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
Let R = (−∞, ∞) and let Q ∈ C 2 : R → R + = [0, ∞) be an even function. Then in this paper we consider the infinite-finite range inequality, an estimate for the Christoffel function, and the Markov-Bernstein inequality with the exponential weights w (x)= |x| e −Q(x) , x ∈ R.
Abstract. Let Q ∈ C 2 : R → [0, ∞) be an even function. Then we will consider the exponential weights w(x) = exp(−Q(x)) in the weight class from [2]. In the paper, we will give some relations among exponential weights in this class and introduce a new weight subclass. In addition, we will investigate some properties of the typical and specific weights in these weight classes.
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