In this paper we present a complete description of ideal projectors from the space of bivariate polynomials F[x, y] onto its subspace F<n[x, y] of polynomials of degree less than n. Several applications are given. In particular, we study small perturbations of ideal projectors as well as the limits of Lagrange projectors. The latter results verify one particular case of a conjecture of Carl de ...

The connection between convergence of product integration rules and mean convergence of Lagrange interpolation in L, (1 <p < 00) has been thoroughly analysed by Sloan and Smith [37]. Motivated by this connection, we investigate mean convergence of Lagrange interpolation at the zeros of orthogonal polynomials associated with Freud weights on R. Our results apply to the weights exp(-x”/2), m = 2,...

In the present paper, we consider two kinds of statistics “number of usegments” and “number of internal u-segments” in Dyck paths. More precisely, using Lagrange inversion formula we present the generating function for the number of Dyck paths according to semilength and our new statistics by the partial Bell polynomials, namely, ∑ D∈Dn ∏ i≥1 t αi(D) i = n ∑ i=1 1 (n− i+ 1)!n,i ( 1!t1, 2!t2, · ...

Lagrange interpolation is a classical method for approximating a continuous function by a polynomial that agrees with the function at a number of chosen points (the “nodes”). However, the accuracy of the approximation is greatly influenced by the location of these nodes. Now, a useful way to measure a given set of nodes to determine whether its Lagrange polynomials are likely to provide good ap...

In this paper, a new trend for improvement in semianalytical method based on scale boundaries in order to solve the 2D elastodynamic problems is provided. In this regard, only the boundaries of the problem domain discretization are by specific subparametric elements. Mapping functions are uses as a class of higherorder Lagrange polynomials, special shape functions, Gauss-LobattoLegendre numeric...

It is well known that there exist continuous functions whose Lagrange interpolation polynomials taken at the roots of the Tchebycheff polynomials T„ (x) diverge everywhere in (-1, + 1) .' On the other hand a few years ago S . Bernstein proved the following result' : Let f(x) be any continuous function ; then to every c > 0 there exists a sequence of polynomials ~p„(x) where ~0 ,(x) is of degree...

Bayesian network classifiers are a powerful machine learning tool. In order to evaluate the expressive power of these models, we compute families of polynomials that sign-represent decision functions induced by Bayesian network classifiers. We prove that those families are linear combinations of products of Lagrange basis polynomials. In absence of V -structures in the predictor sub-graph, we a...

The effective refractive index as a function of vacuum wavelength is approximated by Lagrange interpolation polynomials. The rms value of the chromatic dispersion is then calculated analytically. Itis demonstrated that use of fourth-degree polynomials is far more efficient than the use of second-degree polynomials. The rms value of the chromatic dispersion over the wavelength range (1.25 µm, 1....