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...

We estimate the distribution function of a Lagrange interpolation polynomial and deduce mean boundedness in Lp; p < 1: 1 The Result There is a vast literature on mean convergence of Lagrange interpolation, see [4{ 8] for recent references. In this note, we use distribution functions to investigate mean convergence. We believe the simplicity of the approach merits attention. Recall that if g : R...

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...

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...

In this paper,we deeply research Lagrange interpolation of n-variables and give an application of Cayley-Bacharach theorem for it. We pose the concept of sufficient intersection about s(1 ≤ s ≤ n) algebraic hypersurfaces in n-dimensional complex Euclidean space and discuss the Lagrange interpolation along the algebraic manifold of sufficient intersection. By means of some theorems ( such as Bez...

This paper gives an ecient numerical method for solving the nonlinear systemof Volterra-Fredholm integral equations. A Legendre-spectral method based onthe Legendre integration Gauss points and Lagrange interpolation is proposedto convert the nonlinear integral equations to a nonlinear system of equationswhere the solution leads to the values of unknown functions at collocationpoints.

The authors give a procedure to construct extended interpolation formulae and prove some uniform convergence theorems.

Bounds are proved for the Stieltjes polynomial En+1, and lower bounds are proved for the distances of consecutive zeros of the Stieltjes polynomials and the Legendre polynomials Pn. This sharpens a known interlacing result of Szegö. As a byproduct, bounds are obtained for the Geronimus polynomials Gn. Applying these results, convergence theorems are proved for the Lagrange interpolation process...

In the present paper, both the perfect convergence for the Lagrange interpolation of analytic functions on [ − 1, 1] and the perfect convergence for the trigono-metric interpolation of analytic functions on [ − p, p] with period 2p are discussed.