Convergence of Extended Lagrange Interpolation

Abstract

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

Interlacing of zeros of shifted sequences of one-parameter orthogonal polynomials

“…The following result is a special case of a more general result proven in [6], Theorem 2.2, and was also proved in [5], Corollary 2. Theorem 3.1 Let λ > − 1 2 and let −1 < x 1 < x 2 < · · · < x n < 1 be the zeros of C λ n (x), −1 < y 1 < y 2 < · · · < y n−1 < 1 be the zeros of C λ+1 n−1 (x)…”

“…In the context of our paper, this involves an analysis of the minimum distance between, for example, consecutive zeros of the product of the polynomials L α n L α+t n L α+2 n as t varies continuously between 0 and 2. Interesting work in a related context has been done in [6,7,13] and [14]. The interlacing property of the zeros of classical orthogonal polynomials, such as the Jacobi polynomials, has been used to develop new methods for approximating the finite Hilbert tranform (cf.…”

A mixed scheme of product integration rules in (−1,1)

“…different weight functions, provided that the interpolation zeros are sufficiently "far" among them. Extended interpolation processes have been extensively studied from several authors, in bounded and unbounded intervals, and with estimates of the error in different norms (see for instance [1], [2], [3], [4], [5], [6], [7]). Recently an application in quadrature has been proposed in [8], [9].…”

A new quadrature scheme based on an Extended Lagrange Interpolation process

“…Extended interpolation is of interest in the field of the polynomial approximation since it introduces new systems of "good" interpolation knots (see for instance [4], [3], [2], [5], [27], [29], [16] and the references therein) and the research of "well far apart" zeros of orthogonal polynomials is still an open problem [9]. On the other hand, following a procedure proposed in [26] for the automatic estimate of the numerical error by extended Gaussian rules, extended interpolation can be used in the numerical evaluation of the convergence order in interpolation processes (see for instance [4]). Moreover extended matrices of orthogonal polynomials can be employed, for instance, in the approximation of singular and hypersingular integral transforms [28], [8].…”