Let Pd(C ) denote the space of polynomials of degree at most d in n complex variables. A subset X of C – we will usually speak of configuration or array – is said to be unisolvent for Pd(C ) (or simply unisolvent of degree d) if, for every function f defined on X there exists a unique polynomial P ∈ Pd(C ) such that P(x) = f (x) for every x ∈ X. This polynomial is called the Lagrange interpolat...

We apply multivariate Lagrange interpolation to synthesizing polynomial quantitative loop invariants for probabilistic programs. We reduce the computation of an quantitative loop invariant to solving constraints over program variables and unknown coefficients. Lagrange interpolation allows us to find constraints with less unknown coefficients. Counterexample-guided refinement furthermore genera...

This article explores the development of previous models to determine hubs in a competitive environment. In this paper, by comparing parameters of the ticket price, travel time and the service quality of hub airports, airline hubs are divided into six categories. The degree of importance of travel time and travel cost are determined by a multivariate Lagrange interpolation method, which can pla...

Using the concept of Geometric Weakly Admissible Meshes (see §2 below) together with an algorithm based on the classical QR factorization of matrices, we compute efficient points for discrete multivariate least squares approximation and Lagrange interpolation.

4 Discretization using Polynomial Interpolation Consider a function to be a continuous function defined over and let represent the values of the function at an arbitrary set of points in the domain Another function, say in that assumes values exactly at is called an interpolation function. Most popular form of interpolating functions are polynomials. Polynomial interpolation has many important ...

A classical approach to interpolation of sampled data is polynomial interpolation. However, from the sampling theorem it follows that the ideal approach to interpolation is to convolve the given samples with the sinc function. In this paper we study the properties of the sinc-approximating kernels that can be derived from the Lagrange central interpolation scheme. Both the finite-extent propert...

Let f be analytic on the compact set E ⊂ C , of positive transfinite diameter and let Cr denote the largest equipotential curve of E such that f is analytic within Cr. Generally, the growth of an entire function is measured in terms of its order and type. Here we have established the relations between maximum modulus, maximum term and interpolation error of best uniform approximation to a funct...

We generalize the univariate divided difference to a multivariate setting by considering linear combinations of point evaluations that annihilate the null space of certain differential operators. The relationship between such a linear functional and polynomial interpolation resembles that between the divided difference and Lagrange interpolation. Applying the functional to the shifted multivari...