Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 421340 | https://doi.org/10.1155/2012/421340

E. Berriochoa, A. Cachafeiro, J. M. García Amor, "About Nodal Systems for Lagrange Interpolation on the Circle", Journal of Applied Mathematics, vol. 2012, Article ID 421340, 11 pages, 2012. https://doi.org/10.1155/2012/421340

About Nodal Systems for Lagrange Interpolation on the Circle

Academic Editor: Nicola Guglielmi
Received20 Jul 2011
Accepted12 Oct 2011
Published07 Feb 2012

Abstract

We study the convergence of the Laurent polynomials of Lagrange interpolation on the unit circle for continuous functions satisfying a condition about their modulus of continuity. The novelty of the result is that now the nodal systems are more general than those constituted by the n roots of complex unimodular numbers and the class of functions is different from the usually studied. Moreover, some consequences for the Lagrange interpolation on [1,1] and the Lagrange trigonometric interpolation are obtained.

1. Introduction

The aim of this paper is to study the Lagrange interpolation problem on the unit circle 𝕋={𝑧|𝑧|=1} for nodal systems more general than those constituted by the 𝑛 roots of complex unimodular numbers. This last case has been studied in [1], where there is posed as an open problem its extension to more general nodal systems. Recently a similar problem has been solved in [2] for the Hermite interpolation problem. Now we follow the ideas in [2] to obtain some results for the Lagrange case. Moreover, in [1] there is obtained a result about convergence of the interpolants for continuous functions satisfying a condition related to their modulus of continuity. In the present paper our aim is to obtain a similar result for the new nodal systems and with a different condition on the modulus of continuity for the functions.

The Lagrange interpolation problem on the real line has been widely studied for a long time and many results about convergence are known (see [36]). If we only assume the continuity of the function, it is well known that the behavior is rather irregular. Faber has proved that for each nodal system there exists a continuous function such that the sequence of Lagrange interpolation polynomials is not uniformly convergent. Bernstein has also proved the existence of a continuous function such that the sequence of Lagrange interpolation polynomials is unbounded on a prefixed point. In the case of the nodal systems constituted by the zeros of the Tchebychef polynomials of the first kind, many results are known. Although these last nodal systems are good for interpolation, Grünwald in [7] and Marcinkiewicz in [8] have proved the existence of a continuous function such that the sequence of Lagrange interpolation polynomials, corresponding to the Tchebychef nodal system, is divergent. After this result a natural problem was to obtain an analogous result for an arbitrary nodal system. This result was obtained by Erdös and Vértesi in [9], where they prove that for each nodal system on [1,1] there exists a continuous function such that the sequence of Lagrange interpolation polynomials diverges for almost every point in [1,1]. Thus, to obtain better properties about the convergence of the sequence of Lagrange interpolation polynomials, it is needed to impose some restriction on the function, such as a condition on its modulus of continuity. In the case of Jacobi abscissas, Szegő has obtained important results about convergence by imposing some conditions to the modulus of continuity of the function (see [10]). For example, in the case of the Tchebychef abscissas of first kind, he obtained the uniform convergence to the function on [1,1], under the assumption that its modulus of continuity is 𝑜(|log𝛿|1). Szegő has also obtained uniform convergence of the sequence of Lagrange interpolation polynomials for more general nodal systems, under the assumptions that the nodes are the zeros of the orthogonal polynomials with respect to a weight function 𝑤(𝑥) such that 𝑤(𝑥)1𝑥2𝜇>0,𝑥(1,1) and the modulus of continuity of the functions is 𝑜(𝛿1/2) with 𝛿0.

In the present paper we improve some results about convergence of the Lagrange interpolation polynomials in [1,1], by using the Szegő transformation and the results concerning the unit circle. The organization of the paper is the following. In Section 2 we obtain our main result concerning the uniform convergence of the Laurent polynomial of Lagrange interpolation for nodal systems described in terms of some properties and for continuous functions with modulus of continuity 𝑜(𝛿𝑝) when 𝛿0 and 𝑝1/2. Section 3 is devoted to obtain some consequences of the preceding results concerning the Lagrange interpolation on [1,1]. Finally, in the last section, we obtain some improvements concerning the Lagrange trigonometric interpolation.

2. Lagrange Interpolation in the Space of Laurent Polynomials

Let {𝑧𝑗}𝑛𝑗=1 be a set of complex numbers such that |𝑧𝑗|=1 for all 𝑗=1,,𝑛 and 𝑧𝑖𝑧𝑗 for 𝑖𝑗. Let {𝑢𝑗}𝑛𝑗=1 be a set of arbitrary complex numbers, and let 𝑝(𝑛) and 𝑞(𝑛) be two nondecreasing sequences of nonnegative integers such that 𝑝(𝑛)+𝑞(𝑛)=𝑛1,𝑛2 with lim𝑛𝑝(𝑛)=lim𝑛𝑞(𝑛)=.

We recall that the Lagrange interpolation problem in the space of Laurent polynomials consists in determining the unique Laurent polynomial 𝐿𝑝(𝑛),𝑞(𝑛)(𝑧)Λ𝑝(𝑛),𝑞(𝑛)=span{𝑧𝑘𝑝(𝑛)𝑘𝑞(𝑛)} such that𝐿𝑝(𝑛),𝑞(𝑛)𝑧𝑗=𝑢𝑗,for𝑗=1,,𝑛.(2.1)

If we denote by 𝑊𝑛(𝑧)=𝑛𝑗=1(𝑧𝑧𝑗) the nodal polynomial, then 𝐿𝑝(𝑛),𝑞(𝑛)(𝑧) can be written as follows: 𝐿𝑝(𝑛),𝑞(𝑛)(𝑧)=𝑛𝑗=1𝑙𝑗,𝑛1(𝑧)𝑢𝑗,(2.2) where 𝑙𝑗,𝑛1(𝑧) are the fundamental polynomials of Lagrange interpolation given by 𝑙𝑗,𝑛1(𝑧𝑧)=𝑗𝑝(𝑛)𝑊𝑛(𝑧)𝑊𝑛𝑧𝑗𝑧𝑧𝑗𝑧𝑝(𝑛),for𝑗=1,,𝑛,(2.3) and they are characterized by satisfying 𝑙𝑗,𝑛1(𝑧𝑘)=𝛿𝑗,𝑘, for all 𝑗,𝑘.

We are also going to consider the Lagrange interpolation polynomial for a function 𝐹 defined on 𝕋, that we are going to denote by 𝐿𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧) and which is characterized by fulfilling the conditions 𝐿𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧𝑗)=𝐹(𝑧𝑗) for 𝑗=1,,𝑛.

When the nodal system is constituted by the 𝑛-roots of a complex number with modulus 1, and the function 𝐹 is continuous on 𝕋, and its modulus of continuity satisfies 𝜆(𝐹,𝛿)=𝒪(𝛿𝑝), 𝑝>1/2, the following result about convergence is known (see [1]).

Theorem 2.1. Let 𝐹 be a continuous function on 𝕋, let 𝑝(𝑛) and 𝑞(𝑛) be two nondecreasing sequences of nonnegative integers such that 𝑝(𝑛)+𝑞(𝑛)=𝑛1 and lim𝑛𝑝(𝑛)/(𝑛1)=𝑟 with 0<𝑟<1, and assume that the modulus of continuity of 𝐹 is 𝜆(𝐹,𝛿)=𝒪(𝛿𝑝) for some 𝑝>1/2, if 𝛿0.
Let 𝐿𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧) be the Laurent polynomial of Lagrange interpolation for the function 𝐹 with nodal system {𝑧𝑗}𝑛𝑗=1 being the 𝑛-roots of complex numbers 𝜏𝑛 with |𝜏𝑛|=1.
Then lim𝑛𝐿𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋.

Proof. See [1].

The main tools to prove the preceding result are the explicit expression of the Laurent polynomial of Lagrange interpolation and some properties concerning the nodal system. In [2] the Hermite interpolation problem was studied for general nodal systems satisfying certain properties. Following similar ideas we prove, in the next theorem, a result about the convergence of the Lagrange interpolants for a different class of functions and more general nodal systems.

Theorem 2.2. Let 𝐹 be a continuous function on 𝕋, with modulus of continuity 𝜆(𝐹,𝛿)=𝑜(𝛿1/2), if 𝛿0. Let 𝑝(𝑛) and 𝑞(𝑛) be two nondecreasing sequences of nonnegative integers such that 𝑝(𝑛)+𝑞(𝑛)=𝑛1 and lim𝑛𝑝(𝑛)/(𝑛1)=𝑟 with 0<𝑟<1.
Let {𝑧𝑗}𝑛𝑗=1 be a set of complex numbers such that |𝑧𝑗|=1 for all 𝑗=1,,𝑛 and 𝑧𝑖𝑧𝑗 for 𝑖𝑗 and let 𝑊𝑛(𝑧)=Π𝑛𝑗=1(𝑧𝑧𝑗) be the nodal polynomial. Assume that there exist positive constants 𝐵 and 𝐿 such that for every 𝑧𝕋 and 𝑛 large enough the following relations hold:
(i)𝐵|𝑊𝑛(𝑧)|/𝑛, (ii)|𝑊𝑛(𝑧)|2/𝑛2𝑛𝑗=1(1/|𝑧𝑧𝑗|2)𝐿.
If 𝐿𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧)Λ𝑝(𝑛),𝑞(𝑛) is the Laurent polynomial of Lagrange interpolation related to the nodal system and the function 𝐹, then lim𝑛𝐿𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋.

Proof. First we prove that there exists a positive constant 𝐶 such that 𝑛𝑗=1|𝑙𝑗,𝑛1(𝑧)|𝐶𝑛 for every 𝑧𝕋 and 𝑛 large enough. Indeed, taking into account (2.3) and applying the hypothesis we get 𝑛𝑗=1||𝑙𝑗,𝑛1(||=𝑧)𝑛𝑗=1||𝑊𝑛(𝑧)𝑧𝑗𝑝(𝑛)||||𝑊𝑛𝑧𝑗𝑧𝑧𝑗𝑧𝑝(𝑛)||=𝑛𝑗=1||𝑊𝑛||(𝑧)||𝑊𝑛𝑧𝑗𝑧𝑧𝑗||1𝐵𝑛𝑛𝑗=1||𝑊𝑛||(𝑧)||𝑧𝑧𝑗||1𝐵𝑛𝑛𝑗=1||𝑊𝑛||(𝑧)2||𝑧𝑧𝑗||21/2𝑛𝑗=111/2𝐿𝐵𝑛.(2.4) Let us consider the Laurent polynomial of best uniform approximation to 𝐹, 𝑇𝑝(𝑛),𝑞(𝑛)(𝑧)Λ𝑝(𝑛),𝑞(𝑛). If 𝐸𝑝(𝑛),𝑞(𝑛)(𝐹)=max𝑧𝕋|𝐹(𝑧)𝑇𝑝(𝑛),𝑞(𝑛)(𝑧)|, then it holds that 𝐸𝑝(𝑛),𝑞(𝑛)𝜋(𝐹)2𝜆𝐹,𝑠(𝑛),(2.5) where 𝑠(𝑛)=min(𝑝(𝑛),𝑞(𝑛)) (see [1]). Since lim𝑛𝜋/𝑠(𝑛)=0, then by hypothesis 𝜆(𝐹,𝜋/𝑠(𝑛))=𝑜((𝜋/𝑠(𝑛))1/2).
If we write𝐹(𝑧)𝐿𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧)=𝐹(𝑧)𝑇𝑝(𝑛),𝑞(𝑛)(𝑧)𝐿𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧)+𝑇𝑝(𝑛),𝑞(𝑛)(𝑧)=𝐹(𝑧)𝑇𝑝(𝑛),𝑞(𝑛)(𝑧)𝐿𝑝(𝑛),𝑞(𝑛)𝐹𝑇𝑝(𝑛),𝑞(𝑛);𝑧=𝐹(𝑧)𝑇𝑝(𝑛),𝑞(𝑛)(𝑧)𝑛𝑗=1𝑙𝑗,𝑛1𝐹𝑧(𝑧)𝑗𝑇𝑝(𝑛),𝑞(𝑛)𝑧𝑗,(2.6) then we have ||𝐹(𝑧)𝐿𝑝(𝑛),𝑞(𝑛)||||(𝐹;𝑧)𝐹(𝑧)𝑇𝑝(𝑛),𝑞(𝑛)||+(𝑧)𝑛𝑗=1||𝑙𝑗,𝑛1||||𝐹𝑧(𝑧)𝑗𝑇𝑝(𝑛),𝑞(𝑛)𝑧𝑗||𝐸𝑝(𝑛),𝑞(𝑛)(𝐹)1+𝑛𝑗=1||𝑙𝑗,𝑛1||(𝑧)2𝜆(𝐹,𝜋/𝑠(𝑛))1+𝐶𝑛=2𝜆(𝐹,𝜋/𝑠(𝑛))(𝜋/𝑠(𝑛))1/2𝜋(𝑠(𝑛)/(𝑛1))1/21+𝐶𝑛,𝑛1(2.7) and it is easy to prove that the last expression tends to zero because lim𝑛𝜆(𝐹,𝜋/𝑠(𝑛))/(𝜋/𝑠(𝑛))1/2=0 and lim𝑛𝑠(𝑛)=1𝑛12lim𝑛𝑝(𝑛)𝑛1+lim𝑛𝑞(𝑛)𝑛1lim𝑛||||𝑝(𝑛)𝑞(𝑛)=1𝑛11𝑟,if𝑟2,1,1𝑟,if𝑟0,2.(2.8)

Remark 2.3. (i) Since 𝜆(𝐹,𝛿)=𝑜(𝛿𝑝) for 𝑝>1/2 implies 𝜆(𝐹,𝛿)=𝑜(𝛿1/2), then the preceding result is also valid for functions with modulus of continuity 𝑜(𝛿𝑝), with 𝑝>1/2, if 𝛿0. Hence, in the sequel and for simplicity, we establish all the results with the condition 𝜆(𝐹,𝛿)=𝑜(𝛿1/2).
(ii) Since it is clear that the nodal systems in Theorem 2.1 satisfy the hypothesis of Theorem 2.2, we have that the result given in Theorem 2.1 is also valid for functions with modulus of continuity 𝑜(𝛿1/2), if 𝛿0.

Next we recall a sufficient condition given in [2] in order that the nodal system satisfies the conditions imposed in the previous theorem. We use the so-called para-orthogonal polynomials (see [1113]) and the class of measures satisfying the Szegő condition, (see [10, 1315]). Notice that the nodal systems in Theorem 2.1 are constituted by the 𝑛 roots of complex unimodular numbers, and indeed they are the 𝑛 roots of the para-orthogonal polynomials with respect to the Lebesgue measure on [0,2𝜋].

Theorem 2.4. Let 𝜈 be a measure on [0,2𝜋] in the Szegő class with Szegő function having analytic extension up to |𝑧|>1. Let {𝜙𝑛(𝑧)} be the monic orthogonal polynomial sequence with respect to 𝜈, MOPS(𝜈), and 𝜔𝑛(𝑧,𝜏)=𝜙𝑛(𝑧)+𝜏𝜙𝑛(𝑧), with |𝜏|=1 being the para-orthogonal polynomials. Then there exist positive constants 𝐴,𝐵1,𝐵2, and 𝐿 such that for every 𝑧𝕋 and 𝑛 large enough the following relations hold: (i)|𝜔𝑛(𝑧,𝜏)|𝐴, (ii)𝐵1|𝜔𝑛(𝑧,𝜏)|/𝑛𝐵2, (iii)|𝜔𝑛(𝑧,𝜏)|2/𝑛2𝑛𝑗=1(1/|𝑧𝑧𝑗|2)𝐿, where one assumes that 𝑧1,,𝑧𝑛 are the zeros of 𝜔𝑛(𝑧,𝜏).

Proof. See [2].

Taking into account the preceding results, we are in conditions to prove the following corollary.

Corollary 2.5. Let 𝐹 be a continuous function on 𝕋, with modulus of continuity 𝜆(𝐹,𝛿)=𝑜(𝛿1/2), if 𝛿0. Let 𝑝(𝑛) and 𝑞(𝑛) be two nondecreasing sequences of nonnegative integers such that 𝑝(𝑛)+𝑞(𝑛)=𝑛1 and lim𝑛𝑝(𝑛)/(𝑛1)=𝑟 with 0<𝑟<1.
Let 𝜈 be a measure on [0,2𝜋] in the Szegő class with Szegő function having analytic extension up to |𝑧|>1. Let {𝜙𝑛(𝑧)} be the MOPS(𝜈) and let 𝜔𝑛(𝑧,𝜏)=𝜙𝑛(𝑧)+𝜏𝜙𝑛(𝑧), with |𝜏|=1, be the para-orthogonal polynomials.
If 𝐿𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧)Λ𝑝(𝑛),𝑞(𝑛) is the Laurent polynomial of Lagrange interpolation related to the function 𝐹 and with nodal system the zeros of the para-orthogonal polynomials 𝜔𝑛(𝑧,𝜏), then lim𝑛𝐿𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋.

Proof. Taking into account that the zeros of 𝜔𝑛(𝑧,𝜏) belong to 𝕋 (see [11]), the result is immediate from Theorems 2.2 and 2.4.

Remark 2.6. Notice that the preceding result is valid for the Bernstein-Szegő measures (see [13]).

3. Lagrange Interpolation on [1,1]

In this section we present some consequences of Theorem 2.2 concerning the Lagrange interpolation problems on [1,1]. Let us recall that the Lagrange interpolation polynomial related to a nodal system {𝑥𝑗}𝑛𝑗=1[1,1] and satisfying the conditions {𝑢𝑗}𝑛𝑗=1 is given by 𝑙𝑛1(𝑥)=𝑛𝑗=1(𝑝𝑛(𝑥)/(𝑝𝑛(𝑥𝑗)(𝑥𝑥𝑗)))𝑢𝑗, where 𝑝𝑛(𝑥)=Π𝑛𝑗=1(𝑥𝑥𝑗).

Theorem 3.1. Let 𝑝𝑛(𝑥)=𝑛𝑗=1(𝑥𝑥𝑗) be a nodal system in [1,1] such that 𝑊2𝑛(𝑧)=2𝑛𝑧𝑛𝑝𝑛((𝑧+1/𝑧)/2) satisfies the following inequalities ||𝑊𝐵2𝑛||(𝑧),||𝑊2𝑛2𝑛||(𝑧)2𝑛𝑗=11||𝑧𝑧𝑗||2+1||𝑧𝑧𝑗||2𝐿(2𝑛)2,(3.1) with (𝑧𝑗+(1/𝑧𝑗))/2=𝑥𝑗 for 𝑗=1,,𝑛 and for some positive constants 𝐵 and 𝐿, 𝑛 large enough and every 𝑧𝕋.
Let 𝑓 be a continuous function on [1,1] such that 𝜆(𝑓,𝛿)=𝑜(𝛿1/2), if 𝛿0.
If 𝑙𝑛1(𝑓,𝑥) is the Lagrange interpolation polynomial such that 𝑙𝑛1(𝑓,𝑥𝑗)=𝑓(𝑥𝑗) for 𝑗=1,,𝑛, then 𝑙𝑛1(𝑓,𝑥) converges to 𝑓(𝑥) uniformly on [1,1].

Proof. It is easy to see that the polynomial 𝑊2𝑛(𝑧) has the following expression 𝑊2𝑛(𝑧)=Π𝑛𝑗=1(𝑧𝑧𝑗)(𝑧𝑧𝑗), with (𝑧𝑗+𝑧𝑗)/2=𝑥𝑗.
Let us define a continuous function on 𝕋 by 𝐹(𝑧)=𝐹(𝑧)=𝑓(𝑥), with 𝑥=(𝑧+(1/𝑧))/2 and 𝑧𝕋. It is clear that𝜆(𝐹,𝛿)=sup𝑧1,𝑧2𝕋;|𝑧1𝑧2|<𝛿||𝐹𝑧1𝑧𝐹2||sup𝑥1,𝑥2[1,1];|𝑥1𝑥2|<𝛿||𝑓𝑥1𝑥𝑓2||=𝜆(𝑓,𝛿).(3.2)
If we take 𝑊2𝑛(𝑧) as nodal system on 𝕋, we can consider the following Lagrange interpolation problem: find the Laurent polynomial of Lagrange interpolation 𝐿𝑛,𝑛1(𝐹;𝑧)Λ𝑛,𝑛1 satisfying the interpolation conditions𝐿𝑛,𝑛1𝐹;𝑧𝑗=𝐿𝑛,𝑛1𝐹;𝑧𝑗𝑥=𝑓𝑗,𝑗=1,,𝑛.(3.3) By applying Theorem 2.2 we have that lim𝑛𝐿𝑛,𝑛1(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋.
On the other hand, for 𝑥=(𝑧+(1/𝑧))/2 and 𝑧𝕋 it holds𝐿𝑛,𝑛1(𝐹;𝑧)=𝑛𝑗=1𝑊2𝑛(𝑧)𝑧𝑛𝑗𝑧𝑛𝑊2𝑛𝑧𝑗𝑧𝑧𝑗𝐹𝑧𝑗+𝑛𝑗=1𝑊2𝑛(𝑧)𝑧𝑗𝑛𝑧𝑛𝑊2𝑛𝑧𝑗𝑧𝑧𝑗𝐹𝑧𝑗=𝑛𝑗=1𝑝𝑛(𝑥)𝑝𝑛𝑥𝑗𝑥𝑥𝑗𝑓𝑥𝑗=𝑙𝑛1(𝑓;𝑥).(3.4) Hence lim𝑛𝑙𝑛1(𝑓,𝑥)=𝑓(𝑥) uniformly on [1,1].

As a consequence we obtain, in the next corollary, a result that was proved by Szegő in [10] under weaker conditions. Although our result is not new, we give the proof because the way in which it is obtained is different from Szegő’s proof.

Corollary 3.2. Let 𝑓 be a continuous function on [1,1] such that 𝜆(𝑓,𝛿)=o(𝛿1/2), if 𝛿0. Let 𝑑𝜇(𝑥)=𝑤(𝑥)𝑑𝑥 be a finite positive Borel measure on [1,1] satisfying the Szegő condition 11(log𝑤(𝑥)/1𝑥2)𝑑𝑥>, and let {𝑃𝑛(𝑥)} be the MOPS(𝜇). Assume that the function 𝑤(𝑥)1𝑥2 is positive on [1,1] and it is analytic in an open set containing [1,1].
If 𝑙𝑛1(𝑓,𝑥) is the Lagrange interpolation polynomial satisfying the interpolation conditions 𝑙𝑛1(𝑓,𝑥𝑗)=𝑓(𝑥𝑗),𝑗=1,,𝑛, where {𝑥𝑗}𝑛𝑗=1 are the zeros of the orthogonal polynomial 𝑃𝑛(𝑥), thenlim𝑛𝑙𝑛1(𝑓,𝑥)=𝑓(𝑥)(3.5) uniformly on [1,1].

Proof. By using the Szegő transformation (see [10]), the measure 𝑑𝜇(𝑥) becomes into the measure 𝑑𝜈(𝜃)=(1/2)𝑤(cos𝜃)|sin𝜃|𝑑𝜃, which is in the Szegő class with Szegő function having analytic extension up to |𝑧|>1 (see [14]). If we denote by {𝜙𝑛(𝑧)} the MOPS(𝜈) and by {𝑃𝑛(𝑧)} the MOPS(𝜇), then both sequences are related by𝑃𝑛1(𝑥)=2𝑛1+𝜙2𝑛(𝜙0)2𝑛(𝑧)+𝜙2𝑛(𝑧)𝑧𝑛=12𝑛1+𝜙2𝑛(𝜔0)2𝑛(𝑧,1)𝑧𝑛.(3.6)
The zeros of 𝑃𝑛(𝑥), 𝑥1,,𝑥𝑛, are simple and belong to (1,1) and they are related to the zeros of 𝜔2𝑛(𝑧,1), 𝑧1,,𝑧𝑛,𝑧𝑛+1=𝑧𝑛,,𝑧2𝑛=𝑧1, by 𝑥𝑗=(𝑧𝑗+𝑧𝑗)/2,𝑗=1,,𝑛. By applying Theorem 2.4 we get that the system 𝜔2𝑛(𝑧,1) satisfies the hypothesis of Theorem 3.1. Then we have that 𝑙𝑛1(𝑓,𝑥) converges to 𝑓(𝑥) uniformly on [1,1].

Analogous results can be obtained for other nodal systems related to those given in Corollary 3.2. Let 𝑑𝜇1(𝑥)=𝑤(𝑥)𝑑𝑥 be a finite positive Borel measure on [1,1]. Let us consider the measures𝑑𝜇2(𝑥)=1𝑥2𝑑𝜇1(𝑥),𝑑𝜇3(𝑥)=(1𝑥)𝑑𝜇1(𝑥),𝑑𝜇4(𝑥)=(1+𝑥)𝑑𝜇1(𝑥),(3.7) and let us denote the MOPS with respect to these measures by {𝑃𝑛(𝑥,𝜇𝑖)}4𝑖=1. Let us consider the Szegő transformed measure of 𝑑𝜇1(𝑥), 𝑑𝜈(𝜃)=(1/2)𝑤(cos𝜃)|sin𝜃|𝑑𝜃 with MOPS(𝜈), {𝜙𝑛(𝑧)}. Taking into account the relation between the measures, we can relate the orthogonal sequences as follows (see [16]):𝑃𝑛𝑥,𝜇1=12𝑛1+𝜙2𝑛𝑧(0)𝑛𝑤2𝑛(𝑧,1),1𝑥2𝑃𝑛𝑥,𝜇2=12𝑛+1𝚤1𝜙2𝑛+2𝑧(0)𝑛+1𝑤2𝑛+2(𝑧,1),1𝑥𝑃𝑛𝑥,𝜇3=12𝑛+1/2𝚤1𝜙2𝑛+1𝑧(0)𝑛+(1/2)𝑤2𝑛+1(𝑧,1),1+𝑥𝑃𝑛𝑥,𝜇4=12𝑛+1/21+𝜙2𝑛+1𝑧(0)𝑛+(1/2)𝑤2𝑛+1(𝑧,1).(3.8)

We denote by ±1,𝑥1,,𝑥𝑛 the zeros of 1𝑥2𝑃𝑛(𝑥,𝜇2), by 1,𝑦1,,𝑦𝑛 the zeros of 1𝑥𝑃𝑛(𝑥,𝜇3), and by 1,𝑣1,,𝑣𝑛 the zeros of 1+𝑥𝑃𝑛(𝑥,𝜇4).

If we denote by ±1,𝑧1,,𝑧𝑛,𝑧1,,𝑧𝑛 the zeros of 𝜔2𝑛+2(𝑧,1), by 1,𝑤1,,𝑤𝑛,𝑤1,,𝑤𝑛 the zeros of 𝑤2𝑛+1(𝑧,1), and by 1,𝑢1,,𝑢𝑛,𝑢1,,𝑢𝑛 the zeros of 𝑤2𝑛+1(𝑧,1), then the following relations hold: (𝑧𝑖)=𝑥𝑖,(𝑤𝑖)=𝑦𝑖, and (𝑢𝑖)=𝑣𝑖,𝑖=1,,𝑛. By taking nodal systems related to the zeros of 𝑃𝑛(𝑥,𝜇𝑖), 𝑖=2,3,4, we obtain the next result.

Theorem 3.3. Let 𝑓 be a continuous function on [1,1] such that 𝜆(𝑓,𝛿)=o(𝛿1/2), if 𝛿0. Let 𝜇1 be a finite positive Borel measure on [1,1], 𝑑𝜇1(𝑥)=𝑤(𝑥)d𝑥, satisfying the Szegő condition. Assume that the function 𝑤(𝑥)1𝑥2 is positive in [1,1] and it is analytic in an open set containing [1,1]. Let 𝑑𝜇𝑖(𝑥),𝑖=2,3,4, be the measures given in (3.7). Let us consider the Lagrange interpolation polynomials for the function 𝑓 with the following nodal systems: (i)the zeros of 𝑃𝑛(𝑥,𝜇2) joint with ±1,(ii)the zeros of 𝑃𝑛(𝑥,𝜇3) joint with 1,(iii)the zeros of 𝑃𝑛(𝑥,𝜇4) joint with −1.Then the corresponding Lagrange interpolation polynomials uniformly converge to 𝑓(𝑥) on [1,1].

Proof. By the Szegő transformation the measure 𝑑𝜇1(𝑥) becomes into the measure 𝑑𝜈(𝜃)=(1/2)𝑤(cos𝜃)|sin𝜃|𝑑𝜃, which is in the Szegő class with Szegő function having analytic extension up to |𝑧|>1. We denote by {𝜙𝑛(𝑧)} the MOPS(𝜈). If we define a continuous function 𝐹 on 𝕋 by 𝐹(𝑧)=𝐹(𝑧)=𝑓(𝑥) with 𝑥=(𝑧+1/𝑧)/2 and 𝑧𝕋, then it is clear that 𝜆(𝐹,𝛿)𝜆(𝑓,𝛿).
(i) We consider the para-orthogonal polynomial 𝜔2𝑛+2(𝑧,1), whose zeros are ±1,𝑧1,,𝑧𝑛, 𝑧1,,𝑧𝑛𝕋 and they are related to the zeros of 𝑃𝑛(𝑥,𝜇2) by 𝑥𝑗=(𝑧𝑗+𝑧𝑗)/2;𝑗=1,,𝑛.
Let us consider the following Lagrange interpolation problem: find the Laurent polynomial of Lagrange interpolation 𝐿(𝑛+1),𝑛(𝐹;𝑧) satisfying 𝐿(𝑛+1),𝑛𝐹;𝑧𝑗=𝐿(𝑛+1),𝑛𝐹;𝑧𝑗𝑧=𝐹𝑗𝐿,𝑗=1,,𝑛,(𝑛+1),𝑛(𝐹;1)=𝐹(1),𝐿(𝑛+1),𝑛(𝐹;1)=𝐹(1).(3.9) By applying Corollary 2.5 we have that lim𝑛𝐿(𝑛+1),𝑛(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋. If we take 𝑙𝑛+1𝐿(𝑓,𝑥)=(𝑛+1),𝑛(𝐹;𝑧)+𝐿(𝑛+1),𝑛(𝐹;1/𝑧)2(3.10) for 𝑥=(𝑧+(1/𝑧))/2, then 𝑙𝑛+1(𝑓,𝑥) fulfills 𝑙𝑛+1(𝑓,𝑥𝑗)=𝑓(𝑥𝑗),𝑗=1,,𝑛, and 𝑙𝑛+1(𝑓,±1)=𝑓(±1). Therefore, 𝑙𝑛+1(𝑓,𝑥) is the Lagrange interpolation polynomial for the function 𝑓 and the nodal system given in (i) and lim𝑛𝑙𝑛+1(𝑓,𝑥)=𝑓(𝑥) uniformly on [1,1].
(ii) We consider the para-orthogonal polynomials 𝜔2𝑛+1(𝑧,1) whose zeros, 1,𝑤1,,𝑤𝑛,𝑤1,,𝑤𝑛, are related to the zeros of 𝑃𝑛(𝑥,𝜇3) by 𝑦𝑗=(𝑤𝑗+𝑤𝑗)/2,𝑗=1,,𝑛.
We pose the problem of finding the Laurent polynomial of Lagrange interpolation 𝐿𝑛,𝑛(𝐹;𝑧) satisfying 𝐿𝑛,𝑛𝐹;𝑤𝑗=𝐿𝑛,𝑛𝐹;𝑤𝑗𝑤=𝐹𝑗𝐿,𝑗=1,,𝑛,𝑛,𝑛(𝐹;1)=𝐹(1).(3.11) Since lim𝑛𝐿𝑛,𝑛(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋, if we define 𝑙𝑛(𝑓,𝑥)=(𝐿𝑛,𝑛(𝐹;𝑧)+𝐿𝑛,𝑛(𝐹;1/𝑧))/2 for 𝑥=(𝑧+(1/𝑧))/2 and 𝑧𝕋, then 𝑙𝑛(𝑓,𝑥) fulfills 𝑙𝑛(𝑓,𝑥𝑗)=𝑓(𝑥𝑗),𝑗=1,,𝑛, and 𝑙𝑛(𝑓,1)=𝑓(1). Therefore, lim𝑛𝑙𝑛(𝑓,𝑥)=𝑓(𝑥) uniformly on [1,1].
(iii) It is obtained proceeding the same way as in the previous items.

Remark 3.4. (i) In particular, the preceding result is valid for the following nodal systems: the zeros of the Tchebychef polynomials of the second kind joint with ±1, the zeros of the Tchebychef polynomials of the third kind joint with 1, and the zeros of the Tchebychef polynomials of the fourth kind joint with −1.
(ii) Moreover it is also valid for the polynomial modifications, by positive polynomials, of the Bernstein measures corresponding to the Tchebychef measures mentioned before.

4. Trigonometric Interpolation

Next we obtain some consequences of Theorem 2.2, which are related to the Lagrange trigonometric interpolation. Now the nodal points are in [0,2𝜋] and they are obtained as follows. Let 𝑑𝜇(𝑥)=𝑤(𝑥)𝑑𝑥 be a positive finite Borel measure on [1,1] satisfying the Szegő condition. Assume that the function 𝑤(𝑥)1𝑥2>0 for all 𝑥[1,1] and it is analytic in an open set containing [1,1]. If {𝑃𝑛(𝑥)} is the MOPS(𝜇) and {𝑥𝑗}𝑛𝑗=1 are the zeros of 𝑃𝑛(𝑥), we consider the following nodal system on [0,2𝜋],{𝜃𝑗}2𝑛𝑗=1, such that 𝜃𝑗=arccos𝑥𝑗,𝑗=1,,𝑛 with 0<𝜃𝑗<𝜋 and 𝜃𝑛+𝑗=2𝜋𝜃𝑛𝑗+1 for 𝑗=1,,𝑛; that is, the points are symmetric with respect to 𝜋.

Theorem 4.1. Let 𝑓 be a real continuous function on [0,2𝜋], with modulus of continuity  𝜆(𝑓,𝛿)=𝑜(𝛿1/2), if 𝛿0.
Let the nodal system be {𝜃𝑗}2𝑛𝑗=1 with 𝜃𝑗=arccos𝑥𝑗,𝑗=1,,𝑛 with 0<𝜃𝑗<𝜋 and 𝜃𝑛+𝑗=2𝜋𝜃𝑛𝑗+1 for 𝑗=1,,𝑛, where {𝑥𝑗}𝑛𝑗=1 are the zeros of the orthogonal polynomial 𝑃𝑛(𝑥) with respect to the measure 𝑑𝜇. One also assumes that 𝑑𝜇(𝑥)=𝑤(𝑥)𝑑𝑥 is a positive finite Borel measure on [1,1] satisfying the Szegő condition and such that 𝑤(𝑥)1𝑥2>0 for all 𝑥[1,1] and it is analytic in an open set containing [1,1].
Then there is a Lagrange interpolation trigonometric polynomial of degree 𝑛, 𝜏𝑛(𝜃), such that 𝜏𝑛(𝜃𝑗)=𝑓(𝜃𝑗) for 𝑗=1,,2𝑛 and it satisfies that lim𝑛𝜏𝑛(𝜃)=𝑓(𝜃) uniformly on [0,2𝜋].

Proof. Proceeding like in Corollary 3.2 we obtain that the transformed measure of 𝑑𝜇(𝑥) by the Szegő transformation, 𝑑𝜈(𝜃), satisfies the hypothesis of Theorem 2.4 and the para-orthogonal polynomials satisfy the bound condition of Theorem 2.4.
Let us define 𝐹 by 𝐹(𝑒𝚤𝜃)=𝑓(𝜃) for 𝜃[0,2𝜋]. Let 𝐿𝑛,𝑛1(𝐹,𝑧) be the Lagrange interpolation polynomial such that 𝐿𝑛,𝑛1(𝐹,𝑧𝑗)=𝑓(𝜃𝑗), where 𝑧𝑗=𝑒𝚤𝜃𝑗,𝑗=1,,2𝑛.
Since 𝐹 is continuous on 𝕋 and 𝜆(𝐹,𝛿)=𝑜(𝛿1/2), we can apply Corollary 3.2 and therefore lim𝑛𝐿𝑛,𝑛1(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋.
If we take 𝜏𝑛(𝜃)=(𝐿𝑛,𝑛1(𝐹;𝑒𝚤𝜃)), then 𝜏𝑛(𝜃) satisfies the interpolation conditions, 𝜏𝑛(𝜃𝑗)=𝑓(𝜃𝑗), and lim𝑛𝜏𝑛(𝜃)=𝑓(𝜃).

In the next result we denote the integer part of 𝑥 by [𝑥] and we consider another type of nodal system on [0,2𝜋].

Theorem 4.2. Let 𝜈 be a measure on [0,2𝜋] in the Szegő class with Szegő function having analytic extension up to |𝑧|>1. Let {𝑧𝑗}𝑛𝑗=1 be the zeros of the para-orthogonal polynomials 𝜔𝑛(𝑧)=𝜙𝑛(𝑧)+𝜏𝜙𝑛(𝑧), with |𝜏|=1, and let 𝜃𝑗[0,2𝜋] such that 𝑒𝚤𝜃𝑗=𝑧𝑗,𝑗=1,,𝑛.
If 𝑓 is a continuous function on [0,2𝜋] with 𝜆(𝑓,𝛿)=𝑜(𝛿1/2), if 𝛿0, then there is a Lagrange interpolation trigonometric polynomial of degree [𝑛/2], 𝜏[𝑛/2](𝜃), such that 𝜏[𝑛/2](𝜃𝑗)=𝑓(𝜃𝑗) for 𝑗=1,,𝑛 and it satisfies that lim𝑛𝜏[𝑛/2](𝜃)=𝑓(𝜃) uniformly on [0,2𝜋].

Proof. Let 𝐹 be a continuous function defined by 𝐹(𝑒𝚤𝜃)=𝑓(𝜃). Since 𝜆(𝐹,𝛿)𝜆(𝑓,𝛿), then 𝜆(𝐹,𝛿)=𝑜(𝛿1/2). By applying Corollary 2.5 we obtain for 𝑝(𝑛)+𝑞(𝑛)=𝑛1, with lim𝑛𝑝(𝑛)/(𝑛1)=𝑟 and 0<𝑟<1, there exists 𝐿𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧) such that 𝐿𝑝(𝑛),𝑞(𝑛)(𝐹;𝑒𝚤𝜃𝑗)=𝑓(𝜃𝑗) and lim𝑛𝐿𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋.
We distinguish the two following cases.
(i)If 𝑛 is even, we take 𝑝(𝑛)=𝑛/2 and 𝑞(𝑛)=𝑛/21. Then [𝑛/2]=𝑛/2.(ii)If 𝑛 is odd, we take 𝑝(𝑛)=(𝑛1)/2 and 𝑞(𝑛)=(𝑛1)/2. Then [𝑛/2]=(𝑛1)/2.In any case the real part of 𝐿𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧) is a trigonometric polynomial of degree [𝑛/2], that satisfies the interpolation conditions and the convergence property.

Acknowledgment

The research was supported by Ministerio de Ciencia e Innovación under grant number MTM2011-22713.

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