© Hindawi Publishing Corp. MEAN CONVERGENCE OF GRÜNWALD INTERPOLATION OPERATORS

We investigate weighted Lp mean convergence of Grünwald interpolation operators based on the zeros of orthogonal polynomials with respect to a general weight and generalized Jacobi weights. We give necessary and sufficient conditions for such convergence for all continuous functions.


Introduction.
In this paper, we study weighted L p (0 < p < ∞) mean convergence of Grünwald interpolation operators which was introduced in [3].We first consider the weighted convergence for the Grünwald interpolation operator when a general weight is used.We also consider in particular the weighted L p convergence for the Grünwald interpolation operator when a generalized Jacobi weight is used.Necessary and sufficient conditions for such convergence for a continuous function are presented.In Section 1, we briefly introduce the Hermite-Fejér interpolation and Grünwald interpolation.In Section 2, we review some known results that are closely related to the main results of this paper and will be used in our proof later.We also establish several preliminary results.In Section 3, we state and prove the main results of this paper.
We first introduce the Hermite-Fejér interpolation polynomials and the Grünwald interpolation operator.Let w be a weight function of interval [−1 , 1] and {P n (w, x)} the orthonormal polynomials on [−1, 1] with respect to w. Assume that we are given a system X of nodes The Hermite-Fejér interpolation polynomials of f ∈ C[−1, 1] at X are defined by where , k= 1, 2,...,n, n = 1, 2,..., We use l kn (w, x) and H n (w,f ,x) to denote l kn (X, x) and H n (X,f ,x), respectively, when the set X is chosen to be the zeros of the orthogonal polynomial P n (w, x).For simplicity, we substitute x k for x kn .The Christoffel function is defined by The numbers λ kn (w), defined by are called the Cotes numbers.It is well known (see [2, page 113], [9]) that (1.6) If P is a polynomial of degree at most 2n − 1, then Naturally, when 0 < p < 1, • u,p is not a norm.The function u(x) is called a Jacobi weight function if u(x) can be written as u (a,b) .) The function u(x) is a generalized Jacobi weight function (u ∈ GJ) if u ∈ L 1 and u can be written in the form u(x) = g(x)(1 − x) a (1 + x) b , where g > 0 and g ±1 ∈ L ∞ .The uniform convergence of the corresponding Grünwald interpolation was investigated by several authors [3,6], and L p convergence for such interpolation was studied in [5] with p = 1 only.Here, for convenience, we state the theorem which was proved in [5]. If (1.10)

Auxiliary propositions.
In this section, we obtain some preliminary results.First, we need the following notations.Here and later the symbols const and C denote some positive constants, not necessarily having the same values in different formulas.If A and B are two expressions depending on some variables and indices, then Let w be a generalized Jacobi weight function (w ∈ GJ).In the following, we summarize some results from [8] that will be useful for our development in this paper.Assume that Then we have We also have the estimates for the orthonormal polynomial P n (w, x): uniformly for n ≥ 2, uniformly for n ≥ 2.

Main results.
In this section, we present two results about weighted L p convergence for Grünwald interpolation with the special case in which we obtain a sufficient and necessary condition.
if and only if Proof.(1) It is easy to see that (2) For φ x (t) = (x − t) 2 , we have By Hermite interpolation, we know that Using Lemma 2.2, we find that (3.5) converges to 0. Thus (3.4) converges to 0. According to Lemma 2.1, (3.1) holds if and only if . Let w ∈ GJ with parameters α, β, and u be Jacobi weight function with parameters a, b, where a, b, α, β > −1. ) Proof.Assume that (3.7) holds.
This completes the proof.
As an immediate consequence of Theorem 3.2, we state the following corollary.

Theorem 3 . 1 .
Let w be a weight function.Then