Denote by Ln the projection operator obtained by applying the Lagrange interpolation method, weighted by (1 − x), at the zeros of the Chebyshev polynomial of the second kind of degree n + 1. The norm ‖Ln‖ = max ‖f‖∞≤1 ‖Lnf‖∞, where ‖ · ‖∞ denotes the supremum norm on [−1, 1], is known to be asymptotically the same as the minimum possible norm over all choices of interpolation nodes for unweight...

In this paper, we propose a novel and efficient differential quadrature element based on Lagrange interpolation to solve a sixth order partial differential equations encountered in non-classical beam theories. These non-classical theories render displacement, slope and curvature as degrees of freedom for an Euler-Bernoulli beam. A generalize scheme is presented herein to implementation the mult...

The interpolation by polynomials or other functions is a rather old method in applied mathematics. In the paper of M. Gasca and T. Sauer [9] it is mentioned that the word ''interpolation'' has apparently been introduced by J.Wallis as early as 1655. Compared to this, polynomial interpolation in several variables is a relatively new topic and it probably only started in the second half of 19 cen...

If additional smoothness requirements and boundary conditions are met, the well–known approximation orders of scattered data interpolants by radial functions can roughly be doubled.

We present a novel Facial Motion Cloning method relying on the combination of the radial basis functions (RBF) based scattered data interpolation with the encoding capabilities of the MPEG-4 Facial and Body Animation (FBA) international standard. Beside from an initial manual selection of feature points, our method works fully automatically without user interaction. The produced talking head is...

Radial basis functions (RBFs) form a primary tool for multivariate interpolation, and they are also receiving increased attention for solving PDEs on irregular domains. Traditionally, only non-oscillatory radial functions have been considered. We find here that a certain class of oscillatory radial functions (including Gaussians as their limiting case) leads to non-singular interpolants with in...

This article describes an approach based on hierarchical radial basis functions to reconstruction and modification of motion / image data, which are formulated as problems of scattered data interpolation. Assuming little about the data and choosing highly scalable / well adjustable basis functions, the approach can handle both the reconstruction and the manipulation in a unified and widely appl...