiya (2015)

Applications of Mathematics

We present the error analysis of Lagrange interpolation on triangles. A new a priori error estimate is derived in which the bound is expressed in terms of the diameter and circumradius of a triangle. No geometric conditions on triangles are imposed in order to get this type of error estimates. To derive the new error estimate, we make use of the two key observations. The first is that squeezing a right isosceles triangle perpendicularly does not reduce the approximation property of Lagrange interpolation....

A weighted empirical interpolation method: a priori convergence analysis and applications

Peng Chen, Alfio Quarteroni, Gianluigi Rozza (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We extend the classical empirical interpolation method [M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Compt. Rend. Math. Anal. Num. 339 (2004) 667–672] to a weighted empirical interpolation method in order to approximate nonlinear parametric functions with weighted parameters, e.g. random variables obeying various probability distributions. A priori convergence analysis...

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