Zeros Of Orthonormal Polynomials Research Articles

An Analytic and Numerical Analysis of Weighted Singular Cauchy Integrals with Exponential Weights on ℝ

This article concerns an analytic and numerical analysis of a class of weighted singular Cauchy integrals with exponential weights with finite moments and with smooth external fields with varying smooth convex rate of increase for large argument. Our analysis relies in part on weighted polynomial interpolation at the zeros of orthonormal polynomials with respect to w 2. We also study bounds for the first derivatives of a class of functions of the second kind for w 2.

Lagrange interpolation with exponential weights on [formula omitted

In order to approximate functions defined on (−1,1) with exponential growth for |x|→1, we consider interpolation processes based on the zeros of orthonormal polynomials with respect to exponential weights. Convergence results and error estimates in weighted Lp metric and uniform metric are given. In particular, in some function spaces, the related interpolating polynomials behave essentially like the polynomial of best approximation.

Error estimates of Lagrange interpolation and orthonormal expansions for Freud weights

Let S n [ f] be the nth partial sum of the orthonormal polynomials expansion with respect to a Freud weight. Then we obtain sufficient conditions for the boundedness of S n [ f] and discuss the speed of the convergence of S n [ f] in weighted L p space. We also find sufficient conditions for the boundedness of the Lagrange interpolation polynomial L n [ f], whose nodal points are the zeros of orthonormal polynomials with respect to a Freud weight. In particular, if W( x)=e −(1/2) x 2 is the Hermite weight function, then we obtain sufficient conditions for the inequalities to hold: ∥(S n[f]−f) (k)Wu b∥ L p( R) ⩽C 1 n r−k∥f (r)Wu B∥ L p( R) and ∥(L n[f]−f) (k)Wu b∥ L p( R) ⩽C 1 n r−k∥f (r)W(1+x 2) r/3u B∥ L p( R) , where u γ(x)=(1+|x|) γ, γ∈ R and k=0,1,2…, r.

Uniform or Mean Convergence of Hermite–Fejér Interpolation of Higher Order for Freud Weights

In this paper we show the uniform or mean convergence of Hermite–Fejér interpolation polynomials of higher order based at the zeros of orthonormal polynomials with the typical Freud weight.

Convergence of Modified Lagrange Interpolation toLp-Functions Based on the Zeros of Orthonormal Polynomials with Freud Weights

We consider the “Freud weight”W2Q(x)=exp(−Q(x)). let 1<p<∞, and letL*n(f) be a modified Lagrange interpolation polynomial to a measurable functionf∈{f;esssupx∈R|f(x)|WQ(x)(1+|x|)α<∞},α>0. Then we have limn→∞∫∞−∞[|f(x)−L*n(f; x)|WQ(x)(1+|x|)−Δ]pdx=0, whereΔis a constant depending onpandα.

Convergence of the Derivatives of Hermite-Fejér Interpolation Polynomials of Higher Order Based at the Zeros of Freud Polynomials

We shall prove pointwise convergence of the derivatives of Hermite-Fejér interpolation polynomials of higher order based at the zeros of orthonormal polynomials with respect to Freud weights exp(−xm), m = 2, 4, 6, ....

Pointwise convergence of Lagrange interpolation based at the zeros of orthonormal polynomials with respect to weights on the whole real line

Under various assumptions on a weight W 2, with support R , we obtain rates for the pointwise convergence of Lagrange interpolation based at the zeros of the orthonormal polynomials with respect to W 2, in the case of a uniformly continuous function ƒ(x). The weights considered include W m(x) = exp(− 1 2 ¦x¦ m), m an even positive integer. The technique used generalizes that of Freud, who considered pointwise convergence of Lagrange interpolation in the case of the Hermite weight. However, even for the Hermite weight, our results refine and extend the upper and lower bounds of Freud. We establish as well, as preliminary results, upper and lower bounds for generalized Lebesgue functions and for absolute values of the orthogonal polynomials associated with W m 2( x).