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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 307974, 8 pages
http://dx.doi.org/10.1155/2013/307974
Research Article

-Coherent Pairs on the Unit Circle

1Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo 340, 28045 Colima, COL, Mexico
2Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain

Received 23 July 2013; Accepted 19 September 2013

Academic Editor: Jinde Cao

Copyright © 2013 Luis Garza 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.

Abstract

A pair of Hermitian regular linear functionals on the unit circle is said to be a -coherent pair if their corresponding sequences of monic orthogonal polynomials and satisfy , , , where . In this contribution, we consider the cases when is the linear functional associated with the Lebesgue and Bernstein-Szegő measures, respectively, and we obtain a classification of the situations where is associated with either a positive nontrivial measure or its rational spectral transformation.

1. Introduction

A pair of regular linear functionals on the linear space of polynomials with real coefficients is a -coherent pair if and only if their corresponding sequences of monic orthogonal polynomials (SMOP) and satisfy This concept is a generalization of the notion of coherent pair, for us -coherent pair, introduced by Iserles et al. in [1], where , for every .

In the work by Delgado and Marcellán [2], the notion of a generalized coherent pair of measures, in short, -coherent pair of measures, arose as a necessary and sufficient condition for the existence of an algebraic relation between the SMOP associated with the Sobolev inner product and the SMOP associated with the positive Borel measure in the real line as follows: where are rational functions in . Besides, they obtained the classification of all -coherent pairs of regular functionals and proved that at least one of them must be semiclassical of class at most , and and are related by a rational type expression. This is a generalization of the results of Meijer [3] for the -coherence case (when , ), where either or must be a classical linear functional.

The most general case of the notion of coherent pair was studied by de Jesus et al. in [4] (see also [5]), the so-called -coherent pairs of order , where the derivatives of order and of two SMOP and with respect to the regular linear functionals and are related by where , , , and the real numbers satisfy some natural conditions. They showed that the regular linear functionals and are related by a rational factor, and, when , those linear functionals are semiclassical. Besides, they proved that if is a -coherent pair of order of positive Borel measures on the real line, then holds, where , , , are rational functions in such that for , and is the Sobolev SMOP with respect to the inner product , . Also, they showed that -coherence of order is a necessary condition for the algebraic relation (5). For a historical summary about coherent pairs on the real line, see, for example, the introductory sections in the recent papers of de Jesus et al. [6] and of Marcellán and Pinzón-Cortés [7].

On the other hand, the notion of coherent pair was extended to the theory of orthogonal polynomials in a discrete variable by Area et al. in [810]. They used the difference operator as well as the -derivative operator defined by instead of the usual derivative operator . In this way, they obtained similar results to those by Meijer and similar classification as a limit case when either or , respectively. Likewise, Marcellán and Pinzón-Cortés in [11, 12] studied the analogue of the generalized coherent pairs introduced by Delgado and Marcellán, that is, -coherent pairs and -coherent pairs. Finally, Álvarez-Nodarse et al. [13] analyzed the more general case, -coherent pairs of order and -coherent pairs of order , proving the analogue results to those in [4].

Furthermore, Branquinho et al. in [14] extended the concept of coherent pair to Hermitian linear functionals associated with nontrivial probability measures supported on the unit circle. They studied (3) in the framework of orthogonal polynomials on the unit circle (OPUC). Also, they concluded that if is a -coherent pair of Hermitian regular linear functionals, then is semiclassical and is quasiorthogonal of order at most 6 with respect to the functional , . Besides, they analyzed the cases when either or is the Lebesgue measure or is the Bernstein-Szegő measure.

Later on, Branquinho and Rebocho in [15] obtained that if the sequences and satisfy, for , with , , and some extra conditions, then and are semiclassical sequences of OPUC. Moreover, when for all and under some extra conditions, (8) is a necessary condition for the semiclassical character of . Finally, they analyzed the -coherence case , , , when is the linear functional associated with either the Lebesgue measure or the Bernstein-Szegő measure.

The aim of our contribution is to describe the -coherence pair (, ) when and are regular linear functionals, focusing our attention on the cases when is either the Lebesgue or the Bernstein-Szegő linear functional. The structure of this work is as follows. In Section 2, we state some definitions and basic results which will be useful in the forthcoming sections. In Section 3, we introduce the concept of -coherent pair of Hermitian regular linear functionals, and we obtain some results that will be applied in the sequel. In Section 4, we analyze -coherent pairs when is the linear functional associated with the Lebesgue measure on the unit circle. We determine the cases when the linear functional is associated with a positive measure on the unit circle, or a rational spectral transformation of it. Finally, in Section 5, we deal with a similar analysis for the case when is the linear functional associated with the Bernstein-Szegő measure.

2. Preliminaries

Let us consider the unit circle , the linear space of Laurent polynomials with complex coefficients , and a linear functional . We can associate with a sequence of moments defined by , , and a bilinear form as follows: where , , the linear space of polynomials with complex coefficients. Its Gram matrix with respect to is an infinite Toeplitz matrix with leading principal minors given by , .

The linear functional is said to be Hermitian if , quasidefinite or regular if for all , and positive definite if for all . We will denote by the set of Hermitian linear functionals defined on .

is regular if and only if there exists a (unique) sequence of monic orthogonal polynomials on the unit circle (OPUC) ; this is, it satisfies that and , with , for , . Every monic OPUC has an explicit representation, the so-called Heine’s formula, as follows: Besides, they satisfy the forward and backward Szegő recurrence relations where , , are said to be the Verblunsky (reflection, Schur, Szegő, or Geronimus) coefficients and , , is called the reversed polynomial of . Conversely, if is a sequence of monic polynomials which satisfies (11) and for , then is the sequence of monic OPUC with respect to some Hermitian regular linear functional.

If is a Hermitian regular (resp., positive definite) linear functional, then (see [1618]) (resp., ), for .

A positive definite Hermitian linear functional has an integral representation (see [19]) where is a nontrivial probability measure supported on an infinite subset of . A measure belongs to the Nevai class (see [20, 21]) if .

On the other hand (see [19]), an analytic function , defined on , is said to be a Carathéodory function if and only if and on . If is a probability measure on , then is a Carathéodory function. Conversely, the Herglotz representation theorem claims that every Carathéodory function has a representation given by (13) for a unique probability measure on .

Besides (see [22]), a Carathéodory function (13) admits the expansions where are the moments of the measure associated with .

To complete this section, we state the following definitions. Let be a sequence of monic OPUC with corresponding Verblunsky coefficients , and let . The polynomials defined by are called the associated polynomials of of order . Similarly, given a finite set of complex numbers , with ,  , let us define the new Verblunsky coefficients . Then the monic OPUC defined by the forward Szegő relation associated with are said to be the antiassociated polynomials of of order .

3. -Coherent Pairs on the Unit Circle

A pair of Hermitian regular linear functionals defined on the linear space of Laurent polynomials is said to be a -coherent pair if their corresponding sequences of monic OPUC, and , are related by where , for . In such a case, the pair and is also said to be a -coherent pair. If for every , then is called a -coherent pair.

Lemma 1. If satisfies (16), then, one has the following.(i) if and only if , for every . (ii)For , one has

Proof. From (16) it is easy to check that if and only if there exists , , such that . Also, from (16) and using induction on , it is immediate to prove (17) and (18).

Corollary 2. If is a -coherent pair given by (16), then where whenever .

We will study the -coherence relations when is the linear functional associated with basic positive measures on the unit circle, namely, the Lebesgue and Bernstein-Szegő measures.

The Lebesgue linear functional is the linear functional associated with the Lebesgue measure , and its corresponding sequence of monic OPUC is , for . Besides, the reversed polynomials are ,  , and its Verblunsky coefficients are , for . Furthermore, its moments are , for , and its Carathéodory function is .

The Bernstein-Szegő linear functional is associated with the measure , with and . Its corresponding monic OPUC are for and . Its reversed polynomials are , for , and its Verblunsky coefficients are , for and . Besides, its moments are for , and its Carathéodory function is .

We begin by analyzing the first one.

4. The Lebesgue Linear Functional

Theorem 3. Let be a -coherent pair on the unit circle such that their corresponding monic OPUC satisfy (16), and let be the Lebesgue linear functional. (i)If , then is also the linear functional associated with the Lebesgue measure, and for . (ii)If and ,  , then where is the sequence of moments associated with .

Proof. Since for , then (16) becomes Thus, applying the linear functional on the previous expression, we get
(i) If , then from (25) we have for . Thus, for , and, as a consequence, from (24) we obtain for every .
(ii) From (18), we have Multiplying (26) by and applying , we obtain Thus, multiplying this equation by and adding it to the previous equation for , we get Since , , and , (25) yields for . Thus, from (28), we conclude that for or, equivalently, for . Therefore, (25) becomes (20).
On the other hand, from (26) we obtain (22) and (23). Besides, from the forward Szegő relation and (26), we can obtain another expression for , . By comparing the coefficients of , we get , for . Hence, since and , for , (21) follows.

We are interested in the cases where is also a positive definite linear functional. Notice that, aside from the trivial case when , all of the coherence coefficients are determined from the values of , , and (or, equivalently, , , and ). Not every choice of these parameters will yield a positive definite linear functional . For instance, if and , then we can see from (22) that , , and , . However, it is possible to choose the values of , , and in order to get a positive definite linear functional , or at least its rational spectral transformation. We have the following cases.

Proposition 4. Let be a -coherent pair on the unit circle such that their corresponding monic OPUC satisfy (16), and let be the linear functional associated with the Lebesgue measure. Assume that is normalized (i.e., ). Then, one has the following.(i)Let . If (i.e., ), then and for every . Besides, is the linear functional associated with the Bernstein-Szegő measure with parameter . Furthermore, if for some , then . (ii)If , , and either or holds, then the Carathéodory function associated with is where is the Carathéodory function associated with the Bernstein-Szegő measure with parameter . As a consequence, the orthogonality measure associated with is (iii)For any values of , , the value of can be chosen in such a way that is the linear functional associated with a rational spectral transformation of a Nevai class measure.

Proof. (i) Notice that because . We first prove that if for some , then for . Assume that for some , . From (21), (22), and (23) it follows that and for . Besides, another expression for is , where is given by (23). Thus, the comparison of the coefficients of in both expressions of yields , and thus, . Following the same argument for , we conclude that for and . Therefore, for , , and for . As a consequence, from (21) and (20), it follows that and , . Finally, since , then is the linear functional associated with the Bernstein-Szegő measure.
(ii) From (20), the Carathéodory function associated with is . Since , then (see [19]) the Bernstein-Szegő polynomials of parameter have moments and are orthogonal with respect to the measure , and their associated Carathéodory function is . Therefore, (29) holds. In other words (see [23]), can be obtained by applying a rescaling to the moments of , followed by a perturbation of its first moment (i.e., a diagonal perturbation of the corresponding Toeplitz matrix). Thus, the orthogonality measure associated with is given by (30).
(iii) From (21), given , we have , so we can choose small enough so that is sufficiently close to . Thus, will also be close to , and since will be an increasing sequence and, as a consequence, will be a decreasing sequence. Besides, can be chosen so that converges to a constant , , and therefore the product will also converge to . This shows that , and thus defines a Nevai measure . As a consequence, since has as Verblunsky coefficients, can be expressed as an antiassociated perturbation of order 1 (see [24]) applied to the measure .

5. The Bernstein-Szegő Linear Functional

Now, we proceed to analyze the companion measure when is the Bernstein-Szegő linear functional defined as above.

Theorem 5. Let be the Bernstein-Szegő linear functional, and let be a -coherent pair on the unit circle given by (16). Then, the moments of are where whenever , and the sequence of monic OPUC is given by , , and, for , Furthermore, , , and

Proof. Since , for , then, from (19), we get where whenever . From (36) and using induction on , it is easy to verify that the moments of are given by (32). Besides, from (18) and (33), (34) holds. Furthermore, since is a sequence of monic OPUC, then it follows that , .
On the other hand, from the forward Szegő relation and (33), we can get another expression of , for . Hence, comparing the coefficients of and using (34), (35) follows.

As in the previous section, we are interested in the situations where is also a positive definite linear functional. Notice now that the values of , , , , and determine all other coherence coefficients. We have the following cases.

Proposition 6. Let be the Bernstein-Szegő linear functional, and let be a -coherent pair on the unit circle given by (16). Then, one has the following.(i)If , then and, therefore, and are Lebesgue linear functionals, and   for .(ii)Let .(1) If is normalized (i.e., ) and for some , then ; this is, is the Lebesgue linear functional. As a consequence, , , , and for every . In other words, for , is the linear functional associated with the Bernstein-Szegő measure, with parameter .(2) If , then for ; this is, for ,   is the linear functional associated with the Bernstein-Szegő measure, with parameter .(3)If and , for , then and can be chosen so that is the linear functional associated with an antiassociated perturbation of order 2 applied to a Nevai measure.

Proof. (i) If we multiply (33) by and apply , then we get, for , If we multiply this equation by and we add it to the previous equation for , then we obtain Hence, from (39) and (36), it follows that
On the other hand, if we apply the linear functional to both sides of the -coherence relation (16), we get and Thus, from (39) and (41), we obtain, for ,
Therefore, if , then from (32), the moments of are for , and, as a consequence, (40) becomes and (42) is, for , Then, if , from (43) and (44) it follows that , for , and , for , respectively, which is a contradiction. Thus, if , then ; that is, is the Lebesgue linear functional, and in case the part of Theorem 3 holds.
Now, let us assume .
(ii)(1) From part (i) of Proposition 4, it suffices to show that is the Lebesgue linear functional. Thus, let us prove that if for some (and therefore ), then . Indeed, if for some , then from (33) for ,  , it follows that , for . Furthermore, from the forward Szegő relation and (33) for , we obtain an expression of , for . Hence, comparing the coefficients of this expression and (33) for , we obtain, for , Since , then from (47) it follows that either or . If , then from (46) we get and, as a consequence, from (45) we have . If , then from (46) it follows that either (and thus, from (45), ) or . If and , from (45) it follows that . But if , we can follow a similar argument and conclude that , and since , then we also have , which yields a contradiction. Therefore, .
(ii)(2) If , then from (34) it follows that and, as a consequence, for every . Therefore, from the forward Szegő relation it follows that for .
(ii)(3) From the forward Szegő relation and (33) we obtain an expression of , for . If we compare the coefficients of of this expression and (33), we get and Thus, if , then from (34) it follows that , and, as a consequence, if , , are nonzero, then from (48) we get Besides, from (34), for , and if , then by induction on we can prove that , for , which is (37). Therefore, proceeding as in the proof of Proposition 4, we can choose small enough so that is sufficiently close to . As a consequence, will be an increasing sequence, and hence will be a decreasing sequence. Also, we can choose such that converges to a constant , with . The infinite product will then converge to . Therefore, since are the Verblunsky coefficients of , this linear functional is an antiassociated perturbation of order 2 (see [24]) applied to a Nevai measure .

Acknowledgments

The authors thank the referee the valuable comments. They greatly contributed to improve the contents of the paper. The work of Luis Garza was supported by Conacyt Grant no. 156668 and Beca Santander Iberoamérica para Jóvenes Profesores e Investigadores (Mexico). The work of Francisco Marcellán and Natalia C. Pinzón-Cortés has been supported by Dirección General de Investigación, Desarrollo e Innovación, Ministerio de Economía y Competitividad of Spain, Grant MTM2012-36732-C03-01.

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