About Nodal Systems for Lagrange Interpolation on the Circle

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 [3–6]). 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)√1−𝑥²≥𝜇>0,𝑥∈(−1,1) and the modulus of continuity of the functions is 𝑜(𝛿^1/2) with 𝛿→0.

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(𝛿^𝑝) when 𝛿→0 and 𝑝≥1/2. 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 {𝑧_𝑗}^𝑛𝑗=1 be a set of complex numbers such that |𝑧_𝑗|=1 for all 𝑗=1,…,𝑛 and 𝑧_𝑖≠𝑧_𝑗 for 𝑖≠𝑗. Let {𝑢_𝑗}^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)=𝑛−1,𝑛≥2 with lim_𝑛→∞𝑝(𝑛)=lim_𝑛→∞𝑞(𝑛)=∞.

We recall that the Lagrange interpolation problem in the space of Laurent polynomials consists in determining the unique Laurent polynomial 𝐿_{−𝑝(𝑛),𝑞(𝑛)}(𝑧)∈Λ_{−𝑝(𝑛),𝑞(𝑛)}=span{𝑧^𝑘∶−𝑝(𝑛)≤𝑘≤𝑞(𝑛)} such that𝐿_{−𝑝(𝑛),𝑞(𝑛)}𝑧_𝑗=𝑢_𝑗,for𝑗=1,…,𝑛.(2.1)

If we denote by 𝑊_𝑛(𝑧)=∏^𝑛𝑗=1(𝑧−𝑧_𝑗) the nodal polynomial, then 𝐿_{−𝑝(𝑛),𝑞(𝑛)}(𝑧) can be written as follows: 𝐿_{−𝑝(𝑛),𝑞(𝑛)}(𝑧)=^𝑛_𝑗=1𝑙_{𝑗,𝑛−1}(𝑧)𝑢_𝑗,(2.2) where 𝑙_{𝑗,𝑛−1}(𝑧) are the fundamental polynomials of Lagrange interpolation given by 𝑙_{𝑗,𝑛−1}(𝑧)=𝑧^{𝑗𝑝(𝑛)}𝑊_𝑛(𝑧)𝑊^′_𝑛𝑧_𝑗𝑧−𝑧_𝑗𝑧^𝑝(𝑛),for𝑗=1,…,𝑛,(2.3) and they are characterized by satisfying 𝑙_{𝑗,𝑛−1}(𝑧_𝑘)=𝛿_𝑗,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 𝐿_{−𝑝(𝑛),𝑞(𝑛)}(𝐹;𝑧) and which is characterized by fulfilling the conditions 𝐿_{−𝑝(𝑛),𝑞(𝑛)}(𝐹;𝑧_𝑗)=𝐹(𝑧_𝑗) 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(𝛿^𝑝), 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)=𝑛−1 and lim_𝑛→∞𝑝(𝑛)/(𝑛−1)=𝑟 with 0<𝑟<1, and assume that the modulus of continuity of 𝐹 is 𝜆(𝐹,𝛿)=𝒪(𝛿^𝑝) for some 𝑝>1/2, if 𝛿→0.
Let 𝐿_{−𝑝(𝑛),𝑞(𝑛)}(𝐹;𝑧) be the Laurent polynomial of Lagrange interpolation for the function 𝐹 with nodal system {𝑧_𝑗}^𝑛𝑗=1 being the 𝑛-roots of complex numbers 𝜏_𝑛 with |𝜏_𝑛|=1.
Then lim_𝑛→∞𝐿_{−𝑝(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(𝛿^1/2), if 𝛿→0. Let 𝑝(𝑛) and 𝑞(𝑛) be two nondecreasing sequences of nonnegative integers such that 𝑝(𝑛)+𝑞(𝑛)=𝑛−1 and lim_𝑛→∞𝑝(𝑛)/(𝑛−1)=𝑟 with 0<𝑟<1.
Let {𝑧_𝑗}^𝑛𝑗=1 be a set of complex numbers such that |𝑧_𝑗|=1 for all 𝑗=1,…,𝑛 and 𝑧_𝑖≠𝑧_𝑗 for 𝑖≠𝑗 and let 𝑊_𝑛(𝑧)=Π^𝑛𝑗=1(𝑧−𝑧_𝑗) be the nodal polynomial. Assume that there exist positive constants 𝐵 and 𝐿 such that for every 𝑧∈𝕋 and 𝑛 large enough the following relations hold:
(i)𝐵≤|𝑊^′_𝑛(𝑧)|/𝑛, (ii)|𝑊_𝑛(𝑧)|²/𝑛²∑^𝑛𝑗=1(1/|𝑧−𝑧_𝑗|²)≤𝐿.
If 𝐿_{−𝑝(𝑛),𝑞(𝑛)}(𝐹;𝑧)∈Λ_{−𝑝(𝑛),𝑞(𝑛)} is the Laurent polynomial of Lagrange interpolation related to the nodal system and the function 𝐹, then lim_𝑛→∞𝐿_{−𝑝(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|𝑙_{𝑗,𝑛−1}(𝑧)|≤𝐶√𝑛 for every 𝑧∈𝕋 and 𝑛 large enough. Indeed, taking into account (2.3) and applying the hypothesis we get ^𝑛_𝑗=1||𝑙_{𝑗,𝑛−1}(𝑧)||=^𝑛_𝑗=1||𝑊_𝑛(𝑧)𝑧^{𝑗𝑝(𝑛)}||||𝑊^𝑛𝑧_𝑗𝑧−𝑧_𝑗𝑧^𝑝(𝑛)||=^𝑛_𝑗=1||𝑊_𝑛(𝑧)||||𝑊^𝑛𝑧_𝑗𝑧−𝑧_𝑗||≤𝐵1𝑛^𝑛_𝑗=1||𝑊_𝑛(𝑧)||≤1||𝑧−𝑧_𝑗||𝐵𝑛^𝑛_𝑗=1||𝑊_𝑛(𝑧)||²||𝑧−𝑧_𝑗||²^1/2^𝑛_𝑗=11^1/2≤√𝐿𝐵√𝑛.(2.4) Let us consider the Laurent polynomial of best uniform approximation to 𝐹, 𝑇_{−𝑝(𝑛),𝑞(𝑛)}(𝑧)∈Λ_{−𝑝(𝑛),𝑞(𝑛)}. If 𝐸_{−𝑝(𝑛),𝑞(𝑛)}(𝐹)=max_𝑧∈𝕋|𝐹(𝑧)−𝑇_{−𝑝(𝑛),𝑞(𝑛)}(𝑧)|, then it holds that 𝐸_{−𝑝(𝑛),𝑞(𝑛)}(𝐹)≤2𝜆𝐹,𝜋𝑠(𝑛),(2.5) where 𝑠(𝑛)=min(𝑝(𝑛),𝑞(𝑛)) (see [1]). Since lim_𝑛→∞𝜋/𝑠(𝑛)=0, then by hypothesis 𝜆(𝐹,𝜋/𝑠(𝑛))=𝑜((𝜋/𝑠(𝑛))^1/2).
If we write𝐹(𝑧)−𝐿_{−𝑝(𝑛),𝑞(𝑛)}(𝐹;𝑧)==𝐹𝐹((𝑧𝑧))−−𝑇𝑇_{−𝑝(𝑛),𝑞(𝑛)−𝑝(𝑛),𝑞(𝑛)}(𝑧)−𝐿_{−𝑝(𝑛),𝑞(𝑛)}(𝐹;𝑧)+𝑇_{−𝑝(𝑛),𝑞(𝑛)}(𝑧)=𝐹(𝑧)−𝑇_{−𝑝(𝑛),𝑞(𝑛)}(𝑧)−𝐿_{−𝑝(𝑛),𝑞(𝑛)}𝐹−𝑇_{−𝑝(𝑛),𝑞(𝑛)};𝑧(𝑧)−^𝑛_𝑗=1𝑙_{𝑗,𝑛−1}(𝑧)𝐹𝑧_𝑗−𝑇_{−𝑝(𝑛),𝑞(𝑛)}𝑧_𝑗,(2.6) then we have ||𝐹(𝑧)−𝐿_{−𝑝(𝑛),𝑞(𝑛)}(𝐹;𝑧)||≤||𝐹(𝑧)−𝑇_{−𝑝(𝑛),𝑞(𝑛)}(𝑧)||+^𝑛_𝑗=1≤𝐸_{−𝑝(𝑛),𝑞(𝑛)}||𝑙_{𝑗,𝑛−1}(𝑧)||||𝐹𝑧_𝑗−𝑇_{−𝑝(𝑛),𝑞(𝑛)}𝑧_𝑗||(𝐹)1+^𝑛_𝑗=1=2𝜆(𝐹,𝜋/𝑠(𝑛))||𝑙_{𝑗,𝑛−1}(𝑧)||≤2𝜆(𝐹,𝜋/𝑠(𝑛))1+𝐶√𝑛(𝜋/𝑠(𝑛))^1/2√𝜋(𝑠(𝑛)/(𝑛−1))^1/21+𝐶√𝑛√𝑛−1,(2.7) and it is easy to prove that the last expression tends to zero because lim_𝑛→∞𝜆(𝐹,𝜋/𝑠(𝑛))/(𝜋/𝑠(𝑛))^1/2=0 and lim_𝑛→∞𝑠(𝑛)𝑛−1=21lim^𝑛→∞𝑝(𝑛)𝑛−1+lim^𝑛→∞𝑞(𝑛)𝑛−1−lim^𝑛→∞||𝑝(𝑛)−𝑞(𝑛)||𝑛−1=⎧⎪⎨⎪⎩1−𝑟,if𝑟∈1𝑟,if𝑟∈02,,11,2.(2.8)

Remark 2.3. (i) Since 𝜆(𝐹,𝛿)=𝑜(𝛿^𝑝) for 𝑝>1/2 implies 𝜆(𝐹,𝛿)=𝑜(𝛿^1/2), then the preceding result is also valid for functions with modulus of continuity 𝑜(𝛿^𝑝), with 𝑝>1/2, if 𝛿→0. Hence, in the sequel and for simplicity, we establish all the results with the condition 𝜆(𝐹,𝛿)=𝑜(𝛿^1/2).
(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(𝛿^1/2), if 𝛿→0.

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 [11–13]) and the class of measures satisfying the Szegő condition, (see [10, 13–15]). 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 {𝜙_𝑛(𝑧)} be the monic orthogonal polynomial sequence with respect to 𝜈, MOPS(𝜈), and 𝜔_𝑛(𝑧,𝜏)=𝜙_𝑛(𝑧)+𝜏𝜙^∗𝑛(𝑧), with |𝜏|=1 being the para-orthogonal polynomials. Then there exist positive constants 𝐴,𝐵₁,𝐵₂, and 𝐿 such that for every 𝑧∈𝕋 and 𝑛 large enough the following relations hold: (i)|𝜔_𝑛(𝑧,𝜏)|≤𝐴, (ii)𝐵₁≤|𝜔^′_𝑛(𝑧,𝜏)|/𝑛≤𝐵₂, (iii)|𝜔_𝑛(𝑧,𝜏)|²/𝑛²∑^𝑛𝑗=1(1/|𝑧−𝑧_𝑗|²)≤𝐿, where one assumes that 𝑧₁,…,𝑧_𝑛 are the zeros of 𝜔_𝑛(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(𝛿^1/2), if 𝛿→0. Let 𝑝(𝑛) and 𝑞(𝑛) be two nondecreasing sequences of nonnegative integers such that 𝑝(𝑛)+𝑞(𝑛)=𝑛−1 and lim_𝑛→∞𝑝(𝑛)/(𝑛−1)=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 {𝜙_𝑛(𝑧)} be the MOPS(𝜈) and let 𝜔_𝑛(𝑧,𝜏)=𝜙_𝑛(𝑧)+𝜏𝜙^∗𝑛(𝑧), with |𝜏|=1, be the para-orthogonal polynomials.
If 𝐿_{−𝑝(𝑛),𝑞(𝑛)}(𝐹;𝑧)∈Λ_{−𝑝(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 𝜔_𝑛(𝑧,𝜏), then lim_𝑛→∞𝐿_{−𝑝(𝑛),𝑞(𝑛)}(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋.

Proof. Taking into account that the zeros of 𝜔_𝑛(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 {𝑥_𝑗}^𝑛𝑗=1⊂[−1,1] and satisfying the conditions {𝑢_𝑗}^𝑛𝑗=1 is given by 𝑙_𝑛−1(𝑥)=∑^𝑛𝑗=1(𝑝_𝑛(𝑥)/(𝑝^𝑛(𝑥_𝑗)(𝑥−𝑥_𝑗)))𝑢_𝑗, where 𝑝_𝑛(𝑥)=Π^𝑛𝑗=1(𝑥−𝑥_𝑗).

Theorem 3.1. Let 𝑝_𝑛(𝑥)=∏^𝑛𝑗=1(𝑥−𝑥_𝑗) be a nodal system in [−1,1] such that 𝑊_2𝑛(𝑧)=2^𝑛𝑧^𝑛𝑝_𝑛((𝑧+1/𝑧)/2) satisfies the following inequalities 𝐵≤||𝑊^2𝑛(𝑧)||2𝑛,||𝑊^2𝑛(𝑧)||^2𝑛_𝑗=1||𝑧−1𝑧_𝑗||²+1||𝑧−𝑧_𝑗||²≤𝐿(2𝑛)²,(3.1) with (𝑧_𝑗+(1/𝑧_𝑗))/2=𝑥_𝑗 for 𝑗=1,…,𝑛 and for some positive constants 𝐵 and 𝐿, 𝑛 large enough and every 𝑧∈𝕋.
Let 𝑓 be a continuous function on [−1,1] such that 𝜆(𝑓,𝛿)=𝑜(𝛿^1/2), if 𝛿→0.
If 𝑙_𝑛−1(𝑓,𝑥) is the Lagrange interpolation polynomial such that 𝑙_𝑛−1(𝑓,𝑥_𝑗)=𝑓(𝑥_𝑗) for 𝑗=1,…,𝑛, then 𝑙_𝑛−1(𝑓,𝑥) converges to 𝑓(𝑥) uniformly on [−1,1].

Proof. It is easy to see that the polynomial 𝑊_2𝑛(𝑧) has the following expression 𝑊_2𝑛(𝑧)=Π^𝑛𝑗=1(𝑧−𝑧_𝑗)(𝑧−𝑧_𝑗), with (𝑧_𝑗+𝑧_𝑗)/2=𝑥_𝑗.
Let us define a continuous function on 𝕋 by 𝐹(𝑧)=𝐹(𝑧)=𝑓(𝑥), with 𝑥=(𝑧+(1/𝑧))/2 and 𝑧∈𝕋. It is clear that𝜆(𝐹,𝛿)=sup_{𝑧1,𝑧2∈𝕋;|𝑧1−𝑧2|<𝛿}||𝐹𝑧₁−𝐹𝑧₂||≤sup_{𝑥1,𝑥2∈[−1,1];|𝑥1−𝑥2|<𝛿}||𝑓𝑥₁−𝑓𝑥₂||=𝜆(𝑓,𝛿).(3.2)
If we take 𝑊_2𝑛(𝑧) as nodal system on 𝕋, we can consider the following Lagrange interpolation problem: find the Laurent polynomial of Lagrange interpolation 𝐿_{−𝑛,𝑛−1}(𝐹;𝑧)∈Λ_{−𝑛,𝑛−1} satisfying the interpolation conditions𝐿_{−𝑛,𝑛−1}𝐹;𝑧_𝑗=𝐿_{−𝑛,𝑛−1}𝐹;𝑧_𝑗=𝑓𝑥_𝑗,𝑗=1,…,𝑛.(3.3) By applying Theorem 2.2 we have that lim_𝑛→∞𝐿_{−𝑛,𝑛−1}(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋.
On the other hand, for 𝑥=(𝑧+(1/𝑧))/2 and 𝑧∈𝕋 it holds𝐿_{−𝑛,𝑛−1}(𝐹;𝑧)=^𝑛_𝑗=1𝑊_2𝑛(𝑧)𝑧^𝑛𝑗𝑧^𝑛𝑊^2𝑛𝑧_𝑗𝑧−𝑧_𝑗𝐹𝑧^𝑗+^𝑛_𝑗=1𝑊_2𝑛(𝑧)𝑧_𝑗𝑛𝑧^𝑛𝑊^2𝑛𝑧_𝑗𝑧−𝑧_𝑗𝐹=_𝑛𝑧_𝑗_𝑗=1𝑝_𝑛(𝑥)𝑝^𝑛𝑥_𝑗𝑥−𝑥_𝑗𝑓𝑥^𝑗=𝑙_𝑛−1(𝑓;𝑥).(3.4) Hence lim_𝑛→∞𝑙_𝑛−1(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(𝛿^1/2), if 𝛿→0. Let 𝑑𝜇(𝑥)=𝑤(𝑥)𝑑𝑥 be a finite positive Borel measure on [−1,1] satisfying the Szegő condition ∫¹⁻¹(log𝑤(𝑥)/√1−𝑥²)𝑑𝑥>−∞, and let {𝑃_𝑛(𝑥)} be the MOPS(𝜇). Assume that the function 𝑤(𝑥)√1−𝑥² is positive on [−1,1] and it is analytic in an open set containing [−1,1].
If 𝑙_𝑛−1(𝑓,𝑥) is the Lagrange interpolation polynomial satisfying the interpolation conditions 𝑙_𝑛−1(𝑓,𝑥_𝑗)=𝑓(𝑥_𝑗),𝑗=1,…,𝑛, where {𝑥_𝑗}^𝑛𝑗=1 are the zeros of the orthogonal polynomial 𝑃_𝑛(𝑥), thenlim_𝑛→∞𝑙_𝑛−1(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 {𝜙_𝑛(𝑧)} the MOPS(𝜈) and by {𝑃_𝑛(𝑧)} the MOPS(𝜇), then both sequences are related by𝑃_𝑛(𝑥)=12^𝑛1+𝜙_2𝑛(0)𝜙^2𝑛(𝑧)+𝜙^∗2𝑛(𝑧)𝑧^𝑛=12^𝑛1+𝜙_2𝑛(0)𝜔^2𝑛(𝑧,1)𝑧^𝑛.(3.6)
The zeros of 𝑃_𝑛(𝑥), 𝑥₁,…,𝑥_𝑛, are simple and belong to (−1,1) and they are related to the zeros of 𝜔_2𝑛(𝑧,1), 𝑧₁,…,𝑧_𝑛,𝑧_𝑛+1=𝑧_𝑛,…,𝑧_2𝑛=𝑧₁, by 𝑥_𝑗=(𝑧_𝑗+𝑧_𝑗)/2,𝑗=1,…,𝑛. By applying Theorem 2.4 we get that the system 𝜔_2𝑛(𝑧,1) satisfies the hypothesis of Theorem 3.1. Then we have that 𝑙_𝑛−1(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)=1−𝑥²𝑑𝜇₁(𝑥),𝑑𝜇₃(𝑥)=(1−𝑥)𝑑𝜇₁(𝑥),𝑑𝜇₄(𝑥)=(1+𝑥)𝑑𝜇₁(𝑥),(3.7) and let us denote the MOPS with respect to these measures by {𝑃_𝑛(𝑥,𝜇_𝑖)}^4𝑖=1. Let us consider the Szegő transformed measure of 𝑑𝜇₁(𝑥), 𝑑𝜈(𝜃)=(1/2)𝑤(cos𝜃)|sin𝜃|𝑑𝜃 with MOPS(𝜈), {𝜙_𝑛(𝑧)}. Taking into account the relation between the measures, we can relate the orthogonal sequences as follows (see [16]):𝑃_𝑛𝑥,𝜇₁=12^𝑛1+𝜙_2𝑛(0)𝑧^𝑛√𝑤_2𝑛(𝑧,1),1−𝑥²𝑃_𝑛𝑥,𝜇₂=12^𝑛+1𝚤1−𝜙_2𝑛+2(0)𝑧^𝑛+1√𝑤_2𝑛+2(𝑧,−1),1−𝑥𝑃_𝑛𝑥,𝜇₃=12^𝑛+1/2𝚤1−𝜙_2𝑛+1(0)𝑧^𝑛+(1/2)√𝑤_2𝑛+1(𝑧,−1),1+𝑥𝑃_𝑛𝑥,𝜇₄=12^𝑛+1/21+𝜙_2𝑛+1(0)𝑧^𝑛+(1/2)𝑤_2𝑛+1(𝑧,1).(3.8)

We denote by ±1,𝑥₁,…,𝑥_𝑛 the zeros of √1−𝑥²𝑃_𝑛(𝑥,𝜇₂), by 1,𝑦₁,…,𝑦_𝑛 the zeros of √1−𝑥𝑃_𝑛(𝑥,𝜇₃), and by −1,𝑣₁,…,𝑣_𝑛 the zeros of √1+𝑥𝑃_𝑛(𝑥,𝜇₄).

If we denote by ±1,𝑧₁,…,𝑧_𝑛,𝑧₁,…,𝑧_𝑛 the zeros of 𝜔_2𝑛+2(𝑧,−1), by 1,𝑤₁,…,𝑤_𝑛,𝑤₁,…,𝑤_𝑛 the zeros of 𝑤_2𝑛+1(𝑧,−1), and by −1,𝑢₁,…,𝑢_𝑛,𝑢₁,…,𝑢_𝑛 the zeros of 𝑤_2𝑛+1(𝑧,1), then the following relations hold: ℜ(𝑧_𝑖)=𝑥_𝑖,ℜ(𝑤_𝑖)=𝑦_𝑖, and ℜ(𝑢_𝑖)=𝑣_𝑖,𝑖=1,…,𝑛. By taking nodal systems related to the zeros of 𝑃_𝑛(𝑥,𝜇_𝑖), 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(𝛿^1/2), if 𝛿→0. 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)√1−𝑥² is positive in [−1,1] and it is analytic in an open set containing [−1,1]. Let 𝑑𝜇_𝑖(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 𝑃_𝑛(𝑥,𝜇₂) joint with ±1,(ii)the zeros of 𝑃_𝑛(𝑥,𝜇₃) joint with 1,(iii)the zeros of 𝑃_𝑛(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 {𝜙_𝑛(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 𝑧∈𝕋, then it is clear that 𝜆(𝐹,𝛿)≤𝜆(𝑓,𝛿).
(i) We consider the para-orthogonal polynomial 𝜔_2𝑛+2(𝑧,−1), whose zeros are ±1,𝑧₁,…,𝑧_𝑛, 𝑧₁,…,𝑧_𝑛∈𝕋 and they are related to the zeros of 𝑃_𝑛(𝑥,𝜇₂) by 𝑥_𝑗=(𝑧_𝑗+𝑧_𝑗)/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 𝐿𝐿_{−(𝑛+1),𝑛}𝐹;𝑧_𝑗=𝐿_{−(𝑛+1),𝑛}𝐹;𝑧_𝑗=𝐹𝑧_𝑗,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 lim_𝑛→∞𝐿_{−(𝑛+1),𝑛}(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋. If we take 𝑙_𝑛+1(𝑓,𝑥)=𝐿^{−(𝑛+1),𝑛}(𝐹;𝑧)+𝐿_{−(𝑛+1),𝑛}(𝐹;1/𝑧)2(3.10) for 𝑥=(𝑧+(1/𝑧))/2, then 𝑙_𝑛+1(𝑓,𝑥) fulfills 𝑙_𝑛+1(𝑓,𝑥_𝑗)=𝑓(𝑥_𝑗),𝑗=1,…,𝑛, and 𝑙_𝑛+1(𝑓,±1)=𝑓(±1). Therefore, 𝑙_𝑛+1(𝑓,𝑥) is the Lagrange interpolation polynomial for the function 𝑓 and the nodal system given in (i) and lim_𝑛→∞𝑙_𝑛+1(𝑓,𝑥)=𝑓(𝑥) uniformly on [−1,1].
(ii) We consider the para-orthogonal polynomials 𝜔_2𝑛+1(𝑧,−1) whose zeros, 1,𝑤₁,…,𝑤_𝑛,𝑤₁,…,𝑤_𝑛, are related to the zeros of 𝑃_𝑛(𝑥,𝜇₃) by 𝑦_𝑗=(𝑤_𝑗+𝑤_𝑗)/2,𝑗=1,…,𝑛.
We pose the problem of finding the Laurent polynomial of Lagrange interpolation 𝐿_−𝑛,𝑛(𝐹;𝑧) satisfying 𝐿_−𝑛,𝑛𝐹;𝑤_𝑗=𝐿𝐿_−𝑛,𝑛𝐹;𝑤_𝑗=𝐹𝑤_𝑗,𝑗=1,…,𝑛,_−𝑛,𝑛(𝐹;1)=𝐹(1).(3.11) Since lim_𝑛→∞𝐿_−𝑛,𝑛(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋, if we define 𝑙_𝑛(𝑓,𝑥)=(𝐿_−𝑛,𝑛(𝐹;𝑧)+𝐿_−𝑛,𝑛(𝐹;1/𝑧))/2 for 𝑥=(𝑧+(1/𝑧))/2 and 𝑧∈𝕋, then 𝑙_𝑛(𝑓,𝑥) fulfills 𝑙_𝑛(𝑓,𝑥_𝑗)=𝑓(𝑥_𝑗),𝑗=1,…,𝑛, and 𝑙_𝑛(𝑓,1)=𝑓(1). Therefore, lim_𝑛→∞𝑙_𝑛(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)√1−𝑥²>0 for all 𝑥∈[−1,1] and it is analytic in an open set containing [−1,1]. If {𝑃_𝑛(𝑥)} is the MOPS(𝜇) and {𝑥_𝑗}^𝑛𝑗=1 are the zeros of 𝑃_𝑛(𝑥), we consider the following nodal system on [0,2𝜋],{𝜃_𝑗}^2𝑛𝑗=1, such that 𝜃_𝑗=arccos𝑥_𝑗,𝑗=1,…,𝑛 with 0<𝜃_𝑗<𝜋 and 𝜃_𝑛+𝑗=2𝜋−𝜃_{𝑛−𝑗+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(𝛿^1/2), if 𝛿→0.
Let the nodal system be {𝜃_𝑗}^2𝑛𝑗=1 with 𝜃_𝑗=arccos𝑥_𝑗,𝑗=1,…,𝑛 with 0<𝜃_𝑗<𝜋 and 𝜃_𝑛+𝑗=2𝜋−𝜃_{𝑛−𝑗+1} for 𝑗=1,…,𝑛, where {𝑥_𝑗}^𝑛𝑗=1 are the zeros of the orthogonal polynomial 𝑃_𝑛(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)√1−𝑥²>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, 𝜏_𝑛(𝜃), such that 𝜏_𝑛(𝜃_𝑗)=𝑓(𝜃_𝑗) for 𝑗=1,…,2𝑛 and it satisfies that lim_𝑛→∞𝜏_𝑛(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(𝑒^𝚤𝜃)=𝑓(𝜃) for 𝜃∈[0,2𝜋]. Let 𝐿_{−𝑛,𝑛−1}(𝐹,𝑧) be the Lagrange interpolation polynomial such that 𝐿_{−𝑛,𝑛−1}(𝐹,𝑧_𝑗)=𝑓(𝜃_𝑗), where 𝑧_𝑗=𝑒^𝚤𝜃𝑗,𝑗=1,…,2𝑛.
Since 𝐹 is continuous on 𝕋 and 𝜆(𝐹,𝛿)=𝑜(𝛿^1/2), we can apply Corollary 3.2 and therefore lim_𝑛→∞𝐿_{−𝑛,𝑛−1}(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋.
If we take 𝜏_𝑛(𝜃)=ℜ(𝐿_{−𝑛,𝑛−1}(𝐹;𝑒^𝚤𝜃)), then 𝜏_𝑛(𝜃) satisfies the interpolation conditions, 𝜏_𝑛(𝜃_𝑗)=𝑓(𝜃_𝑗), and lim_𝑛→∞𝜏_𝑛(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 {𝑧_𝑗}^𝑛𝑗=1 be the zeros of the para-orthogonal polynomials 𝜔_𝑛(𝑧)=𝜙_𝑛(𝑧)+𝜏𝜙^∗𝑛(𝑧), with |𝜏|=1, and let 𝜃_𝑗∈[0,2𝜋] such that 𝑒^𝚤𝜃𝑗=𝑧_𝑗,𝑗=1,…,𝑛.
If 𝑓 is a continuous function on [0,2𝜋] with 𝜆(𝑓,𝛿)=𝑜(𝛿^1/2), if 𝛿→0, then there is a Lagrange interpolation trigonometric polynomial of degree ≤[𝑛/2], 𝜏_[𝑛/2](𝜃), such that 𝜏_[𝑛/2](𝜃_𝑗)=𝑓(𝜃_𝑗) for 𝑗=1,…,𝑛 and it satisfies that lim_𝑛→∞𝜏_[𝑛/2](𝜃)=𝑓(𝜃) uniformly on [0,2𝜋].

Proof. Let 𝐹 be a continuous function defined by 𝐹(𝑒^𝚤𝜃)=𝑓(𝜃). Since 𝜆(𝐹,𝛿)≤𝜆(𝑓,𝛿), then 𝜆(𝐹,𝛿)=𝑜(𝛿^1/2). By applying Corollary 2.5 we obtain for 𝑝(𝑛)+𝑞(𝑛)=𝑛−1, with lim_𝑛→∞𝑝(𝑛)/(𝑛−1)=𝑟 and 0<𝑟<1, there exists 𝐿_{−𝑝(𝑛),𝑞(𝑛)}(𝐹;𝑧) such that 𝐿_{−𝑝(𝑛),𝑞(𝑛)}(𝐹;𝑒^𝚤𝜃𝑗)=𝑓(𝜃_𝑗) and lim_𝑛→∞𝐿_{−𝑝(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/2−1. Then [𝑛/2]=𝑛/2.(ii)If 𝑛 is odd, we take 𝑝(𝑛)=(𝑛−1)/2 and 𝑞(𝑛)=(𝑛−1)/2. Then [𝑛/2]=(𝑛−1)/2.In any case the real part of 𝐿_{−𝑝(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.