Pointwise convergence of Hermite-Fejér interpolation of higher order for Freud weights

Abstract

This paper is concerned with the approximation by Hermite-Fejer interpolation of higher order based at the zeros of orthogonal polynomials with respect to the typical Freud weight. We will prove a convergence result for even order and a divergence result for odd order. Introduction.The purpose of this paper is to investigate the pointwise convergence of Hermite-Fejer interpolation of higher order based at the zeros of orthogonal polynomials with respect to a Freud weight of the form exp( -x m ) with an even positive integer m.Let Q(x) = x m /2 and let w(x) = exp(-Q(x)), where m = 2, 4, 6,... . The orthonormal polynomials p n (w 2 x)=p n (x) = y n x n +, where y π >0, are defined by the relation ί:These polynomials were investigated first by Freud, e.g. [Frl], [Fr2], and recently by many authors in connection with approximation theory. For detailed references and an extensive survey, readers may refer to Nevai [Ne3].We denote the zeros of p n (x) by x kn , k= 1, 2,..., n 9 whereLet v be a positive integer. For a function/(x) defined on the real line R, the Hermite-Fejer interpolation polynomial L π (v;/, x) of order v based at the zeros x ln ,..., x nn is defined to be the unique algebraic polynomial of degree at most vn -1 which satisfies L n (y;/, x kn

Specific Examples of Exponential Weights

“…For the future works such as the differential relation of orthogonal polynomials with respect to the exponential weights, we need further assumptions with respect to Q(x) (see [1,3]). In the following, we introduce a new weight subclass of the weight class in Definition 1.1.…”

Higher order Hermite-Fejér interpolation polynomials with Laguerre-type weights

“…In Section 2, we will introduce the classes F (C 2 ) and F (C 2 +) , and then, we will obtain some relations of the factors derived from the classes F (C 2 ) , F (C 2 +) and the classes L(C 2 +) , L(C 2 +) . Finally, we will prove the main theorems using known results in [1][2][3][4][5], in Section 3. We say that f : ℝ ℝ + is quasi-increasing if there exists C > 0 such that f(x) ≤ Cf(y)…”

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

“…We see that L n ( f ; x)=L n (1, f; x) is the Lagrange interpolation polynomial and L n (2, f ; x) is the ordinary Hermite Feje r interpolation polynomial. In [3] we showed the following. Recently, for the Lagrange interpolation polynomial L n ( f ; x), Lubinsky and Matjila [6] obtained the following nice result.…”

“…1Â2 1(mÂ2)Â1(m+1Â2)] 1Âm be Freud's constant, and let :=m(mÂ2) (m&1)Âm ( m&2 mÂ2&1 ) ; m&1 . In [3], we showed that the proposition held for x kn # [%, 3], where % and 3 are positive constants. We omit the proof of Proposition 5.3, because we can show it by careful repeating the same line of the consideration as one in [3].…”

“…( 1.6) and the coefficients e jk can be calculated by (1.4) (see [3]). We see that L n ( f ; x)=L n (1, f; x) is the Lagrange interpolation polynomial and L n (2, f ; x) is the ordinary Hermite Feje r interpolation polynomial.…”