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. GarcΓ­a 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 [βˆ’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 π•‹βˆΆ={π‘§βˆΆ|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, GrΓΌnwald 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 ErdΓΆs and VΓ©rtesi in [9], where they prove that for each nodal system on [βˆ’1,1] there exists a continuous function such that the sequence of Lagrange interpolation polynomials diverges for almost every point in [βˆ’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, SzegΕ‘ 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 [βˆ’1,1], under the assumption that its modulus of continuity is π‘œ(|log𝛿|βˆ’1). SzegΕ‘ 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 𝑀(π‘₯) such that βˆšπ‘€(π‘₯)1βˆ’π‘₯2β‰₯πœ‡>0,π‘₯∈(βˆ’1,1) and the modulus of continuity of the functions is π‘œ(𝛿1/2) with 𝛿→0.

In the present paper we improve some results about convergence of the Lagrange interpolation polynomials in [βˆ’1,1], by using the SzegΕ‘ 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 π‘œ(𝛿𝑝) when 𝛿→0 and 𝑝β‰₯1/2. Section 3 is devoted to obtain some consequences of the preceding results concerning the Lagrange interpolation on [βˆ’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 𝑧𝑖≠𝑧𝑗 for 𝑖≠𝑗. Let {𝑒𝑗}𝑛𝑗=1 be a set of arbitrary complex numbers, and let 𝑝(𝑛) and π‘ž(𝑛) be two nondecreasing sequences of nonnegative integers such that 𝑝(𝑛)+π‘ž(𝑛)=π‘›βˆ’1,𝑛β‰₯2 with limπ‘›β†’βˆžπ‘(𝑛)=limπ‘›β†’βˆžπ‘ž(𝑛)=∞.

We recall that the Lagrange interpolation problem in the space of Laurent polynomials consists in determining the unique Laurent polynomial πΏβˆ’π‘(𝑛),π‘ž(𝑛)(𝑧)βˆˆΞ›βˆ’π‘(𝑛),π‘ž(𝑛)=span{π‘§π‘˜βˆΆβˆ’π‘(𝑛)β‰€π‘˜β‰€π‘ž(𝑛)} such thatπΏβˆ’π‘(𝑛),π‘ž(𝑛)𝑧𝑗=𝑒𝑗,for𝑗=1,…,𝑛.(2.1)

If we denote by π‘Šπ‘›βˆ(𝑧)=𝑛𝑗=1(π‘§βˆ’π‘§π‘—) the nodal polynomial, then πΏβˆ’π‘(𝑛),π‘ž(𝑛)(𝑧) can be written as follows: πΏβˆ’π‘(𝑛),π‘ž(𝑛)(𝑧)=𝑛𝑗=1𝑙𝑗,π‘›βˆ’1(𝑧)𝑒𝑗,(2.2) where 𝑙𝑗,π‘›βˆ’1(𝑧) are the fundamental polynomials of Lagrange interpolation given by 𝑙𝑗,π‘›βˆ’1(𝑧𝑧)=𝑗𝑝(𝑛)π‘Šπ‘›(𝑧)π‘Šβ€²π‘›ξ€·π‘§π‘—ξ€Έξ€·π‘§βˆ’π‘§π‘—ξ€Έπ‘§π‘(𝑛),for𝑗=1,…,𝑛,(2.3) and they are characterized by satisfying 𝑙𝑗,π‘›βˆ’1(π‘§π‘˜)=𝛿𝑗,π‘˜, for all 𝑗,π‘˜.

We are also going to consider the Lagrange interpolation polynomial for a function 𝐹 defined on 𝕋, that we are going to denote by πΏβˆ’π‘(𝑛),π‘ž(𝑛)(𝐹;𝑧) and which is characterized by fulfilling the conditions πΏβˆ’π‘(𝑛),π‘ž(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 πœ†(𝐹,𝛿)=π’ͺ(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 π‘ž(𝑛) be two nondecreasing sequences of nonnegative integers such that 𝑝(𝑛)+π‘ž(𝑛)=π‘›βˆ’1 and limπ‘›β†’βˆžπ‘(𝑛)/(π‘›βˆ’1)=π‘Ÿ with 0<π‘Ÿ<1, and assume that the modulus of continuity of 𝐹 is πœ†(𝐹,𝛿)=π’ͺ(𝛿𝑝) for some 𝑝>1/2, if 𝛿→0.
Let πΏβˆ’π‘(𝑛),π‘ž(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 πœπ‘› with |πœπ‘›|=1.
Then limπ‘›β†’βˆžπΏβˆ’π‘(𝑛),π‘ž(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 πœ†(𝐹,𝛿)=π‘œ(𝛿1/2), if 𝛿→0. Let 𝑝(𝑛) and π‘ž(𝑛) be two nondecreasing sequences of nonnegative integers such that 𝑝(𝑛)+π‘ž(𝑛)=π‘›βˆ’1 and limπ‘›β†’βˆžπ‘(𝑛)/(π‘›βˆ’1)=π‘Ÿ with 0<π‘Ÿ<1.
Let {𝑧𝑗}𝑛𝑗=1 be a set of complex numbers such that |𝑧𝑗|=1 for all 𝑗=1,…,𝑛 and 𝑧𝑖≠𝑧𝑗 for 𝑖≠𝑗 and let π‘Šπ‘›(𝑧)=Π𝑛𝑗=1(π‘§βˆ’π‘§π‘—) be the nodal polynomial. Assume that there exist positive constants 𝐡 and 𝐿 such that for every π‘§βˆˆπ•‹ and 𝑛 large enough the following relations hold:
(i)𝐡≀|π‘Šβ€²π‘›(𝑧)|/𝑛, (ii)|π‘Šπ‘›(𝑧)|2/𝑛2βˆ‘π‘›π‘—=1(1/|π‘§βˆ’π‘§π‘—|2)≀𝐿.
If πΏβˆ’π‘(𝑛),π‘ž(𝑛)(𝐹;𝑧)βˆˆΞ›βˆ’π‘(𝑛),π‘ž(𝑛) is the Laurent polynomial of Lagrange interpolation related to the nodal system and the function 𝐹, then limπ‘›β†’βˆžπΏβˆ’π‘(𝑛),π‘ž(𝑛)(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋.

Proof. First we prove that there exists a positive constant 𝐢 such that βˆ‘π‘›π‘—=1|𝑙𝑗,π‘›βˆ’1√(𝑧)|≀𝐢𝑛 for every π‘§βˆˆπ•‹ and 𝑛 large enough. Indeed, taking into account (2.3) and applying the hypothesis we get 𝑛𝑗=1||𝑙𝑗,π‘›βˆ’1(||=𝑧)𝑛𝑗=1||π‘Šπ‘›(𝑧)𝑧𝑗𝑝(𝑛)||||π‘Šξ…žπ‘›ξ€·π‘§π‘—ξ€Έξ€·π‘§βˆ’π‘§π‘—ξ€Έπ‘§π‘(𝑛)||=𝑛𝑗=1||π‘Šπ‘›||(𝑧)||π‘Šξ…žπ‘›ξ€·π‘§π‘—ξ€Έξ€·π‘§βˆ’π‘§π‘—ξ€Έ||≀1𝐡𝑛𝑛𝑗=1||π‘Šπ‘›||(𝑧)||π‘§βˆ’π‘§π‘—||≀1𝐡𝑛𝑛𝑗=1||π‘Šπ‘›||(𝑧)2||π‘§βˆ’π‘§π‘—||2ξƒͺ1/2𝑛𝑗=11ξƒͺ1/2β‰€βˆšπΏπ΅βˆšπ‘›.(2.4) Let us consider the Laurent polynomial of best uniform approximation to 𝐹, π‘‡βˆ’π‘(𝑛),π‘ž(𝑛)(𝑧)βˆˆΞ›βˆ’π‘(𝑛),π‘ž(𝑛). If πΈβˆ’π‘(𝑛),π‘ž(𝑛)(𝐹)=maxπ‘§βˆˆπ•‹|𝐹(𝑧)βˆ’π‘‡βˆ’π‘(𝑛),π‘ž(𝑛)(𝑧)|, then it holds that πΈβˆ’π‘(𝑛),π‘ž(𝑛)ξ‚΅πœ‹(𝐹)≀2πœ†πΉ,𝑠(𝑛),(2.5) where 𝑠(𝑛)=min(𝑝(𝑛),π‘ž(𝑛)) (see [1]). Since limπ‘›β†’βˆžπœ‹/𝑠(𝑛)=0, then by hypothesis πœ†(𝐹,πœ‹/𝑠(𝑛))=π‘œ((πœ‹/𝑠(𝑛))1/2).
If we write𝐹(𝑧)βˆ’πΏβˆ’π‘(𝑛),π‘ž(𝑛)(𝐹;𝑧)=𝐹(𝑧)βˆ’π‘‡βˆ’π‘(𝑛),π‘ž(𝑛)(𝑧)βˆ’πΏβˆ’π‘(𝑛),π‘ž(𝑛)(𝐹;𝑧)+π‘‡βˆ’π‘(𝑛),π‘ž(𝑛)(𝑧)=𝐹(𝑧)βˆ’π‘‡βˆ’π‘(𝑛),π‘ž(𝑛)(𝑧)βˆ’πΏβˆ’π‘(𝑛),π‘ž(𝑛)ξ€·πΉβˆ’π‘‡βˆ’π‘(𝑛),π‘ž(𝑛)ξ€Έ;𝑧=𝐹(𝑧)βˆ’π‘‡βˆ’π‘(𝑛),π‘ž(𝑛)(𝑧)βˆ’π‘›ξ“π‘—=1𝑙𝑗,π‘›βˆ’1𝐹𝑧(𝑧)π‘—ξ€Έβˆ’π‘‡βˆ’π‘(𝑛),π‘ž(𝑛)𝑧𝑗,ξ€Έξ€Έ(2.6) then we have ||𝐹(𝑧)βˆ’πΏβˆ’π‘(𝑛),π‘ž(𝑛)||≀||(𝐹;𝑧)𝐹(𝑧)βˆ’π‘‡βˆ’π‘(𝑛),π‘ž(𝑛)||+(𝑧)𝑛𝑗=1||𝑙𝑗,π‘›βˆ’1||||𝐹𝑧(𝑧)π‘—ξ€Έβˆ’π‘‡βˆ’π‘(𝑛),π‘ž(𝑛)𝑧𝑗||β‰€πΈβˆ’π‘(𝑛),π‘ž(𝑛)(𝐹)1+𝑛𝑗=1||𝑙𝑗,π‘›βˆ’1||ξƒͺξ‚€βˆš(𝑧)≀2πœ†(𝐹,πœ‹/𝑠(𝑛))1+𝐢𝑛=2πœ†(𝐹,πœ‹/𝑠(𝑛))(πœ‹/𝑠(𝑛))1/2βˆšπœ‹(𝑠(𝑛)/(π‘›βˆ’1))1/2√1+πΆπ‘›βˆš,π‘›βˆ’1(2.7) and it is easy to prove that the last expression tends to zero because limπ‘›β†’βˆžπœ†(𝐹,πœ‹/𝑠(𝑛))/(πœ‹/𝑠(𝑛))1/2=0 and limπ‘›β†’βˆžπ‘ (𝑛)=1π‘›βˆ’12ξ‚΅limπ‘›β†’βˆžπ‘(𝑛)π‘›βˆ’1+limπ‘›β†’βˆžπ‘ž(𝑛)π‘›βˆ’1βˆ’limπ‘›β†’βˆž||||𝑝(𝑛)βˆ’π‘ž(𝑛)ξ‚Ά=⎧βŽͺ⎨βŽͺβŽ©ξ‚€1π‘›βˆ’11βˆ’π‘Ÿ,ifπ‘Ÿβˆˆ2,ξ‚€1,1π‘Ÿ,ifπ‘Ÿβˆˆ0,2ξ‚„.(2.8)

Remark 2.3. (i) Since πœ†(𝐹,𝛿)=π‘œ(𝛿𝑝) for 𝑝>1/2 implies πœ†(𝐹,𝛿)=π‘œ(𝛿1/2), then the preceding result is also valid for functions with modulus of continuity π‘œ(𝛿𝑝), with 𝑝>1/2, if 𝛿→0. Hence, in the sequel and for simplicity, we establish all the results with the condition πœ†(𝐹,𝛿)=π‘œ(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 π‘œ(𝛿1/2), 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 SzegΕ‘ 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œ‹].

Theorem 2.4. Let 𝜈 be a measure on [0,2πœ‹] in the SzegΕ‘ class with SzegΕ‘ function having analytic extension up to |𝑧|>1. Let {πœ™π‘›(𝑧)} be the monic orthogonal polynomial sequence with respect to 𝜈, MOPS(𝜈), and πœ”π‘›(𝑧,𝜏)=πœ™π‘›(𝑧)+πœπœ™βˆ—π‘›(𝑧), with |𝜏|=1 being the para-orthogonal polynomials. Then there exist positive constants 𝐴,𝐡1,𝐡2, and 𝐿 such that for every π‘§βˆˆπ•‹ and 𝑛 large enough the following relations hold: (i)|πœ”π‘›(𝑧,𝜏)|≀𝐴, (ii)𝐡1≀|πœ”β€²π‘›(𝑧,𝜏)|/𝑛≀𝐡2, (iii)|πœ”π‘›(𝑧,𝜏)|2/𝑛2βˆ‘π‘›π‘—=1(1/|π‘§βˆ’π‘§π‘—|2)≀𝐿, where one assumes that 𝑧1,…,𝑧𝑛 are the zeros of πœ”π‘›(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 πœ†(𝐹,𝛿)=π‘œ(𝛿1/2), if 𝛿→0. Let 𝑝(𝑛) and π‘ž(𝑛) be two nondecreasing sequences of nonnegative integers such that 𝑝(𝑛)+π‘ž(𝑛)=π‘›βˆ’1 and limπ‘›β†’βˆžπ‘(𝑛)/(π‘›βˆ’1)=π‘Ÿ with 0<π‘Ÿ<1.
Let 𝜈 be a measure on [0,2πœ‹] in the SzegΕ‘ class with SzegΕ‘ function having analytic extension up to |𝑧|>1. Let {πœ™π‘›(𝑧)} be the MOPS(𝜈) and let πœ”π‘›(𝑧,𝜏)=πœ™π‘›(𝑧)+πœπœ™βˆ—π‘›(𝑧), with |𝜏|=1, be the para-orthogonal polynomials.
If πΏβˆ’π‘(𝑛),π‘ž(𝑛)(𝐹;𝑧)βˆˆΞ›βˆ’π‘(𝑛),π‘ž(𝑛) is the Laurent polynomial of Lagrange interpolation related to the function 𝐹 and with nodal system the zeros of the para-orthogonal polynomials πœ”π‘›(𝑧,𝜏), then limπ‘›β†’βˆžπΏβˆ’π‘(𝑛),π‘ž(𝑛)(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋.

Proof. Taking into account that the zeros of πœ”π‘›(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-SzegΕ‘ measures (see [13]).

3. Lagrange Interpolation on [βˆ’1,1]

In this section we present some consequences of Theorem 2.2 concerning the Lagrange interpolation problems on [βˆ’1,1]. Let us recall that the Lagrange interpolation polynomial related to a nodal system {π‘₯𝑗}𝑛𝑗=1βŠ‚[βˆ’1,1] and satisfying the conditions {𝑒𝑗}𝑛𝑗=1 is given by π‘™π‘›βˆ’1βˆ‘(π‘₯)=𝑛𝑗=1(𝑝𝑛(π‘₯)/(π‘ξ…žπ‘›(π‘₯𝑗)(π‘₯βˆ’π‘₯𝑗)))𝑒𝑗, where 𝑝𝑛(π‘₯)=Π𝑛𝑗=1(π‘₯βˆ’π‘₯𝑗).

Theorem 3.1. Let π‘π‘›βˆ(π‘₯)=𝑛𝑗=1(π‘₯βˆ’π‘₯𝑗) be a nodal system in [βˆ’1,1] such that π‘Š2𝑛(𝑧)=2𝑛𝑧𝑛𝑝𝑛((𝑧+1/𝑧)/2) satisfies the following inequalities ||π‘Šπ΅β‰€ξ…ž2𝑛||(𝑧),||π‘Š2𝑛2𝑛||(𝑧)2𝑛𝑗=11||π‘§βˆ’π‘§π‘—||2+1||π‘§βˆ’π‘§π‘—||2ξƒͺ≀𝐿(2𝑛)2,(3.1) with (𝑧𝑗+(1/𝑧𝑗))/2=π‘₯𝑗 for 𝑗=1,…,𝑛 and for some positive constants 𝐡 and 𝐿, 𝑛 large enough and every π‘§βˆˆπ•‹.
Let 𝑓 be a continuous function on [βˆ’1,1] such that πœ†(𝑓,𝛿)=π‘œ(𝛿1/2), if 𝛿→0.
If π‘™π‘›βˆ’1(𝑓,π‘₯) is the Lagrange interpolation polynomial such that π‘™π‘›βˆ’1(𝑓,π‘₯𝑗)=𝑓(π‘₯𝑗) for 𝑗=1,…,𝑛, then π‘™π‘›βˆ’1(𝑓,π‘₯) converges to 𝑓(π‘₯) uniformly on [βˆ’1,1].

Proof. It is easy to see that the polynomial π‘Š2𝑛(𝑧) has the following expression π‘Š2𝑛(𝑧)=Π𝑛𝑗=1(π‘§βˆ’π‘§π‘—)(π‘§βˆ’π‘§π‘—), with (𝑧𝑗+𝑧𝑗)/2=π‘₯𝑗.
Let us define a continuous function on 𝕋 by 𝐹(𝑧)=𝐹(𝑧)=𝑓(π‘₯), with π‘₯=(𝑧+(1/𝑧))/2 and π‘§βˆˆπ•‹. It is clear thatπœ†(𝐹,𝛿)=sup𝑧1,𝑧2βˆˆπ•‹;|𝑧1βˆ’π‘§2|<𝛿||𝐹𝑧1ξ€Έξ€·π‘§βˆ’πΉ2ξ€Έ||≀supπ‘₯1,π‘₯2∈[βˆ’1,1];|π‘₯1βˆ’π‘₯2|<𝛿||𝑓π‘₯1ξ€Έξ€·π‘₯βˆ’π‘“2ξ€Έ||=πœ†(𝑓,𝛿).(3.2)
If we take π‘Š2𝑛(𝑧) as nodal system on 𝕋, we can consider the following Lagrange interpolation problem: find the Laurent polynomial of Lagrange interpolation πΏβˆ’π‘›,π‘›βˆ’1(𝐹;𝑧)βˆˆΞ›βˆ’π‘›,π‘›βˆ’1 satisfying the interpolation conditionsπΏβˆ’π‘›,π‘›βˆ’1𝐹;𝑧𝑗=πΏβˆ’π‘›,π‘›βˆ’1𝐹;𝑧𝑗π‘₯=𝑓𝑗,𝑗=1,…,𝑛.(3.3) By applying Theorem 2.2 we have that limπ‘›β†’βˆžπΏβˆ’π‘›,π‘›βˆ’1(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋.
On the other hand, for π‘₯=(𝑧+(1/𝑧))/2 and π‘§βˆˆπ•‹ it holdsπΏβˆ’π‘›,π‘›βˆ’1(𝐹;𝑧)=𝑛𝑗=1π‘Š2𝑛(𝑧)π‘§π‘›π‘—π‘§π‘›π‘Šξ…ž2π‘›ξ€·π‘§π‘—ξ€Έξ€·π‘§βˆ’π‘§π‘—ξ€ΈπΉξ€·π‘§π‘—ξ€Έ+𝑛𝑗=1π‘Š2𝑛(𝑧)π‘§π‘—π‘›π‘§π‘›π‘Šξ…ž2π‘›ξ€·π‘§π‘—ξ€Έξ€·π‘§βˆ’π‘§π‘—ξ€ΈπΉξ€·π‘§π‘—ξ€Έ=𝑛𝑗=1𝑝𝑛(π‘₯)π‘ξ…žπ‘›ξ€·π‘₯𝑗π‘₯βˆ’π‘₯𝑗𝑓π‘₯𝑗=π‘™π‘›βˆ’1(𝑓;π‘₯).(3.4) Hence limπ‘›β†’βˆžπ‘™π‘›βˆ’1(𝑓,π‘₯)=𝑓(π‘₯) uniformly on [βˆ’1,1].

As a consequence we obtain, in the next corollary, a result that was proved by SzegΕ‘ 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 [βˆ’1,1] such that πœ†(𝑓,𝛿)=o(𝛿1/2), if 𝛿→0. Let π‘‘πœ‡(π‘₯)=𝑀(π‘₯)𝑑π‘₯ be a finite positive Borel measure on [βˆ’1,1] satisfying the SzegΕ‘ condition ∫1βˆ’1√(log𝑀(π‘₯)/1βˆ’π‘₯2)𝑑π‘₯>βˆ’βˆž, and let {𝑃𝑛(π‘₯)} be the MOPS(πœ‡). Assume that the function βˆšπ‘€(π‘₯)1βˆ’π‘₯2 is positive on [βˆ’1,1] and it is analytic in an open set containing [βˆ’1,1].
If π‘™π‘›βˆ’1(𝑓,π‘₯) is the Lagrange interpolation polynomial satisfying the interpolation conditions π‘™π‘›βˆ’1(𝑓,π‘₯𝑗)=𝑓(π‘₯𝑗),𝑗=1,…,𝑛, where {π‘₯𝑗}𝑛𝑗=1 are the zeros of the orthogonal polynomial 𝑃𝑛(π‘₯), thenlimπ‘›β†’βˆžπ‘™π‘›βˆ’1(𝑓,π‘₯)=𝑓(π‘₯)(3.5) uniformly on [βˆ’1,1].

Proof. By using the SzegΕ‘ transformation (see [10]), the measure π‘‘πœ‡(π‘₯) becomes into the measure π‘‘πœˆ(πœƒ)=(1/2)𝑀(cosπœƒ)|sinπœƒ|π‘‘πœƒ, which is in the SzegΕ‘ class with SzegΕ‘ function having analytic extension up to |𝑧|>1 (see [14]). If we denote by {πœ™π‘›(𝑧)} the MOPS(𝜈) and by {𝑃𝑛(𝑧)} the MOPS(πœ‡), then both sequences are related by𝑃𝑛1(π‘₯)=2𝑛1+πœ™2𝑛(ξ€Έπœ™0)2𝑛(𝑧)+πœ™βˆ—2𝑛(𝑧)𝑧𝑛=12𝑛1+πœ™2𝑛(ξ€Έπœ”0)2𝑛(𝑧,1)𝑧𝑛.(3.6)
The zeros of 𝑃𝑛(π‘₯), π‘₯1,…,π‘₯𝑛, are simple and belong to (βˆ’1,1) and they are related to the zeros of πœ”2𝑛(𝑧,1), 𝑧1,…,𝑧𝑛,𝑧𝑛+1=𝑧𝑛,…,𝑧2𝑛=𝑧1, by π‘₯𝑗=(𝑧𝑗+𝑧𝑗)/2,𝑗=1,…,𝑛. By applying Theorem 2.4 we get that the system πœ”2𝑛(𝑧,1) satisfies the hypothesis of Theorem 3.1. Then we have that π‘™π‘›βˆ’1(𝑓,π‘₯) converges to 𝑓(π‘₯) uniformly on [βˆ’1,1].

Analogous results can be obtained for other nodal systems related to those given in Corollary 3.2. Let π‘‘πœ‡1(π‘₯)=𝑀(π‘₯)𝑑π‘₯ be a finite positive Borel measure on [βˆ’1,1]. Let us consider the measuresπ‘‘πœ‡2ξ€·(π‘₯)=1βˆ’π‘₯2ξ€Έπ‘‘πœ‡1(π‘₯),π‘‘πœ‡3(π‘₯)=(1βˆ’π‘₯)π‘‘πœ‡1(π‘₯),π‘‘πœ‡4(π‘₯)=(1+π‘₯)π‘‘πœ‡1(π‘₯),(3.7) and let us denote the MOPS with respect to these measures by {𝑃𝑛(π‘₯,πœ‡π‘–)}4𝑖=1. Let us consider the SzegΕ‘ transformed measure of π‘‘πœ‡1(π‘₯), π‘‘πœˆ(πœƒ)=(1/2)𝑀(cosπœƒ)|sinπœƒ|π‘‘πœƒ with MOPS(𝜈), {πœ™π‘›(𝑧)}. Taking into account the relation between the measures, we can relate the orthogonal sequences as follows (see [16]):𝑃𝑛π‘₯,πœ‡1ξ€Έ=12𝑛1+πœ™2𝑛𝑧(0)𝑛𝑀2π‘›βˆš(𝑧,1),1βˆ’π‘₯2𝑃𝑛π‘₯,πœ‡2ξ€Έ=12𝑛+1πš€ξ€·1βˆ’πœ™2𝑛+2𝑧(0)𝑛+1𝑀2𝑛+2√(𝑧,βˆ’1),1βˆ’π‘₯𝑃𝑛π‘₯,πœ‡3ξ€Έ=12𝑛+1/2πš€ξ€·1βˆ’πœ™2𝑛+1𝑧(0)𝑛+(1/2)𝑀2𝑛+1√(𝑧,βˆ’1),1+π‘₯𝑃𝑛π‘₯,πœ‡4ξ€Έ=12𝑛+1/2ξ€·1+πœ™2𝑛+1𝑧(0)𝑛+(1/2)𝑀2𝑛+1(𝑧,1).(3.8)

We denote by Β±1,π‘₯1,…,π‘₯𝑛 the zeros of √1βˆ’π‘₯2𝑃𝑛(π‘₯,πœ‡2), by 1,𝑦1,…,𝑦𝑛 the zeros of √1βˆ’π‘₯𝑃𝑛(π‘₯,πœ‡3), and by βˆ’1,𝑣1,…,𝑣𝑛 the zeros of √1+π‘₯𝑃𝑛(π‘₯,πœ‡4).

If we denote by Β±1,𝑧1,…,𝑧𝑛,𝑧1,…,𝑧𝑛 the zeros of πœ”2𝑛+2(𝑧,βˆ’1), by 1,𝑀1,…,𝑀𝑛,𝑀1,…,𝑀𝑛 the zeros of 𝑀2𝑛+1(𝑧,βˆ’1), and by βˆ’1,𝑒1,…,𝑒𝑛,𝑒1,…,𝑒𝑛 the zeros of 𝑀2𝑛+1(𝑧,1), then the following relations hold: β„œ(𝑧𝑖)=π‘₯𝑖,β„œ(𝑀𝑖)=𝑦𝑖, and β„œ(𝑒𝑖)=𝑣𝑖,𝑖=1,…,𝑛. By taking nodal systems related to the zeros of 𝑃𝑛(π‘₯,πœ‡π‘–), 𝑖=2,3,4, we obtain the next result.

Theorem 3.3. Let 𝑓 be a continuous function on [βˆ’1,1] such that πœ†(𝑓,𝛿)=o(𝛿1/2), if 𝛿→0. Let πœ‡1 be a finite positive Borel measure on [βˆ’1,1], π‘‘πœ‡1(π‘₯)=𝑀(π‘₯)dπ‘₯, satisfying the SzegΕ‘ condition. Assume that the function βˆšπ‘€(π‘₯)1βˆ’π‘₯2 is positive in [βˆ’1,1] and it is analytic in an open set containing [βˆ’1,1]. Let π‘‘πœ‡π‘–(π‘₯),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(π‘₯,πœ‡2) joint with Β±1,(ii)the zeros of 𝑃𝑛(π‘₯,πœ‡3) joint with 1,(iii)the zeros of 𝑃𝑛(π‘₯,πœ‡4) joint with βˆ’1.Then the corresponding Lagrange interpolation polynomials uniformly converge to 𝑓(π‘₯) on [βˆ’1,1].

Proof. By the SzegΕ‘ transformation the measure π‘‘πœ‡1(π‘₯) becomes into the measure π‘‘πœˆ(πœƒ)=(1/2)𝑀(cosπœƒ)|sinπœƒ|π‘‘πœƒ, which is in the SzegΕ‘ class with SzegΕ‘ function having analytic extension up to |𝑧|>1. We denote by {πœ™π‘›(𝑧)} the MOPS(𝜈). If we define a continuous function 𝐹 on 𝕋 by 𝐹(𝑧)=𝐹(𝑧)=𝑓(π‘₯) with π‘₯=(𝑧+1/𝑧)/2 and π‘§βˆˆπ•‹, then it is clear that πœ†(𝐹,𝛿)β‰€πœ†(𝑓,𝛿).
(i) We consider the para-orthogonal polynomial πœ”2𝑛+2(𝑧,βˆ’1), whose zeros are Β±1,𝑧1,…,𝑧𝑛, 𝑧1,…,π‘§π‘›βˆˆπ•‹ and they are related to the zeros of 𝑃𝑛(π‘₯,πœ‡2) by π‘₯𝑗=(𝑧𝑗+𝑧𝑗)/2;𝑗=1,…,𝑛.
Let us consider the following Lagrange interpolation problem: find the Laurent polynomial of Lagrange interpolation πΏβˆ’(𝑛+1),𝑛(𝐹;𝑧) satisfying πΏβˆ’(𝑛+1),𝑛𝐹;𝑧𝑗=πΏβˆ’(𝑛+1),𝑛𝐹;𝑧𝑗𝑧=𝐹𝑗𝐿,𝑗=1,…,𝑛,βˆ’(𝑛+1),𝑛(𝐹;1)=𝐹(1),πΏβˆ’(𝑛+1),𝑛(𝐹;βˆ’1)=𝐹(βˆ’1).(3.9) By applying Corollary 2.5 we have that limπ‘›β†’βˆžπΏβˆ’(𝑛+1),𝑛(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋. If we take 𝑙𝑛+1𝐿(𝑓,π‘₯)=βˆ’(𝑛+1),𝑛(𝐹;𝑧)+πΏβˆ’(𝑛+1),𝑛(𝐹;1/𝑧)2(3.10) for π‘₯=(𝑧+(1/𝑧))/2, then 𝑙𝑛+1(𝑓,π‘₯) fulfills 𝑙𝑛+1(𝑓,π‘₯𝑗)=𝑓(π‘₯𝑗),𝑗=1,…,𝑛, and 𝑙𝑛+1(𝑓,Β±1)=𝑓(Β±1). Therefore, 𝑙𝑛+1(𝑓,π‘₯) is the Lagrange interpolation polynomial for the function 𝑓 and the nodal system given in (i) and limπ‘›β†’βˆžπ‘™π‘›+1(𝑓,π‘₯)=𝑓(π‘₯) uniformly on [βˆ’1,1].
(ii) We consider the para-orthogonal polynomials πœ”2𝑛+1(𝑧,βˆ’1) whose zeros, 1,𝑀1,…,𝑀𝑛,𝑀1,…,𝑀𝑛, are related to the zeros of 𝑃𝑛(π‘₯,πœ‡3) by 𝑦𝑗=(𝑀𝑗+𝑀𝑗)/2,𝑗=1,…,𝑛.
We pose the problem of finding the Laurent polynomial of Lagrange interpolation πΏβˆ’π‘›,𝑛(𝐹;𝑧) satisfying πΏβˆ’π‘›,𝑛𝐹;𝑀𝑗=πΏβˆ’π‘›,𝑛𝐹;𝑀𝑗𝑀=𝐹𝑗𝐿,𝑗=1,…,𝑛,βˆ’π‘›,𝑛(𝐹;1)=𝐹(1).(3.11) Since limπ‘›β†’βˆžπΏβˆ’π‘›,𝑛(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋, if we define 𝑙𝑛(𝑓,π‘₯)=(πΏβˆ’π‘›,𝑛(𝐹;𝑧)+πΏβˆ’π‘›,𝑛(𝐹;1/𝑧))/2 for π‘₯=(𝑧+(1/𝑧))/2 and π‘§βˆˆπ•‹, then 𝑙𝑛(𝑓,π‘₯) fulfills 𝑙𝑛(𝑓,π‘₯𝑗)=𝑓(π‘₯𝑗),𝑗=1,…,𝑛, and 𝑙𝑛(𝑓,1)=𝑓(1). Therefore, limπ‘›β†’βˆžπ‘™π‘›(𝑓,π‘₯)=𝑓(π‘₯) uniformly on [βˆ’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 Β±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 βˆ’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œ‹] and they are obtained as follows. Let π‘‘πœ‡(π‘₯)=𝑀(π‘₯)𝑑π‘₯ be a positive finite Borel measure on [βˆ’1,1] satisfying the SzegΕ‘ condition. Assume that the function βˆšπ‘€(π‘₯)1βˆ’π‘₯2>0 for all π‘₯∈[βˆ’1,1] and it is analytic in an open set containing [βˆ’1,1]. If {𝑃𝑛(π‘₯)} is the MOPS(πœ‡) and {π‘₯𝑗}𝑛𝑗=1 are the zeros of 𝑃𝑛(π‘₯), we consider the following nodal system on [0,2πœ‹],{πœƒπ‘—}2𝑛𝑗=1, such that πœƒπ‘—=arccosπ‘₯𝑗,𝑗=1,…,𝑛 with 0<πœƒπ‘—<πœ‹ and πœƒπ‘›+𝑗=2πœ‹βˆ’πœƒπ‘›βˆ’π‘—+1 for 𝑗=1,…,𝑛; that is, the points are symmetric with respect to πœ‹.

Theorem 4.1. Let 𝑓 be a real continuous function on [0,2πœ‹], with modulus of continuityβ€‰β€‰πœ†(𝑓,𝛿)=π‘œ(𝛿1/2), if 𝛿→0.
Let the nodal system be {πœƒπ‘—}2𝑛𝑗=1 with πœƒπ‘—=arccosπ‘₯𝑗,𝑗=1,…,𝑛 with 0<πœƒπ‘—<πœ‹ and πœƒπ‘›+𝑗=2πœ‹βˆ’πœƒπ‘›βˆ’π‘—+1 for 𝑗=1,…,𝑛, where {π‘₯𝑗}𝑛𝑗=1 are the zeros of the orthogonal polynomial 𝑃𝑛(π‘₯) with respect to the measure π‘‘πœ‡. One also assumes that π‘‘πœ‡(π‘₯)=𝑀(π‘₯)𝑑π‘₯ is a positive finite Borel measure on [βˆ’1,1] satisfying the SzegΕ‘ condition and such that βˆšπ‘€(π‘₯)1βˆ’π‘₯2>0 for all π‘₯∈[βˆ’1,1] and it is analytic in an open set containing [βˆ’1,1].
Then there is a Lagrange interpolation trigonometric polynomial of degree ≀𝑛, πœπ‘›(πœƒ), such that πœπ‘›(πœƒπ‘—)=𝑓(πœƒπ‘—) for 𝑗=1,…,2𝑛 and it satisfies that limπ‘›β†’βˆžπœπ‘›(πœƒ)=𝑓(πœƒ) uniformly on [0,2πœ‹].

Proof. Proceeding like in Corollary 3.2 we obtain that the transformed measure of π‘‘πœ‡(π‘₯) by the SzegΕ‘ transformation, π‘‘πœˆ(πœƒ), 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(π‘’πš€πœƒ)=𝑓(πœƒ) for πœƒβˆˆ[0,2πœ‹]. Let πΏβˆ’π‘›,π‘›βˆ’1(𝐹,𝑧) be the Lagrange interpolation polynomial such that πΏβˆ’π‘›,π‘›βˆ’1(𝐹,𝑧𝑗)=𝑓(πœƒπ‘—), where 𝑧𝑗=π‘’πš€πœƒπ‘—,𝑗=1,…,2𝑛.
Since 𝐹 is continuous on 𝕋 and πœ†(𝐹,𝛿)=π‘œ(𝛿1/2), we can apply Corollary 3.2 and therefore limπ‘›β†’βˆžπΏβˆ’π‘›,π‘›βˆ’1(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋.
If we take πœπ‘›(πœƒ)=β„œ(πΏβˆ’π‘›,π‘›βˆ’1(𝐹;π‘’πš€πœƒ)), then πœπ‘›(πœƒ) satisfies the interpolation conditions, πœπ‘›(πœƒπ‘—)=𝑓(πœƒπ‘—), and limπ‘›β†’βˆžπœπ‘›(πœƒ)=𝑓(πœƒ).

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

Theorem 4.2. Let 𝜈 be a measure on [0,2πœ‹] in the SzegΕ‘ class with SzegΕ‘ function having analytic extension up to |𝑧|>1. Let {𝑧𝑗}𝑛𝑗=1 be the zeros of the para-orthogonal polynomials πœ”π‘›(𝑧)=πœ™π‘›(𝑧)+πœπœ™βˆ—π‘›(𝑧), with |𝜏|=1, and let πœƒπ‘—βˆˆ[0,2πœ‹] such that π‘’πš€πœƒπ‘—=𝑧𝑗,𝑗=1,…,𝑛.
If 𝑓 is a continuous function on [0,2πœ‹] with πœ†(𝑓,𝛿)=π‘œ(𝛿1/2), if 𝛿→0, then there is a Lagrange interpolation trigonometric polynomial of degree ≀[𝑛/2], 𝜏[𝑛/2](πœƒ), such that 𝜏[𝑛/2](πœƒπ‘—)=𝑓(πœƒπ‘—) for 𝑗=1,…,𝑛 and it satisfies that limπ‘›β†’βˆžπœ[𝑛/2](πœƒ)=𝑓(πœƒ) uniformly on [0,2πœ‹].

Proof. Let 𝐹 be a continuous function defined by 𝐹(π‘’πš€πœƒ)=𝑓(πœƒ). Since πœ†(𝐹,𝛿)β‰€πœ†(𝑓,𝛿), then πœ†(𝐹,𝛿)=π‘œ(𝛿1/2). By applying Corollary 2.5 we obtain for 𝑝(𝑛)+π‘ž(𝑛)=π‘›βˆ’1, with limπ‘›β†’βˆžπ‘(𝑛)/(π‘›βˆ’1)=π‘Ÿ and 0<π‘Ÿ<1, there exists πΏβˆ’π‘(𝑛),π‘ž(𝑛)(𝐹;𝑧) such that πΏβˆ’π‘(𝑛),π‘ž(𝑛)(𝐹;π‘’πš€πœƒπ‘—)=𝑓(πœƒπ‘—) and limπ‘›β†’βˆžπΏβˆ’π‘(𝑛),π‘ž(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 π‘ž(𝑛)=𝑛/2βˆ’1. Then [𝑛/2]=𝑛/2.(ii)If 𝑛 is odd, we take 𝑝(𝑛)=(π‘›βˆ’1)/2 and π‘ž(𝑛)=(π‘›βˆ’1)/2. Then [𝑛/2]=(π‘›βˆ’1)/2.In any case the real part of πΏβˆ’π‘(𝑛),π‘ž(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 InnovaciΓ³n under grant number MTM2011-22713.

References

  1. L. Daruis and P. GonzΓ‘lez-Vera, β€œSome results about interpolation with nodes on the unit circle,” Indian Journal of Pure and Applied Mathematics, vol. 31, no. 10, pp. 1273–1296, 2000. View at: Google Scholar | Zentralblatt MATH
  2. E. Berriochoa, A. Cachafeiro, and E. MartΔΊnez, β€œAbout measures and nodal systems for which the Hermite interpolants uniformly converge to continuous functions on the circle and interval,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 4813–4824, 2012. View at: Google Scholar
  3. J.-P. Berrut and L. N. Trefethen, β€œBarycentric Lagrange interpolation,” SIAM Review, vol. 46, no. 3, pp. 501–517, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  4. T. J. Rivlin, The Chebyshev Polynomials, Wiley-Interscience, New York, NY, USA, 1974.
  5. J. Szabados and P. VΓ©rtesi, Interpolation of Functions, World Scientific Publishing, Teaneck, NJ, USA, 1990.
  6. P. TurΓ‘n, β€œOn some open problems of approximation theory,” Journal of Approximation Theory, vol. 29, no. 1, pp. 23–85, 1980. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. G. GrΓΌnwald, β€œOn the theory of interpolation,” Acta Mathematica, vol. 75, pp. 219–245, 1943. View at: Google Scholar | Zentralblatt MATH
  8. J. Marcinkiewicz, β€œSur la divergence des polynmes d'interpolation,” Acta Scientiarum Mathematicarum, vol. 8, pp. 131–135, 1937. View at: Google Scholar
  9. P. ErdΓΆs and P. VΓ©rtesi, β€œOn the almost everywhere divergence of Lagrange interpolation,” in Approximation and Function Spaces (GdaΕ„sk, 1979), pp. 270–278, North-Holland, Amsterdam, The Netherlands, 1981. View at: Google Scholar
  10. G. SzegΕ‘, Orthogonal Polynomials, American Mathematical Society Colloquium Publications, vol. 23, American Mathematical Society, Providence, RI, USA, 4th edition, 1975.
  11. W. B. Jones, O. NjΓ₯stad, and W. J. Thron, β€œMoment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle,” The Bulletin of the London Mathematical Society, vol. 21, no. 2, pp. 113–152, 1989. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  12. L. Golinskii, β€œQuadrature formula and zeros of para-orthogonal polynomials on the unit circle,” Acta Mathematica Hungarica, vol. 96, no. 3, pp. 169–186, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  13. B. Simon, Orthogonal Polynomials on the Unit Circle, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, USA, 2005.
  14. P. Nevai and V. Totik, β€œOrthogonal polynomials and their zeros,” Acta Scientiarum Mathematicarum, vol. 53, no. 1-2, pp. 99–104, 1989. View at: Google Scholar | Zentralblatt MATH
  15. L. Ya. Geronimus, Orthogonal Polynomials: Estimates, Asymptotic Formulas, and Series of Polynomials Orthogonal on the Unit Circle and on an Interval, Authorized Translation from the Russian, Consultants Bureau, New York, NY, USA, 1961.
  16. J. M. GarcΓ­a Amor, Ortogonalidad Bernstein-Chebyshev en la recta real, Doctoral Dissertation, Universidad de Vigo, 2003.

Copyright Β© 2012 E. Berriochoa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views846
Downloads464
Citations

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.