On summability of weighted Lagrange interpolation. I

The paper is devoted to the study of summability of weighted Lagrange interpolation on the roots of orthogonal polynomials with respect to a weight function w. Starting from the Lagrange interpolation polynomials we shall construct a wide class of discrete processes (using summations) which are uniformly convergent in a suitable Banach space (C,·) of continuous functions (w denotes a weight). We shall give such conditions with respect to w, , (C,·) and to summation methods for which the uniform convergence holds. Error estimates for the approximation will also be considered.     

On summability of weighted Lagrange interpolation. III

Starting from the Lagrange interpolation on the roots of Jacobi polynomials, a wide class of discrete linear processes is constructed using summations. Some special cases are also considered, such as the Fejér, de la Vallée Poussin, Cesàro, Riesz and Rogosinski summations. The aim of this note is to show that the sequences of this type of polynomials are uniformly convergent on the whole interval [-1,1] in suitable weighted spaces of continuous functions. Order of convergence will also be investigated. Some statements of this paper can be obtained as corollaries of our general results proved in [15].     

On the summability of weighted Lagrange interpolation on the roots of Jacobi polynomials

The aim of this paper is to investigate the summability of weighted Lagrange interpolation on the roots of Jacobi polynomials. Starting from the Lagrange interpolation polynomials, we shall construct a wide class of discrete processes which are uniformly convergent in a suitable weighted space of continuous functions. Error estimates for the approximation will also be considered. This revised version was published online in June 2006 with corrections to the Cover Date.     

Extended Lagrange interpolation in weighted uniform norm

The author studies the uniform convergence of extended Lagrange interpolation processes based on the zeros of Generalized Laguerre polynomials.     

Mean convergence of Lagrange interpolation, II

The purpose of the paper is to investigate weighted L^p convergence of Lagrange interpolation taken at the zeros of Hermite polynomials. It is shown that if a continuous function satisfies some growth conditions, then the corresponding Lagrange interpolation process converges in every L^p (1 < p < ∞) provided that the weight function is chosen in a suitable way.     

Necessary conditions for weighted mean convergence of Lagrange interpolation for exponential weights

Given a continuous real-valued function f which vanishes outside a fixed finite interval, we establish necessary conditions for weighted mean convergence of Lagrange interpolation for a general class of even weights w which are of exponential decay on the real line or at the endpoints of (−1,1).     

Necessary and sufficient Tauberian conditions for certain weighted mean methods of summability. II

We prove [1, Theorems 1 and 2] under weaker conditions and in a simpler way than we did in the cited paper. This revised version was published online in June 2006 with corrections to the Cover Date.     

On Lagrange and Hermite interpolation in R k

Summary A method for the construction of a set of data of interpolation in several variables is given. The resulting data, which are either function values or directional derivatives values, give rise to a space of polynomials, in such a way that unisolvence is guaranteed. The interpolating polynomial is calculated using a procedure which generalizes the Newton divided differences formula for a single variable.     

Lagrange interpolation with constraints

Uniform convergence of Lagrange interpolation at the zeros of Jacobi polynomials in the presence of constraints is investigated. We show that by a simple procedure it is always possible to transform the matrices of these zeros into matrices such that the corresponding Lagrange interpolating polynomial respecting the given constraints well approximates a given function and its derivatives.     

Translativity of absolute weighted mean summability

In this paper, we give necessary and sufficient conditions on (p _n) for |R,p _n| _k , k 1, to be translative. So we extend the known results of Al-Madi [1] and Cesco [4] to the case k > 1.     

An Erdős type convergence process in weighted interpolation. II

The aim of this paper is to continue the investigation of the second author started in [14], where a weighted version of a classical result of P. Erdős was proved using Freud type weights. We shall show that an analogous statement is true for weighted interpolation if we consider exponential weights on [-1,1]. This revised version was published online in June 2006 with corrections to the Cover Date.     

A study on weighted mean summability

In the present paper, a general theorem on summability factors of infinite series has been proved. Also we have obtained a new result concerning the |C,1;δ| _k summability factors.     

Stieltjes polynomials and Lagrange interpolation

Bounds are proved for the Stieltjes polynomial , and lower bounds are proved for the distances of consecutive zeros of the Stieltjes polynomials and the Legendre polynomials . This sharpens a known interlacing result of Szegö. As a byproduct, bounds are obtained for the Geronimus polynomials . Applying these results, convergence theorems are proved for the Lagrange interpolation process with respect to the zeros of , and for the extended Lagrange interpolation process with respect to the zeros of in the uniform and weighted norms. The corresponding Lebesgue constants are of optimal order.

     

On the Lagrange interpolation for a subset ofC k functions

We study the optimal order of approximation forC ^k piecewise analytic functions (cf. Definition 1.2) by Lagrange interpolation associated with the Chebyshev extremal points. It is proved that the Jackson order of approximation is attained, and moreover, ifx is away from the singular points, the local order of approximation atx can be improved byO(n ^?1). Such improvement of the local order of approximation is also shown to be sharp. These results extend earlier results of Mastroianni and Szabados on the order of approximation for continuous piecewise polynomial functions (splines) by the Lagrange interpolation, and thus solve a problem of theirs (about the order of approximation for |x|³) in a much more general form.