H. S. Jung

L ∞ convergence of interpolation and associated product integration for exponential weights

investigate convergence in a weighted L∞ norm of Hermite, Hermite, and Grunwald interpolations at zeros of orthogonal polynomials with respect to exponential weights such as Freud, Erdos, and exponential weight on (-1, 1). Convergence of product integration rules induced by the various approximation processes is deduced. We also give more precise weight conditions for convergence of interpolations with respect to above three types of weights, respectively.

On mean convergence of Hermite–Fejér and Hermite interpolation for Erdős weights

Abstract We investigate convergence of Hermite–Fejer and Hermite interpolation polynomials in L p (0 for Erdos weights.

Convergence of Hermite and Hermite-Fejér Interpolation of Higher Order for Freud Weights

We investigate weighted Lp(0<p<∞) convergence of Hermite and Hermite–Fejer interpolation polynomials of higher order at the zeros of Freud orthogonal polynomials on the real line. Our results cover as special cases Lagrange, Hermite–Fejer and Krylov–Stayermann interpolation polynomials.

Mean Convergence of Extended Lagrange Interpolation for Exponential Weights

In this paper, we complete our investigations of mean convergence of Lagrange interpolation for fast decaying even and smooth exponential weights on the line. In doing so, we also present a summary of recent related work on the line and [−1,1] by the authors, Szabados, Vertesi, Lubinsky and Matjila. We also emphasize the important and fundamental ideas, applied in our proofs, that were developed by Erdős, Turan, Askey, Freud, Nevai, Szabados, Vértesi and their students and collaborators. These methods include forward quadrature estimates, orthogonal expansions, Hilbert transforms, bounds on Lebesgue functions and the uniform boundedness principle.

A note on mean convergence of Lagrange interpolation in L p (0 < p ≤ 1)

Abstract Let w ≔exp(− Q ), where Q is of faster than smooth polynomial growth at ∞, for example, w k,α (x)≔ exp (− exp k (|x| α )), α>1 . We obtain a necessary and sufficient condition for mean convergence of Lagrange interpolation for such weights in L p (0 completing earlier investigations by the first author and D.S. Lubinsky in L p (1 .

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

Abstract 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 conditions of convergence of Hermite-Fejér interpolation polynomials for exponential weights

This paper gives the conditions necessary for weighted convergence of Hermite-Fejer interpolation for a general class of even weights which are of exponential decay on the real line or at the end points of (-1, 1 ). The results of this paper guarantee that the conditions of Theorem 2.3 in [11] are optimal.

Estimates for the first and second derivatives of the Stieltjes polynomials

Let wλ(x) = (1 - x2)λ-1/2 and Pn(λ) be the ultraspherical polynomials with respect to wλ(x). Then we denote En-1(λ) the Stieltjes polynomials with respect to wλ(x) satisfying ∫11wλ(x) Pn(λ)(x)En+1(λ)(x)xmdx { =0, 0 ≤ m ≤ n + 1, ≠ = 0, m = n + 1. In this paper, we give estimates for the first and second derivatives of the Stielijes polynomials En-1(λ) and the product En+1(λ)Pn(λ) by obtaining the asymptotic differential relations. Moreover, using these differential relations we estimate the second derivatives of En+1(λ)(x) and En+1(λ)(x)Pn(λ)(x) at the zeros of En+1(λ)(x) and the product En+1(λ)(x)Pn(λ)(x) respectively.

Convergence of a quadrature formula for variable-signed weight functions

A quadrature formula for a variable-signed weight function w ( x ) is constructed using Hermite interpolating polynomials. We show its mean and quadratic mean convergence. We also discuss the rate of convergence in terms of the modulus of continuity for higher order derivatives with respect to the sup norm.