For n 1, let fxjngnj=1 be n distinct points and let Ln[ ] denote the corresponding Lagrange Interpolation operator. Let W : R ! [0;1). What conditions on the array fxjng1 j n; n 1 ensure the existence of p > 0 such that lim n!1 k (f Ln[f ])W b kLp(R)= 0 for every continuous f : R ! Rwith suitably restricted growth, and some “weighting factor” ? We obtain a necessary and su¢ cient condition for ...

Let be an arbitrary regular triangulation of a simply connected compact polygonal domain R 2 and let S 1 q (() denote the space of bivariate polynomial splines of degree q and smoothness 1 with respect to. We develop an algorithm for constructing point sets admissible for Lagrange interpolation by S 1 q (() if q 4. In the case q = 4 it may be necessary to slightly modify , but only if exeptiona...

Throughout this study, we continue the analysis of a recently found out Gibbs–Wilbraham phenomenon, being related to behavior Lagrange interpolation polynomials continuous absolute value function. Our study establishes error polynomial interpolants function |x| on [−1,1], using Chebyshev and Chebyshev–Lobatto nodal systems with an even number points. Moreover, respect odd cases, relevant change...

We develop a fast discrete algorithm for computing the sparse Fourier expansion of a function of d dimension. For this purpose, we introduce a sparse multiscale Lagrange interpolation method for the function. Using this interpolation method, we then design a quadrature scheme for evaluating the Fourier coefficients of the sparse Fourier expansion. This leads to a fast discrete algorithm for com...

Given a finite set {(x~, Yi)} of ordered pairs from X × Y where X, Y are Hilbert spaces over the same field, there are numerous techniques for constructing a function, f, on X to Y such that f(xi) = y¢ • However, when X, 2" have a causality structure and f must be causal then the data interpolation problem is much more complicated. In this paper two interpolation methods, namely linear interpol...

This report is concerned with the computation cost of the determinant of a (bivariate) polynomial matrix required in the guaranteed accuracy L∞-norm computation. The obtained computation cost is in terms of word operations, unlike most results available in the literature where the computation cost is provided in terms of arithmetic operations. The proposed method employs multivariate Lagrange i...

In this paper, using the Newton's formula of Lagrange interpolation, we present a new proof of the anisotropic error bounds for Lagrange interpolation of any order on the triangle, rectangle, tetrahedron and cube in a unified way. It is known that the polynomial interpolations are the foundations of construction the finite elements and the interpolation error estimates play a key role in derivi...

A simple algorithm for the construction of the unique Hermite interpolating polynomial (in the special case, the Lagrange interpolating polynomial) is given. The interpolation matches preassigned data of function and consecutive derivatives on a set of points laying on several radial rays. This algorithm is realized in the software package Mathematica.

We describe an algorithm for constructing point sets which admit unique Lagrange and Hermite interpolation from the space S 1 3 (() of C 1 splines of degree 3 deened on a general class of triangulations. The triangulations consist of nested polygons whose vertices are connected by line segments. In particular, we have to determine the dimension of S 1 3 (() which is not known for arbitrary tria...

Given a set of scattered data, we usually use a minimal energy method to find Lagrange interpolation based on bivariate spline spaces over a triangulation of the scattered data locations. It is known that the approximation order of the minimal energy spline interpolation is only 2 in terms of the size of triangulation. To improve this order of approximation, we propose several new schemes in th...