Verblunsky coefficients related with periodic real sequences and associated measures on the unit circle (original) (raw)

Verblunsky coefficients related with periodic real sequences and associated measures on the unit circle ⊙{ }^{\odot}

Cleonice F. Bracciali a { }^{\text {a }}, Jairo S. Silva b,c { }^{\text {b,c }}, A. Sri Ranga a { }^{\text {a }}, Daniel O. Veronese d,c,s{ }^{\mathrm{d}, \mathrm{c}, \mathrm{s}}
a{ }^{a} Depto de Matemática Aplicada, IBILCE, UNESP - Univ Estadual Paulista, 15054-000, São José do Rio Preto, SP, Brazil
b { }^{\text {b }} Depto de Matemática, Universidade Federal do Maranhão, 65080-805, São Luis, MA, Brazil
c { }^{\text {c }} Pós-Graduação em Matemática, IBILCE, UNESP - Univ Estadual Paulista, 15054-000, São José do Rio Preto, SP, Brazil
d { }^{\text {d }} ICTE, Universidade Federal do Triângulo Mineiro, 38064-200, Uberaba, MG, Brazil

A R T I C L E I N F O

Article history:

Received 19 April 2016
Available online xxxx
Submitted by K. Driver

Keywords:

Probability measures
Periodic Verblunsky coefficients
Chain sequences
Periodic real sequences

A B S T R A C T

It is known that given a pair of real sequences {{cn}n=1∞,{dn}n=1∞}\left\{\left\{c_{n}\right\}_{n=1}^{\infty},\left\{d_{n}\right\}_{n=1}^{\infty}\right\}, with {dn}n=1∞\left\{d_{n}\right\}_{n=1}^{\infty} a positive chain sequence, we can associate a unique nontrivial probability measure μ\mu on the unit circle. Precisely, the measure is such that the corresponding Verblunsky coefficients {αn}n=0∞\left\{\alpha_{n}\right\}_{n=0}^{\infty} are given by the relation

αn−1=ρˉn−1[1−2mn−icn1−icn],n≥1\alpha_{n-1}=\bar{\rho}_{n-1}\left[\frac{1-2 m_{n}-i c_{n}}{1-i c_{n}}\right], \quad n \geq 1

where ρ0=1,ρn=∏k=1n(1−ick)/(1+ick),n≥1\rho_{0}=1, \rho_{n}=\prod_{k=1}^{n}\left(1-i c_{k}\right) /\left(1+i c_{k}\right), n \geq 1 and {mn}n=0∞\left\{m_{n}\right\}_{n=0}^{\infty} is the minimal parameter sequence of {dn}n=1∞\left\{d_{n}\right\}_{n=1}^{\infty}. In this paper we consider the space, denoted by NpN_{p}, of all nontrivial probability measures such that the associated real sequences {cn}n=1∞\left\{c_{n}\right\}_{n=1}^{\infty} and {mn}n=1∞\left\{m_{n}\right\}_{n=1}^{\infty} are periodic with period pp, for p∈Np \in \mathbb{N}. By assuming an appropriate metric on the space of all nontrivial probability measures on the unit circle, we show that there exists a homeomorphism gpg_{p} between the metric subspaces NpN_{p} and VpV_{p}, where VpV_{p} denotes the space of nontrivial probability measures with associated pp-periodic Verblunsky coefficients. Moreover, it is shown that the set FpF_{p} of fixed points of gpg_{p} is exactly Vp∩NpV_{p} \cap N_{p} and this set is characterized by a (p−1)(p-1)-dimensional submanifold of Rp\mathbb{R}^{p}. We also prove that the study of probability measures in NpN_{p} is equivalent to the study of probability measures in VpV_{p}. Furthermore, it is shown that the pure points of measures in NpN_{p} are, in fact, zeros of associated para-orthogonal polynomials of degree pp. We also look at the essential support of probability measures in the limit periodic case, i.e., when the sequences {cn}n=1∞\left\{c_{n}\right\}_{n=1}^{\infty} and {mn}n=1∞\left\{m_{n}\right\}_{n=1}^{\infty} are limit periodic with period pp. Finally, we give some examples to illustrate the results obtained.
© 2016 Elsevier Inc. All rights reserved.

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1. ∘{ }^{\circ} The first and third authors are supported by funds from FAPESP (2014/22571-2) and CNPq (475502/2013-2, 305073/2014-1, 305208/2015-2) of Brazil. The second and fourth authors are supported by grants from CAPES of Brazil.
   • Corresponding author at: ICTE, Universidade Federal do Triângulo Mineiro, 38064-200, Uberaba, MG, Brazil.
     E-mail addresses: cleonice@ibilce.unesp.br (C.F. Bracciali), jairo.santos@ufma.br (J.S. Silva), ranga@ibilce.unesp.br (A. Sri Ranga), veronese@icte.uftm.edu.br (D.O. Veronese). ↩︎

1. Introduction

Given a nontrivial probability measure μ(z)=μ(eiθ)\mu(z)=\mu\left(e^{i \theta}\right) on the unit circle T={z=eiθ:0≤θ≤2π}\mathbb{T}=\left\{z=e^{i \theta}: 0 \leq \theta \leq 2 \pi\right\}, the associated sequence of orthogonal polynomials on the unit circle (OPUC, in short) {ϕn(z)}n=0∞\left\{\phi_{n}(z)\right\}_{n=0}^{\infty} is that with the property

∫Tzˉjϕn(z)dμ(z)=∫02πe−ijθϕn(eiθ)dμ(eiθ)=0,0≤j≤n−1,n≥1\int_{\mathbb{T}} \bar{z}^{j} \phi_{n}(z) d \mu(z)=\int_{0}^{2 \pi} e^{-i j \theta} \phi_{n}\left(e^{i \theta}\right) d \mu\left(e^{i \theta}\right)=0, \quad 0 \leq j \leq n-1, \quad n \geq 1

where ϕ0(z)=1\phi_{0}(z)=1. Letting κn−2=∥ϕn∥2=∫T∣ϕn(z)∣2dμ(z)\kappa_{n}^{-2}=\left\|\phi_{n}\right\|^{2}=\int_{\mathbb{T}}\left|\phi_{n}(z)\right|^{2} d \mu(z), the orthonormal polynomials on the unit circle are φn(z)=κnϕn(z),n≥0\varphi_{n}(z)=\kappa_{n} \phi_{n}(z), n \geq 0.

The polynomials ϕn(z),n≥0\phi_{n}(z), n \geq 0 (assumed here as monic polynomials) satisfy the relations

ϕn(z)=zϕn−1(z)−αˉn−1ϕn−1∗(z),n≥1ϕn(z)=(1−∣αn−1∣2)zϕn−1(z)−αˉn−1ϕn∗(z),\begin{array}{ll} \phi_{n}(z)=z \phi_{n-1}(z)-\bar{\alpha}_{n-1} \phi_{n-1}^{*}(z), & n \geq 1 \\ \phi_{n}(z)=\left(1-\left|\alpha_{n-1}\right|^{2}\right) z \phi_{n-1}(z)-\bar{\alpha}_{n-1} \phi_{n}^{*}(z), & \end{array}

where αn−1=−ϕn(0)‾\alpha_{n-1}=-\overline{\phi_{n}(0)} and ϕn∗(z)=znϕn(1/z)‾\phi_{n}^{*}(z)=z^{n} \overline{\phi_{n}(1 / z)} denotes the reversed (reciprocal) polynomial of ϕn(z)\phi_{n}(z). The complex numbers αn\alpha_{n}, in recent years, have been referred to as Verblunsky coefficients. It is known that these coefficients are such that ∣αn∣<1,n≥0\left|\alpha_{n}\right|<1, n \geq 0. On the other hand, Verblunsky’s Theorem shows that given any sequence of complex numbers with modulus less than one there exists a unique associated nontrivial probability measure on the unit circle (see [18, Theorem 1.7.11]). Therefore, the OPUC and the associated measure are completely determined from these coefficients. A very nice and short constructive proof of this last statement can be found in [10]. For more details on the classical theory of orthogonal polynomials on the unit circle we refer to [18−20][18-20].

Recently, it was shown in [7] that given a pair of real sequences {{cn}n=1∞,{dn}n=1∞}\left\{\left\{c_{n}\right\}_{n=1}^{\infty},\left\{d_{n}\right\}_{n=1}^{\infty}\right\}, with {dn}n=1∞\left\{d_{n}\right\}_{n=1}^{\infty} a positive chain sequence, we can associate a unique nontrivial probability measure μ\mu on the unit circle. The associated measure is such that the corresponding Verblunsky coefficients {αn}n=0∞\left\{\alpha_{n}\right\}_{n=0}^{\infty} are given by the relation

αn−1=ρˉn−1[1−2mn−icn1−icn],n≥1\alpha_{n-1}=\bar{\rho}_{n-1}\left[\frac{1-2 m_{n}-i c_{n}}{1-i c_{n}}\right], \quad n \geq 1

where ρ0=1,ρn=∏k=1n(1−ick)/(1+ick),n≥1\rho_{0}=1, \rho_{n}=\prod_{k=1}^{n}\left(1-i c_{k}\right) /\left(1+i c_{k}\right), n \geq 1 and {mn}n=0∞\left\{m_{n}\right\}_{n=0}^{\infty} is the minimal parameter sequence of {dn}n=1∞\left\{d_{n}\right\}_{n=1}^{\infty}. In Theorem 2.1 we have given the complete information regarding this statement and its reciprocal.

Thus, we can look at the measure μ\mu from the sequence of its associated Verblunsky coefficients or, equivalently, from the real sequences {cn}n=1∞\left\{c_{n}\right\}_{n=1}^{\infty} and {mn}n=1∞\left\{m_{n}\right\}_{n=1}^{\infty}, where {mn}n=0∞\left\{m_{n}\right\}_{n=0}^{\infty} is the minimal parameter sequence of the chain sequence {dn}n=1∞\left\{d_{n}\right\}_{n=1}^{\infty}. From the theory of chain sequences (see [6]) we have

dn=(1−mn−1)mn,n≥1d_{n}=\left(1-m_{n-1}\right) m_{n}, \quad n \geq 1

with 0<mn<1,n≥10<m_{n}<1, n \geq 1 and m0=0m_{0}=0.
There are many results on classical OPUC theory regarding nontrivial probability measures with associated pp-periodic sequence of Verblunsky coefficients. In the first section we summarize some of them. We denote by VpV_{p} the space of all nontrivial probability measures with associated pp-periodic sequence of Verblunsky coefficients, i.e.,

μ∈Vp if and only if αn+p=αn,n≥0\mu \in V_{p} \quad \text { if and only if } \quad \alpha_{n+p}=\alpha_{n}, \quad n \geq 0

An important fact is that we can have a measure with periodic Verblunsky coefficients and such that the sequences of the associated pair {{cn}n=1∞,{mn}n=1∞}\left\{\left\{c_{n}\right\}_{n=1}^{\infty},\left\{m_{n}\right\}_{n=1}^{\infty}\right\} are not periodic (see, for example, [8]). For this reason,

Fig. 1. Relations between the spaces Np,VpN_{p}, V_{p} and UpU_{p}.
we introduce the notation NpN_{p} for the space of all nontrivial probability measures for which the sequences of the associated pair {{cn}n=1∞,{mn}n=1∞}\left\{\left\{c_{n}\right\}_{n=1}^{\infty},\left\{m_{n}\right\}_{n=1}^{\infty}\right\} are periodic with period pp, i.e.,

μ∈Np if and only if cn+p=cn and mn+p=mn,n≥1\mu \in N_{p} \quad \text { if and only if } \quad c_{n+p}=c_{n} \text { and } m_{n+p}=m_{n}, \quad n \geq 1

Following Simon [19], we say that a sequence of Verblunsky coefficients is periodic up to a phase if there exist λ∈T\lambda \in \mathbb{T} and p∈Np \in \mathbb{N} such that

αn+p=λαn,n≥0\alpha_{n+p}=\lambda \alpha_{n}, \quad n \geq 0

Throughout in this paper, UpU_{p} will denote the space of all nontrivial probability measures such that its associated sequence of Verblunsky coefficients satisfies (1.3).

Clearly, any measure belonging to VpV_{p} also belongs to UpU_{p}. Even though the spaces VpV_{p} and UpU_{p} are very known we do not have any knowledge about the new space NpN_{p} introduced by the condition (1.2). In the following, we formulate some natural questions around this space.
Q.1.- What is the relation between the spaces Np,VpN_{p}, V_{p} and UpU_{p} ?
Q.2.- Is there any measure belonging to Np∩VpN_{p} \cap V_{p} ? Which are these measures?
Q.3.- What is the behavior of probability measures belonging to the NpN_{p} space?

The aim of this paper is to provide answers to the questions formulated above and other related questions. The main results are presented in Sections 3, 4 and 5. One of them can be stated as follows.

Theorem 1.1. Let p∈Np \in \mathbb{N}. For the spaces Np,VpN_{p}, V_{p} and UpU_{p} the following relations hold:
(i) The topological spaces NpN_{p} and VpV_{p} are homeomorphic.
(ii) Np⫋UpN_{p} \varsubsetneqq U_{p};
(iii) Np∩Vp≠∅N_{p} \cap V_{p} \neq \emptyset;
(iv) Np\Vp≠∅N_{p} \backslash V_{p} \neq \emptyset and Vp\Np≠∅V_{p} \backslash N_{p} \neq \emptyset.

If we denote by P(T)\mathcal{P}(\mathbb{T}) the space of all nontrivial probability measures on the unit circle, some of the relations presented in Theorem 1.1 can be illustrated as in Fig. 1.

Remark 1.1. While each measure in the space NpN_{p} is essentially characterized by an element of Rp×(0,1)p\mathbb{R}^{p} \times(0,1)^{p}, we will prove in Section 3 that each measure in the intersection Vp∩NpV_{p} \cap N_{p} is essentially characterized by an element of Xp×(0,1)pX_{p} \times(0,1)^{p}, where XpX_{p} is a (p−1)(p-1)-dimensional submanifold of Rp\mathbb{R}^{p}.

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1. Please cite this article in press as: C.F. Bracciali et al., Verblunsky coefficients related with periodic real sequences and associated measures on the unit circle, J. Math. Anal. Appl. (2016), http://dx.doi.org/10.1016/j.jmaa.2016.08.009 ↩︎

Another main result that we prove in this paper shows that every measure in NpN_{p} (except the Lebesgue measure) is equivalent by rotation to exactly pp measures in VpV_{p} and reciprocally.

Theorem 1.2. Let dν(z)=(2πiz)−1dzd \nu(z)=(2 \pi i z)^{-1} d z be the Lebesgue measure defined on the unit circle. Then, the following hold:
(i) For each μ∈Np\{ν}\mu \in N_{p} \backslash\{\nu\} there exist exactly pp values w1,w2,…,wpw_{1}, w_{2}, \ldots, w_{p}, solutions of wp=λw^{p}=\lambda for certain λ∈T\lambda \in \mathbb{T}, such that μ(wj‾z)∈Vp,j=1,2,…,p\mu\left(\overline{w_{j}} z\right) \in V_{p}, j=1,2, \ldots, p.
(ii) Reciprocally, for each measure μ~∈Vp\{ν}\tilde{\mu} \in V_{p} \backslash\{\nu\} there exist exactly pp values w1,w2,…,wpw_{1}, w_{2}, \ldots, w_{p}, solutions of ϕp(μ~;w)−ϕp∗(μ~;w)=0\phi_{p}(\tilde{\mu} ; w)-\phi_{p}^{*}(\tilde{\mu} ; w)=0, such that μ~(wjz)∈Np,j=1,2,…,p\tilde{\mu}\left(w_{j} z\right) \in N_{p}, j=1,2, \ldots, p. Here, ϕp(μ~;z)\phi_{p}(\tilde{\mu} ; z) denotes the orthogonal polynomial of degree pp associated with μ~\tilde{\mu} and ϕp∗(μ~;z)\phi_{p}^{*}(\tilde{\mu} ; z) its reciprocal conjugate.

Remark 1.2. As we will see in the proof of Theorem 1.2, the number λ\lambda that appears in item (i) is directly related with the measure μ\mu and can be given in terms of the sequence {cn}n=1∞\left\{c_{n}\right\}_{n=1}^{\infty}. In Section 2 we will see that the solutions of the equation ϕp(μ~;z)−ϕp∗(μ~;z)=0\phi_{p}(\tilde{\mu} ; z)-\phi_{p}^{*}(\tilde{\mu} ; z)=0 in item (ii) are corresponding to the possible pure points of the measure μ~\tilde{\mu}.

By a combination of the Radon-Nikodym and Lebesgue decomposition theorems, any nontrivial measure μ\mu can be decomposed into an absolutely continuous part dμac(eiθ)=w(θ)dθ2sd \mu_{a c}\left(e^{i \theta}\right)=w(\theta) \frac{d \theta}{2 s} and in a singular part which we will denote by dμsd \mu_{s}.

A point z0∈Tz_{0} \in \mathbb{T} is in the essential support of μ\mu if and only if for any open set AA about z0,A∩supp⁡(μ)z_{0}, A \cap \operatorname{supp}(\mu) is an infinite set. Here, supp⁡(μ)\operatorname{supp}(\mu) denotes the support of the measure μ\mu. Throughout in this paper, σess(μ)\sigma_{e s s}(\mu) will denote the essential support of μ\mu.

Taking into account the informations mentioned above, the behavior of probability measures in VpV_{p} (see Theorem 2.3) and Theorem 1.2, we can also state the following.

Corollary 1.2.1. Let dμ=w(θ)dθ2s+dμsd \mu=w(\theta) \frac{d \theta}{2 s}+d \mu_{s} be a probability measure in NpN_{p}. Then, there exist associated closed sets B1,…,Bp\mathcal{B}_{1}, \ldots, \mathcal{B}_{p} such that σess(μ)=∪Bj\sigma_{e s s}(\mu)=\cup \mathcal{B}_{j} and dμs[∪Bj]=∅d \mu_{s}\left[\cup \mathcal{B}_{j}\right]=\emptyset. Moreover, in each disjoint open arc on T\∪j=1pBj\mathbb{T} \backslash \cup_{j=1}^{p} \mathcal{B}_{j}, μ\mu has either no support or a single pure point.

The paper is organized as follows. In Section 2 we give a summary of some required theoretical results. Section 3 deals with the results concerning the space NpN_{p} and relations with the spaces VpV_{p} and UpU_{p}. Some additional results on measures in NpN_{p} and associated para-orthogonal polynomials on the unit circle (POPUC, in short) are presented in Section 4. In Section 5 we study the essential support of measures associated with a pair of limit periodic sequences {{cn}n=1∞,{mn}n=1∞}\left\{\left\{c_{n}\right\}_{n=1}^{\infty},\left\{m_{n}\right\}_{n=1}^{\infty}\right\}. Finally, in Section 6 we give some examples to illustrate the main results.

2. Some preliminary results

In this section we present some results concerning nontrivial probability measures and positive chain sequences (for more details on chain sequences we refer to [6] and [21]). Furthermore, some results about periodic Verblunsky coefficients are presented.

We begin with two theorems established in [7]. The first theorem provides a characterization for nontrivial probability measures in terms of two sequences {cn}n=1∞\left\{c_{n}\right\}_{n=1}^{\infty} and {dn}n=1∞\left\{d_{n}\right\}_{n=1}^{\infty}.

Theorem 2.1. (a) Given a nontrivial probability measure μ\mu on the unit circle, then associated with it there exists a unique pair of real sequences {{cn}n=1∞,{dn}n=1∞}\left\{\left\{c_{n}\right\}_{n=1}^{\infty},\left\{d_{n}\right\}_{n=1}^{\infty}\right\}, where {dn}n=1∞\left\{d_{n}\right\}_{n=1}^{\infty} is also a positive chain sequence. Specifically, if {αn}n=0∞\left\{\alpha_{n}\right\}_{n=0}^{\infty} is the associated sequence of Verblunsky coefficients and if the sequence ρn\rho_{n} is such that

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1. Please cite this article in press as: C.F. Bracciali et al., Verblunsky coefficients related with periodic real sequences and associated measures on the unit circle, J. Math. Anal. Appl. (2016), http://dx.doi.org/10.1016/j.jmaa.2016.08.009 ↩︎

ρ0=1 and ρn=ρn−11−ρˉn−1αˉn−11−ρn−1αn−1,n≥1\rho_{0}=1 \quad \text { and } \quad \rho_{n}=\rho_{n-1} \frac{1-\bar{\rho}_{n-1} \bar{\alpha}_{n-1}}{1-\rho_{n-1} \alpha_{n-1}}, \quad n \geq 1

then m0=0m_{0}=0,

cn=−Im(ρn−1αn−1)1−Re(ρn−1αn−1) and mn=12∣1−ρn−1αn−1∣2[1−Re(ρn−1αn−1)],n≥1c_{n}=\frac{-\mathcal{I} m\left(\rho_{n-1} \alpha_{n-1}\right)}{1-\mathcal{R} e\left(\rho_{n-1} \alpha_{n-1}\right)} \quad \text { and } \quad m_{n}=\frac{1}{2} \frac{\left|1-\rho_{n-1} \alpha_{n-1}\right|^{2}}{\left[1-\mathcal{R} e\left(\rho_{n-1} \alpha_{n-1}\right)\right]}, \quad n \geq 1

where {mn}n=0∞\left\{m_{n}\right\}_{n=0}^{\infty} is the minimal parameter sequence of {dn}n=1∞\left\{d_{n}\right\}_{n=1}^{\infty}. Moreover, the maximal parameter sequence {Mn}n=0∞\left\{M_{n}\right\}_{n=0}^{\infty} of {dn}n=1∞\left\{d_{n}\right\}_{n=1}^{\infty} is such that M0M_{0} is the value of the jump in the measure at z=1z=1.
(b) Conversely, given a pair of real sequences {{cn}n=1∞,{dn}n=1∞}\left\{\left\{c_{n}\right\}_{n=1}^{\infty},\left\{d_{n}\right\}_{n=1}^{\infty}\right\}, where {dn}n=1∞\left\{d_{n}\right\}_{n=1}^{\infty} is also a positive chain sequence then associated with this pair there exists a unique nontrivial probability measure μ\mu supported on the unit circle. Specifically, if {mn}n=0∞\left\{m_{n}\right\}_{n=0}^{\infty} is the minimal parameter sequence of {dn}n=1∞\left\{d_{n}\right\}_{n=1}^{\infty}, then ρ0=1\rho_{0}=1,

αn−1=ρˉn−1[1−2mn−icn1−icn] and ρn=1−icn1+icnρn−1,n≥1\alpha_{n-1}=\bar{\rho}_{n-1}\left[\frac{1-2 m_{n}-i c_{n}}{1-i c_{n}}\right] \quad \text { and } \quad \rho_{n}=\frac{1-i c_{n}}{1+i c_{n}} \rho_{n-1}, \quad n \geq 1

Moreover, the measure has a jump M0M_{0} at z=1z=1, where {Mn}n=0∞\left\{M_{n}\right\}_{n=0}^{\infty} is the maximal parameter sequence of {dn}n=1∞\left\{d_{n}\right\}_{n=1}^{\infty}.

The next theorem gives information regarding the pure points. This result is obtained as a consequence of Wall’s criterion for maximal parameter sequence of positive chain sequences.

Theorem 2.2. The probability measure μ\mu, with associated Verblunsky coefficients {αj}j=1∞\left\{\alpha_{j}\right\}_{j=1}^{\infty}, has a pure point at w(∣w∣=1)w(|w|=1) if and only if

∑n=1∞[∏j=1n∣1−wτj−1(w)αj−1∣21−∣αj−1∣2]=s(w)<∞\sum_{n=1}^{\infty}\left[\prod_{j=1}^{n} \frac{\left|1-w \tau_{j-1}(w) \alpha_{j-1}\right|^{2}}{1-\left|\alpha_{j-1}\right|^{2}}\right]=s(w)<\infty

Moreover, the size of the mass at the point z=wz=w is equal to t=[1+s(w)]−1t=[1+s(w)]^{-1}. Here, τ0(w)=1\tau_{0}(w)=1 and

τj+1(w)=ϕj+1(w)ϕj+1∗(w)=wτj(w)−αˉj1−wτj(w)αj,j≥0\tau_{j+1}(w)=\frac{\phi_{j+1}(w)}{\phi_{j+1}^{*}(w)}=\frac{w \tau_{j}(w)-\bar{\alpha}_{j}}{1-w \tau_{j}(w) \alpha_{j}}, \quad j \geq 0

Remark 2.1. From the results established in [7] we also have that ρn=τn(1),n≥0\rho_{n}=\tau_{n}(1), n \geq 0, where ρn\rho_{n} is given by (2.1) and τn(1)\tau_{n}(1) by (2.2) with w=1w=1.

We now present a review of basic results on measures with associated periodic Verblunsky coefficients. For more details regarding these results we refer to [11,16,17,19][11,16,17,19].

Let {αn}n=0∞\left\{\alpha_{n}\right\}_{n=0}^{\infty} be a pp-periodic sequence (αn+p=αn,n≥0)\left(\alpha_{n+p}=\alpha_{n}, n \geq 0\right) of Verblunsky coefficients associated with the measure denoted by μ(p)\mu^{(p)} (here, pp is a fixed natural number). Consider the discriminant function Δ(z)=\Delta(z)= z−p/2Tr⁡(Tp(z))z^{-p / 2} \operatorname{Tr}\left(T_{p}(z)\right) where

Tp(z)=A(αp−1,z)…A(α0,z)A(αj,z)=(1−∣αj∣2)−1/2(z−αˉj−αjz1),j=0,1,…,p−1\begin{gathered} T_{p}(z)=A\left(\alpha_{p-1}, z\right) \ldots A\left(\alpha_{0}, z\right) \\ A\left(\alpha_{j}, z\right)=\left(1-\left|\alpha_{j}\right|^{2}\right)^{-1 / 2}\left(\begin{array}{cc} z & -\bar{\alpha}_{j} \\ -\alpha_{j} z & 1 \end{array}\right), \quad j=0,1, \ldots, p-1 \end{gathered}

and Tr⁡(Tp(z))\operatorname{Tr}\left(T_{p}(z)\right) denotes the trace of Tp(z)T_{p}(z).
It is well known that all the pp distinct solutions of the equation Δ(z)=2\Delta(z)=2, which we denote by z1+,…,zp+z_{1}^{+}, \ldots, z_{p}^{+}, lie on the unit circle T\mathbb{T}. In the same way, the pp distinct solutions of the equation Δ(z)=−2\Delta(z)=-2, denoted by

z1−,…zp−z_{1}^{-}, \ldots z_{p}^{-}, also lie on T\mathbb{T}. Using these solutions it is possible to show that the unit circle can be decomposed into 2p2 p alternating sets G1,B1,G2,…,BpG_{1}, B_{1}, G_{2}, \ldots, B_{p} with each gap, GjG_{j}, open and each band, BjB_{j}, closed. Moreover, each band BjB_{j} is given by Bj={z∈T∣arg⁡(zjσj)≤arg⁡(z)≤arg⁡(zj−σj)}B_{j}=\left\{z \in \mathbb{T} \mid \arg \left(z_{j}^{\sigma_{j}}\right) \leq \arg (z) \leq \arg \left(z_{j}^{-\sigma_{j}}\right)\right\} with σj=(−1)j+1,j=1,2,…,p\sigma_{j}=(-1)^{j+1}, j=1,2, \ldots, p.

The following theorem summarizes some basic results which can be found in [19].
Theorem 2.3. Let {αj}j=0∞\left\{\alpha_{j}\right\}_{j=0}^{\infty} be a periodic sequence of Verblunsky coefficients of period pp and let dμ(p)=d \mu^{(p)}= w(θ)dθ2π+dμs(p)w(\theta) \frac{d \theta}{2 \pi}+d \mu_{s}^{(p)} be the associated probability measure. Then, if B1,…,BpB_{1}, \ldots, B_{p} are the corresponding bands, we have that σcss(μ(p))=∪Bj\sigma_{c s s}\left(\mu^{(p)}\right)=\cup B_{j} and dμs(p)[∪Bj]=∅d \mu_{s}^{(p)}\left[\cup B_{j}\right]=\emptyset. Moreover, in each disjoint open arc on T\∪j=1pBj,μ(p)\mathbb{T} \backslash \cup_{j=1}^{p} B_{j}, \mu^{(p)} has either no support or a single pure point.

It can be shown that the possible pure points of the measure μ(p)\mu^{(p)} in Theorem 2.3 are the pp distinct solutions for the equation ϕp(z)−ϕp∗(z)=0\phi_{p}(z)-\phi_{p}^{*}(z)=0. Moreover, the weight function associated with μ(p)\mu^{(p)} can be given explicitly (see [19, Chapter 11]).

3. The NpN_{p} space

Our first purpose in this section is to prove Theorem 1.1. Before this, let us establish some preliminary results.

In the space P(T)\mathcal{P}(\mathbb{T}) we can define a metric DD by

D(μ1,μ2):=sup⁡{∣αj(μ1)−αj(μ2)∣:j∈N∪{0}}D\left(\mu_{1}, \mu_{2}\right):=\sup \left\{\left|\alpha_{j}^{\left(\mu_{1}\right)}-\alpha_{j}^{\left(\mu_{2}\right)}\right|: j \in \mathbb{N} \cup\{0\}\right\}

where {αj(μ1)}j=0∞\left\{\alpha_{j}^{\left(\mu_{1}\right)}\right\}_{j=0}^{\infty} and {αj(μ2)}j=0∞\left\{\alpha_{j}^{\left(\mu_{2}\right)}\right\}_{j=0}^{\infty} are the respective sequences of Verblunsky coefficients associated with the measures μ1\mu_{1} and μ2\mu_{2}. Since VpV_{p} and NpN_{p} are subsets of P(T)\mathcal{P}(\mathbb{T}) both are metric subspaces.

By considering the metric DD, we have the following result.
Lemma 3.1. Let {μn}n=1∞\left\{\mu_{n}\right\}_{n=1}^{\infty} be a sequence of measures in P(T)\mathcal{P}(\mathbb{T}) and, for each n∈Nn \in \mathbb{N}, let {αj(μn)}j=0∞\left\{\alpha_{j}^{\left(\mu_{n}\right)}\right\}_{j=0}^{\infty}, {{cj(μn)}j=1∞,{dj(μn)}j=1∞}\left\{\left\{c_{j}^{\left(\mu_{n}\right)}\right\}_{j=1}^{\infty},\left\{d_{j}^{\left(\mu_{n}\right)}\right\}_{j=1}^{\infty}\right\} be the corresponding sequences as in Theorem 2.1. Then, if μ∈P(T)\mu \in \mathcal{P}(\mathbb{T}), the following statements are equivalent.
(i) lim⁡n→∞D(μn,μ)=0\lim _{n \rightarrow \infty} D\left(\mu_{n}, \mu\right)=0.
(ii) For each j≥0,lim⁡n→∞αj(μn)=αj(μ)j \geq 0, \lim _{n \rightarrow \infty} \alpha_{j}^{\left(\mu_{n}\right)}=\alpha_{j}^{(\mu)}.
(iii) For each j≥1,lim⁡n→∞cj(μn)=cj(μ)j \geq 1, \lim _{n \rightarrow \infty} c_{j}^{\left(\mu_{n}\right)}=c_{j}^{(\mu)} and lim⁡n→∞dj(μn)=dj(μ)\lim _{n \rightarrow \infty} d_{j}^{\left(\mu_{n}\right)}=d_{j}^{(\mu)}.

Proof. (i) ⇔\Leftrightarrow (ii) Follows directly from the definition of the metric DD.
(ii) ⇒\Rightarrow (iii) From Theorem 2.1 item a) we have, for each n∈Nn \in \mathbb{N},

ρ0(μn)=1 and ρj(μn)=ρj−1(μn)1−ρˉj−1(μn)αj−1(μn)‾1−ρj−1(μn)αj−1(μn),j≥1\rho_{0}^{\left(\mu_{n}\right)}=1 \quad \text { and } \quad \rho_{j}^{\left(\mu_{n}\right)}=\rho_{j-1}^{\left(\mu_{n}\right)} \frac{1-\bar{\rho}_{j-1}^{\left(\mu_{n}\right)} \overline{\alpha_{j-1}^{\left(\mu_{n}\right)}}}{1-\rho_{j-1}^{\left(\mu_{n}\right)} \alpha_{j-1}^{\left(\mu_{n}\right)}}, \quad j \geq 1

Since ρ0(μn)=ρ0(μ)=1,n≥1\rho_{0}^{\left(\mu_{n}\right)}=\rho_{0}^{(\mu)}=1, n \geq 1, it follows that ρ0(μn)→ρ0(μ)\rho_{0}^{\left(\mu_{n}\right)} \rightarrow \rho_{0}^{(\mu)}. Suppose that ρk(μn)→ρk(μ)\rho_{k}^{\left(\mu_{n}\right)} \rightarrow \rho_{k}^{(\mu)}. Then, using (3.1) and the fact that αk(μn)→αk(μ)\alpha_{k}^{\left(\mu_{n}\right)} \rightarrow \alpha_{k}^{(\mu)} we see that ρk+1(μn)→ρk+1(μ)\rho_{k+1}^{\left(\mu_{n}\right)} \rightarrow \rho_{k+1}^{(\mu)}. Thus, by mathematical induction, we have proved that

ρj(μn)→ρj(μ),j≥0\rho_{j}^{\left(\mu_{n}\right)} \rightarrow \rho_{j}^{(\mu)}, \quad j \geq 0

Now, using (2.1), (3.2) and the assumption (ii) we clearly have that

cj(μn)→cj(μ) and mj(μn)→mj(μ),j≥1c_{j}^{\left(\mu_{n}\right)} \rightarrow c_{j}^{(\mu)} \text { and } m_{j}^{\left(\mu_{n}\right)} \rightarrow m_{j}^{(\mu)}, j \geq 1

Consequently, dj(μn)=(1−mj−1(μn))mj(μn)→(1−mj−1(μ))mj(μ)=dj(μ),j≥1d_{j}^{\left(\mu_{n}\right)}=\left(1-m_{j-1}^{\left(\mu_{n}\right)}\right) m_{j}^{\left(\mu_{n}\right)} \rightarrow\left(1-m_{j-1}^{(\mu)}\right) m_{j}^{(\mu)}=d_{j}^{(\mu)}, j \geq 1.
(iii) ⇒\Rightarrow (ii) First, we observe that d1(μn)=m1(μn)d_{1}^{\left(\mu_{n}\right)}=m_{1}^{\left(\mu_{n}\right)} and d1(μ)=m1(μ)d_{1}^{(\mu)}=m_{1}^{(\mu)}. Hence, since d1(μn)→d1(μ)d_{1}^{\left(\mu_{n}\right)} \rightarrow d_{1}^{(\mu)}, it follows that m1(μn)→m1(μ)m_{1}^{\left(\mu_{n}\right)} \rightarrow m_{1}^{(\mu)}. Consequently, since dj(μn)=(1−mj−1(μn))mj(μn)d_{j}^{\left(\mu_{n}\right)}=\left(1-m_{j-1}^{\left(\mu_{n}\right)}\right) m_{j}^{\left(\mu_{n}\right)} and dj(μ)=(1−mj−1(μ))mj(μ)d_{j}^{(\mu)}=\left(1-m_{j-1}^{(\mu)}\right) m_{j}^{(\mu)}, it also follows that mj(μn)→mj(μ),j≥2m_{j}^{\left(\mu_{n}\right)} \rightarrow m_{j}^{(\mu)}, j \geq 2. Thus, from the relation (2.1) the required result holds.

Remark 3.1. From the first equivalence on Lemma 3.1, we clearly have that convergence in the metric DD is equivalent to weak convergence on P(T)\mathcal{P}(\mathbb{T}) (see [18, Theorem 1.5.6]). In other words, D(μn,μ)→0D\left(\mu_{n}, \mu\right) \rightarrow 0 if and only if for every continuous complex-valued function ff on T\mathbb{T},

∫Tfdμn→∫Tfdμ\int_{\mathbb{T}} f d \mu_{n} \rightarrow \int_{\mathbb{T}} f d \mu

Now we obtain necessary and sufficient conditions for a measure belongs to the space NpN_{p}. Consider μ∈P(T)\mu \in \mathcal{P}(\mathbb{T}) with its associated pair {{cj(μ)}j=1∞,{mj(μ)}j=1∞}\left\{\left\{c_{j}^{(\mu)}\right\}_{j=1}^{\infty},\left\{m_{j}^{(\mu)}\right\}_{j=1}^{\infty}\right\}. Consider also its associated sequence of Verblunsky coefficients {αj(μ)}j=0∞\left\{\alpha_{j}^{(\mu)}\right\}_{j=0}^{\infty}. From relations (2.1) we have

ρj−1(μ)αj−1(μ)=1−2mj(μ)−icj(μ)1−icj(μ) and ρj(μ)=1−icj(μ)1+icj(μ)ρj−1(μ),j≥1\rho_{j-1}^{(\mu)} \alpha_{j-1}^{(\mu)}=\frac{1-2 m_{j}^{(\mu)}-i c_{j}^{(\mu)}}{1-i c_{j}^{(\mu)}} \quad \text { and } \quad \rho_{j}^{(\mu)}=\frac{1-i c_{j}^{(\mu)}}{1+i c_{j}^{(\mu)}} \rho_{j-1}^{(\mu)}, \quad j \geq 1

with ρ0(μ)=1\rho_{0}^{(\mu)}=1.
Then, if we take λˉ=ρp(μ)\bar{\lambda}=\rho_{p}^{(\mu)}, we can state the following theorem.
Theorem 3.2. Let μ∈P(T)\mu \in \mathcal{P}(\mathbb{T}) and {αj(μ)}j=0∞\left\{\alpha_{j}^{(\mu)}\right\}_{j=0}^{\infty} be its associated sequence of Verblunsky coefficients. Then, μ∈Np\mu \in N_{p} if and only if αj+p(μ)=λαj(μ),j≥0\alpha_{j+p}^{(\mu)}=\lambda \alpha_{j}^{(\mu)}, j \geq 0, with λˉ=ρp(μ)\bar{\lambda}=\rho_{p}^{(\mu)}.

Proof. Assume that αj+p(μ)=λαj(μ),j≥0\alpha_{j+p}^{(\mu)}=\lambda \alpha_{j}^{(\mu)}, j \geq 0, where λˉ=ρp(μ)\bar{\lambda}=\rho_{p}^{(\mu)}. From relations (2.1) and from αp(μ)=λα0(μ)\alpha_{p}^{(\mu)}=\lambda \alpha_{0}^{(\mu)} we easily conclude that cp+1(μ)=c1(μ)c_{p+1}^{(\mu)}=c_{1}^{(\mu)} and mp+1(μ)=m1(μ)m_{p+1}^{(\mu)}=m_{1}^{(\mu)}. Suppose that

cj+p(μ)=cj(μ) and mj+p(μ)=mj(μ),1≤j≤kc_{j+p}^{(\mu)}=c_{j}^{(\mu)} \quad \text { and } \quad m_{j+p}^{(\mu)}=m_{j}^{(\mu)}, \quad 1 \leq j \leq k

Then, from (2.1) we have

αk+p(μ)=∏n=1p[1+icn(μ)1−icn(μ)]∏n=p+1p+k[1+icn(μ)1−icn(μ)][1−2mk+p+1(μ)−ick+p+1(μ)1−ick+p+1(μ)]\alpha_{k+p}^{(\mu)}=\prod_{n=1}^{p}\left[\frac{1+i c_{n}^{(\mu)}}{1-i c_{n}^{(\mu)}}\right] \prod_{n=p+1}^{p+k}\left[\frac{1+i c_{n}^{(\mu)}}{1-i c_{n}^{(\mu)}}\right]\left[\frac{1-2 m_{k+p+1}^{(\mu)}-i c_{k+p+1}^{(\mu)}}{1-i c_{k+p+1}^{(\mu)}}\right]

and

αk(μ)=ρˉk(μ)[1−2mk+1(μ)−ick+1(μ)1−ick+1(μ)]\alpha_{k}^{(\mu)}=\bar{\rho}_{k}^{(\mu)}\left[\frac{1-2 m_{k+1}^{(\mu)}-i c_{k+1}^{(\mu)}}{1-i c_{k+1}^{(\mu)}}\right]

Hence, since αk+p(μ)=λαk(μ)\alpha_{k+p}^{(\mu)}=\lambda \alpha_{k}^{(\mu)}, from (3.3), (3.4) and (3.5) we conclude that

1−2mk+p+1(μ)−ick+p+1(μ)1−ick+p+1(μ)=1−2mk+1(μ)−ick+1(μ)1−ick+1(μ)\frac{1-2 m_{k+p+1}^{(\mu)}-i c_{k+p+1}^{(\mu)}}{1-i c_{k+p+1}^{(\mu)}}=\frac{1-2 m_{k+1}^{(\mu)}-i c_{k+1}^{(\mu)}}{1-i c_{k+1}^{(\mu)}}

and, consequently,

ck+p+1(μ)=ck+1(μ) and mk+p+1(μ)=mk+1(μ)c_{k+p+1}^{(\mu)}=c_{k+1}^{(\mu)} \quad \text { and } \quad m_{k+p+1}^{(\mu)}=m_{k+1}^{(\mu)}

Hence, by mathematical induction, we see that cj+p(μ)=cj(μ)c_{j+p}^{(\mu)}=c_{j}^{(\mu)} and mj+p(μ)=mj(μ),j≥1m_{j+p}^{(\mu)}=m_{j}^{(\mu)}, j \geq 1. This shows that μ\mu belongs to NpN_{p}.

Reciprocally, if μ∈Np\mu \in N_{p} then, using the pp-periodicity of {cj(μ)}j=1∞\left\{c_{j}^{(\mu)}\right\}_{j=1}^{\infty} we see that

ρj+p(μ)=∏k=1j+p1−ick(μ)1+ick(μ)=λˉ∏k=p+1j+p1−ick(μ)1+ick(μ)=λˉ∏k=1j1−ick(μ)1+ick(μ)=λˉρj(μ),j≥1\rho_{j+p}^{(\mu)}=\prod_{k=1}^{j+p} \frac{1-i c_{k}^{(\mu)}}{1+i c_{k}^{(\mu)}}=\bar{\lambda} \prod_{k=p+1}^{j+p} \frac{1-i c_{k}^{(\mu)}}{1+i c_{k}^{(\mu)}}=\bar{\lambda} \prod_{k=1}^{j} \frac{1-i c_{k}^{(\mu)}}{1+i c_{k}^{(\mu)}}=\bar{\lambda} \rho_{j}^{(\mu)}, j \geq 1

Hence, using (2.1), (3.6) and the pp-periodicity of the pair {{cj(μ)}j=1∞,{mj(μ)}j=1∞}\left\{\left\{c_{j}^{(\mu)}\right\}_{j=1}^{\infty},\left\{m_{j}^{(\mu)}\right\}_{j=1}^{\infty}\right\}, we get

αj+p(μ)=ρˉj+p(μ)[1−2mj+p+1(μ)−icj+p+1(μ)1−icj+p+1(μ)]=λρˉj(μ)[1−2mj+1(μ)−icj+1(μ)1−icj+1(μ)]=λαj(μ),j≥0\alpha_{j+p}^{(\mu)}=\bar{\rho}_{j+p}^{(\mu)}\left[\frac{1-2 m_{j+p+1}^{(\mu)}-i c_{j+p+1}^{(\mu)}}{1-i c_{j+p+1}^{(\mu)}}\right]=\lambda \bar{\rho}_{j}^{(\mu)}\left[\frac{1-2 m_{j+1}^{(\mu)}-i c_{j+1}^{(\mu)}}{1-i c_{j+1}^{(\mu)}}\right]=\lambda \alpha_{j}^{(\mu)}, \quad j \geq 0

This completes the proof.
Corollary 3.2.1. Let μ∈Np\mu \in N_{p}. If there exists μ~∈Vp\tilde{\mu} \in V_{p} such that σess(μ)=σess(μ~)\sigma_{e s s}(\mu)=\sigma_{e s s}(\tilde{\mu}) then μ∈Vp∩Np\mu \in V_{p} \cap N_{p}.
Proof. Assume first that μ~∈Vp\tilde{\mu} \in V_{p} is not the Lebesgue measure. Then, we have αj+p(μ~)=βαj(μ~),j≥0\alpha_{j+p}^{(\tilde{\mu})}=\beta \alpha_{j}^{(\tilde{\mu})}, j \geq 0, with β\beta exactly equal to 1 . On the other hand, since μ∈Np\mu \in N_{p}, from Theorem 3.2 it follows that αj+p(μ)=λαj(μ),j≥0\alpha_{j+p}^{(\mu)}=\lambda \alpha_{j}^{(\mu)}, j \geq 0, with λˉ=ρp(μ)\bar{\lambda}=\rho_{p}^{(\mu)}.

Hence, since σess(μ)=σess(μ~)\sigma_{e s s}(\mu)=\sigma_{e s s}(\tilde{\mu}) we must have λ=β=1\lambda=\beta=1 (see [19, Corollary 11.4.12]). This shows that μ∈Vp∩Np\mu \in V_{p} \cap N_{p}.

Now, if μ~\tilde{\mu} is the Lebesgue measure, then we have σess(μ)=σess(μ~)=T\sigma_{e s s}(\mu)=\sigma_{e s s}(\tilde{\mu})=\mathbb{T}. Consequently, from Rakhmanov’s Theorem (see, for example, [19, Corollary 9.1.11]) it follows that

lim⁡n→∞∣αn(μ)∣=0\lim _{n \rightarrow \infty}\left|\alpha_{n}^{(\mu)}\right|=0

which is, from (2.1), equivalent to

lim⁡n→∞cn(μ)=0 and lim⁡n→∞mn(μ)=12\lim _{n \rightarrow \infty} c_{n}^{(\mu)}=0 \quad \text { and } \quad \lim _{n \rightarrow \infty} m_{n}^{(\mu)}=\frac{1}{2}

Thus, from the periodicity of the pair of sequences {{cj(μ)}j=1∞,{mj(μ)}j=1∞}\left\{\left\{c_{j}^{(\mu)}\right\}_{j=1}^{\infty},\left\{m_{j}^{(\mu)}\right\}_{j=1}^{\infty}\right\}, and from (3.7) one must have

cn(μ)=0 and mn(μ)=12,n≥1c_{n}^{(\mu)}=0 \quad \text { and } \quad m_{n}^{(\mu)}=\frac{1}{2}, \quad n \geq 1

or, equivalently, αn(μ)=0,n≥0\alpha_{n}^{(\mu)}=0, n \geq 0.
Hence, we conclude that μ=μ~∈Vp∩Np\mu=\tilde{\mu} \in V_{p} \cap N_{p}. This completes the proof.
In [4] it was shown how to get measures in Vp∩NpV_{p} \cap N_{p} ( pp even) by imposing some restrictions of sign on the sequence {cn}n=1∞\left\{c_{n}\right\}_{n=1}^{\infty}. To be precise, if {{cn}n=1∞,{mn}n=1∞}\left\{\left\{c_{n}\right\}_{n=1}^{\infty},\left\{m_{n}\right\}_{n=1}^{\infty}\right\} is a pair of pp-periodic sequences ( pp even) and {cn}n=1∞\left\{c_{n}\right\}_{n=1}^{\infty} has the property c2n=−c2n−1,n≥1c_{2 n}=-c_{2 n-1}, n \geq 1, then the associated measure μ\mu belongs to Vp∩NpV_{p} \cap N_{p}.

Fig. 2. Submanifold X3X_{3}.

We are interested in characterizing the general choices of {{cn}n=1∞,{mn}n=1∞}\left\{\left\{c_{n}\right\}_{n=1}^{\infty},\left\{m_{n}\right\}_{n=1}^{\infty}\right\} which are associated with measures in Vp∩NpV_{p} \cap N_{p}. In order to do this, let us consider, for ℓ≥2\ell \geq 2, the set

Xℓ:={(x1,…,xℓ)∈Rℓ:xℓ=−Im∏j=1ℓ−1(1+ixj)Re∏j=1ℓ−1(1+ixj)}X_{\ell}:=\left\{\left(x_{1}, \ldots, x_{\ell}\right) \in \mathbb{R}^{\ell}: x_{\ell}=-\frac{\mathcal{I} m \prod_{j=1}^{\ell-1}\left(1+i x_{j}\right)}{\mathcal{R} e \prod_{j=1}^{\ell-1}\left(1+i x_{j}\right)}\right\}

For ℓ=1\ell=1 we define X1:={0}⊂RX_{1}:=\{0\} \subset \mathbb{R}.
Clearly, for any ℓ≥1\ell \geq 1, the set XℓX_{\ell} is a submanifold of Rℓ\mathbb{R}^{\ell} with dimension ℓ−1\ell-1. The Fig. 2 illustrates the submanifold XℓX_{\ell} when ℓ=3\ell=3.

Furthermore, for the submanifolds Xℓ,ℓ≥1X_{\ell}, \ell \geq 1, the following holds.
Proposition 3.3. For ℓ≥1\ell \geq 1, let XℓX_{\ell} be defined as before. Then, Xℓ×{0}⊂Xℓ+1X_{\ell} \times\{0\} \subset X_{\ell+1}.
Proof. Firstly, we notice that

(x1,…,xℓ)∈Xℓ if, and only if ∏j=1ℓ(1+ixj)∈R,ℓ≥1\left(x_{1}, \ldots, x_{\ell}\right) \in X_{\ell} \quad \text { if, and only if } \quad \prod_{j=1}^{\ell}\left(1+i x_{j}\right) \in \mathbb{R}, \quad \ell \geq 1

Hence, if (x1,…,xℓ)∈Xℓ\left(x_{1}, \ldots, x_{\ell}\right) \in X_{\ell} then ∏j=1ℓ(1+ixj)∈R\prod_{j=1}^{\ell}\left(1+i x_{j}\right) \in \mathbb{R} and, consequently,

[∏j=1ℓ(1+ixj)][1+i0]∈R\left[\prod_{j=1}^{\ell}\left(1+i x_{j}\right)\right][1+i 0] \in \mathbb{R}

Thus, (x1,…,xℓ,0)∈Xℓ+1\left(x_{1}, \ldots, x_{\ell}, 0\right) \in X_{\ell+1}, which completes the proof.
In the next theorem we use the submanifold XpX_{p} to characterize the set Vp∩Np,p≥1V_{p} \cap N_{p}, p \geq 1.
Theorem 3.4. Let μ∈Np\mu \in N_{p} (or VpV_{p} ). Then, μ∈Vp∩Np\mu \in V_{p} \cap N_{p} if and only if (c1(μ),…,cp(μ))∈Xp\left(c_{1}^{(\mu)}, \ldots, c_{p}^{(\mu)}\right) \in X_{p}.
Proof. Firstly, from Theorem 3.2, it is not hard to see that μ∈Vp∩Np\mu \in V_{p} \cap N_{p} if and only if

∏j=1p(1−icj(μ))∏j=1p(1+icj(μ))=1\frac{\prod_{j=1}^{p}\left(1-i c_{j}^{(\mu)}\right)}{\prod_{j=1}^{p}\left(1+i c_{j}^{(\mu)}\right)}=1

[1]

1. Please cite this article in press as: C.F. Bracciali et al., Verblunsky coefficients related with periodic real sequences and associated measures on the unit circle, J. Math. Anal. Appl. (2016), http://dx.doi.org/10.1016/j.jmaa.2016.08.009 ↩︎

or, equivalently, ∏j=1p(1+icj(μ))∈R\prod_{j=1}^{p}\left(1+i c_{j}^{(\mu)}\right) \in \mathbb{R}. Now, the result follows by observing that ∏j=1p(1+icj(μ))∈R\prod_{j=1}^{p}\left(1+i c_{j}^{(\mu)}\right) \in \mathbb{R} if and only if (c1(μ),…,cp(μ))∈Xp\left(c_{1}^{(\mu)}, \ldots, c_{p}^{(\mu)}\right) \in X_{p}.

Remark 3.2. In Example 2, see (6.4), we present a family of measures that satisfies the condition of Theorem 3.4 with p=2p=2.

We say that μ\mu is a symmetric measure if dμ(z)=−dμ(1/z),z∈Td \mu(z)=-d \mu(1 / z), z \in \mathbb{T}. If S={μ∈P(T):μS=\{\mu \in \mathcal{P}(\mathbb{T}): \mu is symmetric }\} then the following corollary holds.

Corollary 3.4.1. S∩Np⊆Vp∩Np,p≥1S \cap N_{p} \subseteq V_{p} \cap N_{p}, p \geq 1. Moreover, equality holds if and only if p=1p=1.
Proof. From results established in [5] one can observe that μ∈S\mu \in S if and only if cn(μ)=0,n≥1c_{n}^{(\mu)}=0, n \geq 1.
Hence, if μ∈S∩Np\mu \in S \cap N_{p} we have (c1(μ),c2(μ),…,cp(μ))=(0,0,…,0)∈Xp\left(c_{1}^{(\mu)}, c_{2}^{(\mu)}, \ldots, c_{p}^{(\mu)}\right)=(0,0, \ldots, 0) \in X_{p}, and, by Theorem 3.4 it follows that S∩Np⊆Vp∩Np,p≥1S \cap N_{p} \subseteq V_{p} \cap N_{p}, p \geq 1.

Moreover, from Theorem 3.4 and Proposition 3.3 we see that equality holds if and only if p=1p=1.
Now, with the use of the results already obtained in this section, we are able to prove the Theorem 1.1.
Proof of Theorem 1.1. (i) Define gp:Np→Vpg_{p}: N_{p} \rightarrow V_{p} by the following relation

gp(μ):=μ~⇔αj(μ~)=αj(μ),j=0,1,…,p−1g_{p}(\mu):=\tilde{\mu} \Leftrightarrow \alpha_{j}^{(\tilde{\mu})}=\alpha_{j}^{(\mu)}, j=0,1, \ldots, p-1

where {αj(μ)}j=0∞\left\{\alpha_{j}^{(\mu)}\right\}_{j=0}^{\infty} and {αj(μ~)}j=0∞\left\{\alpha_{j}^{(\tilde{\mu})}\right\}_{j=0}^{\infty} are the sequences of Verblunsky coefficients associated with μ\mu and μ~\tilde{\mu}, respectively. Let us prove that gpg_{p} is a bijection.

Consider μ1,μ2∈Np\mu_{1}, \mu_{2} \in N_{p} such that gp(μ1)=gp(μ2)g_{p}\left(\mu_{1}\right)=g_{p}\left(\mu_{2}\right). If gp(μ1)=μ~1g_{p}\left(\mu_{1}\right)=\tilde{\mu}_{1} and gp(μ2)=μ~2g_{p}\left(\mu_{2}\right)=\tilde{\mu}_{2}, from the definition of gpg_{p}, we have

αj(μ1)=αj(μ~1)=αj(μ~2)=αj(μ2),j=0,1,…,p−1\alpha_{j}^{\left(\mu_{1}\right)}=\alpha_{j}^{\left(\tilde{\mu}_{1}\right)}=\alpha_{j}^{\left(\tilde{\mu}_{2}\right)}=\alpha_{j}^{\left(\mu_{2}\right)}, \quad j=0,1, \ldots, p-1

From the relation (2.1), this implies that

cj(μ1)=cj(μ2) and mj(μ1)=mj(μ2),j=1,2,…,pc_{j}^{\left(\mu_{1}\right)}=c_{j}^{\left(\mu_{2}\right)} \quad \text { and } \quad m_{j}^{\left(\mu_{1}\right)}=m_{j}^{\left(\mu_{2}\right)}, \quad j=1,2, \ldots, p

Now, μ1=μ2\mu_{1}=\mu_{2} follows from μ1,μ2∈Np\mu_{1}, \mu_{2} \in N_{p}. This shows that gpg_{p} is injective.
To prove that gpg_{p} is surjective for each μ~∈Vp\tilde{\mu} \in V_{p} we can take μ∈Np\mu \in N_{p} such that

cj(μ)=cj(μ~) and mj(μ)=mj(μ~),j=1,2,…,pc_{j}^{(\mu)}=c_{j}^{(\tilde{\mu})} \quad \text { and } \quad m_{j}^{(\mu)}=m_{j}^{(\tilde{\mu})}, \quad j=1,2, \ldots, p

Then, from (2.1), we clearly have that gp(μ)=μ~g_{p}(\mu)=\tilde{\mu}. Hence, we have proved that gpg_{p} is a bijection.
Now, we need to prove that gpg_{p} and gp−1g_{p}^{-1} are continuous functions in the metric DD. Let us start with gpg_{p}. Consider a sequence of measures μn\mu_{n} which converges to a measure μ\mu in NpN_{p}. Then, D(μn,μ)→0D\left(\mu_{n}, \mu\right) \rightarrow 0 as n→∞n \rightarrow \infty. Hence, if gp(μn)=μ~ng_{p}\left(\mu_{n}\right)=\tilde{\mu}_{n} and gp(μ)=μ~g_{p}(\mu)=\tilde{\mu}, by the definition of gpg_{p}, we have

αj(μ~n)=αj(μn) and αj(μ~)=αj(μ),j=0,1,…,p−1\alpha_{j}^{\left(\tilde{\mu}_{n}\right)}=\alpha_{j}^{\left(\mu_{n}\right)} \quad \text { and } \quad \alpha_{j}^{(\tilde{\mu})}=\alpha_{j}^{(\mu)}, \quad j=0,1, \ldots, p-1

Thus, by Lemma 3.1, we conclude that

lim⁡η→∞αj(μ~n)=lim⁡η→∞αj(μn)=αj(μ)=αj(μ~),j=0,1,…,p−1\lim _{\eta \rightarrow \infty} \alpha_{j}^{\left(\tilde{\mu}_{n}\right)}=\lim _{\eta \rightarrow \infty} \alpha_{j}^{\left(\mu_{n}\right)}=\alpha_{j}^{(\mu)}=\alpha_{j}^{(\tilde{\mu})}, \quad j=0,1, \ldots, p-1

Please cite this article in press as: C.F. Bracciali et al., Verblunsky coefficients related with periodic real sequences and associated measures on the unit circle, J. Math. Anal. Appl. (2016), http://dx.doi.org/10.1016/j.jmaa.2016.08.009

Moreover, since μ~∈Vp\tilde{\mu} \in V_{p} and μ~n∈Vp,n≥0\tilde{\mu}_{n} \in V_{p}, n \geq 0, we see that (3.8) holds for j≥0j \geq 0. This shows (once more from Lemma 3.1) that D(gp(μn),gp(μ))→0D\left(g_{p}\left(\mu_{n}\right), g_{p}(\mu)\right) \rightarrow 0. Hence, gpg_{p} is continuous.

To prove the continuity of gp−1g_{p}^{-1}, we consider a sequence of measures μ~n\tilde{\mu}_{n} which converges to a measure μ~\tilde{\mu} in VpV_{p}. If gp−1(μ~n)=μng_{p}^{-1}\left(\tilde{\mu}_{n}\right)=\mu_{n} and gp−1(μ~)=μg_{p}^{-1}(\tilde{\mu})=\mu, by the definition of gp−1g_{p}^{-1} and Lemma 3.1 we clearly have

lim⁡n→∞αj(μn)=αj(μ),j=0,1,…,p−1\lim _{n \rightarrow \infty} \alpha_{j}^{\left(\mu_{n}\right)}=\alpha_{j}^{(\mu)}, \quad j=0,1, \ldots, p-1

Hence, by part a) of Theorem 2.1, we see that

lim⁡n→∞cj(μn)=cj(μ) and lim⁡n→∞dj(μn)=dj(μ),j=1,2,…,p\lim _{n \rightarrow \infty} c_{j}^{\left(\mu_{n}\right)}=c_{j}^{(\mu)} \quad \text { and } \quad \lim _{n \rightarrow \infty} d_{j}^{\left(\mu_{n}\right)}=d_{j}^{(\mu)}, \quad j=1,2, \ldots, p

and the result follows applying again Lemma 3.1.
(ii) From Theorem 3.2 it is immediate that Np⊂UpN_{p} \subset U_{p}. To see that Np≠UpN_{p} \neq U_{p} let us construct a measure belonging to Up\NpU_{p} \backslash N_{p}. Choose pp complex numbers α0,α1,…,αp−1\alpha_{0}, \alpha_{1}, \ldots, \alpha_{p-1} such that ∣αj∣<1,j=0,1,…,p−1\left|\alpha_{j}\right|<1, j=0,1, \ldots, p-1. Now, consider ρ0,ρ1,…,ρp\rho_{0}, \rho_{1}, \ldots, \rho_{p} generated by

ρ0=1 and ρn=ρn−11−ρˉn−1αˉn−11−ρn−1αn−1,n=1,2,…,p\rho_{0}=1 \quad \text { and } \quad \rho_{n}=\rho_{n-1} \frac{1-\bar{\rho}_{n-1} \bar{\alpha}_{n-1}}{1-\rho_{n-1} \alpha_{n-1}}, \quad n=1,2, \ldots, p

Let μ\mu be the measure such that its associated Verblunsky coefficients satisfy

αn+p=βαn,n≥0\alpha_{n+p}=\beta \alpha_{n}, \quad n \geq 0

with β≠ρp,∣β∣=1\beta \neq \rho_{p},|\beta|=1. From Theorem 2.1, ρp(μ)=ρp\rho_{p}^{(\mu)}=\rho_{p} which means that the constructed measure μ\mu belongs to Up\NpU_{p} \backslash N_{p}.

Finally, (iii) and (iv) follow directly from Theorem 3.4.
Remark 3.3. From Theorem 3.2 and the definition of gpg_{p}, clearly we have that μ\mu is a fixed point of gpg_{p} if and only if λˉ=ρp(μ)=1\bar{\lambda}=\rho_{p}^{(\mu)}=1. Then, if we denote the set of fixed points of gpg_{p} by FpF_{p}, we immediately get that Fp=Vp∩NpF_{p}=V_{p} \cap N_{p}.

On the space P(T)\mathcal{P}(\mathbb{T}) we can define an equivalence relation by

μ1∼μ2⇔∃w∈T such that μ1(z)=μ2(wz),∀z∈T\mu_{1} \sim \mu_{2} \Leftrightarrow \exists w \in \mathbb{T} \quad \text { such that } \quad \mu_{1}(z)=\mu_{2}(w z), \quad \forall z \in \mathbb{T}

In this case, we say that μ1\mu_{1} and μ2\mu_{2} are equivalent by rotation. The equivalence class of μ1\mu_{1} is denoted by [μ1]\left[\mu_{1}\right]. Now, our aim is to prove Theorem 1.2 which shows that

{[μ~]:μ~∈Vp}={[μ]:μ∈Np}\left\{[\tilde{\mu}]: \tilde{\mu} \in V_{p}\right\}=\left\{[\mu]: \mu \in N_{p}\right\}

In the following ν\nu denotes the Lebesgue measure, i.e., dν(z)=(2πiz)−1dzd \nu(z)=(2 \pi i z)^{-1} d z.
Lemma 3.5. Let μ∈Np\{ν}\mu \in N_{p} \backslash\{\nu\} and μ~(z)=μ(wˉz),w∈T\tilde{\mu}(z)=\mu(\bar{w} z), w \in \mathbb{T}. Then, μ~∈Vp\{ν}\tilde{\mu} \in V_{p} \backslash\{\nu\} if and only if wp=λw^{p}=\lambda, where λˉ=ρp(μ)\bar{\lambda}=\rho_{p}^{(\mu)}.

Proof. Assume that μ~∈Vp\{ν}\tilde{\mu} \in V_{p} \backslash\{\nu\}. Since μ∈Np\{ν}\mu \in N_{p} \backslash\{\nu\}, from Theorem 3.2 we have

αn+p(μ)=λαn(μ),n≥0\alpha_{n+p}^{(\mu)}=\lambda \alpha_{n}^{(\mu)}, \quad n \geq 0

with λˉ=ρp(μ)\bar{\lambda}=\rho_{p}^{(\mu)}. On the other hand, from μ~(z)=μ(wˉz)\tilde{\mu}(z)=\mu(\bar{w} z), it is well known that

αn(μ~)=w−n−1αn(μ),n≥0\alpha_{n}^{(\tilde{\mu})}=w^{-n-1} \alpha_{n}^{(\mu)}, \quad n \geq 0

Since μ≠ν\mu \neq \nu, there exists k∈{0,1,2,…,p−1}k \in\{0,1,2, \ldots, p-1\} such that αk(μ)≠0\alpha_{k}^{(\mu)} \neq 0. Therefore, by observing that αk+p(μ~)=αk(μ~)\alpha_{k+p}^{(\tilde{\mu})}=\alpha_{k}^{(\tilde{\mu})}, from (3.9) and (3.10), we conclude that wp=λw^{p}=\lambda.

Reciprocally, if wp=λw^{p}=\lambda, from Theorem 3.2 we have αn+p(μ)=wpαn(μ),n≥0\alpha_{n+p}^{(\mu)}=w^{p} \alpha_{n}^{(\mu)}, n \geq 0. Hence, since αn+p(μ)=\alpha_{n+p}^{(\mu)}= wpαn(μ),n≥0w^{p} \alpha_{n}^{(\mu)}, n \geq 0, and μ~(z)=μ(wˉz)\tilde{\mu}(z)=\mu(\bar{w} z), we immediately obtain that αn+p(μ~)=αn(μ~),n≥0\alpha_{n+p}^{(\tilde{\mu})}=\alpha_{n}^{(\tilde{\mu})}, n \geq 0, that is, μ~∈Vp\{ν}\tilde{\mu} \in V_{p} \backslash\{\nu\}.

Lemma 3.5 shows that there exist exactly pp measures in Vp\{ν}V_{p} \backslash\{\nu\} which are equivalent by rotation to a given measure in Np\{ν}N_{p} \backslash\{\nu\}. This result is a natural consequence of Theorem 3.2.

The next result deals with the opposite problem.
Lemma 3.6. Let μ~∈Vp\{ν}\tilde{\mu} \in V_{p} \backslash\{\nu\} and μ(z)=μ~(wz),w∈T\mu(z)=\tilde{\mu}(w z), w \in \mathbb{T}. Then, μ∈Np\{ν}\mu \in N_{p} \backslash\{\nu\} if and only if τp(μ~)(w)=1\tau_{p}^{(\tilde{\mu})}(w)=1, where τp(μ~)(w)\tau_{p}^{(\tilde{\mu})}(w) is given by (2.2)(2.2).

Proof. Assume that μ∈Np\{ν}\mu \in N_{p} \backslash\{\nu\}. By noticing that μ~(z)=μ(wˉz)\tilde{\mu}(z)=\mu(\bar{w} z), from Lemma 3.5, we have wˉτ=λˉ=ρp(μ)\bar{w}^{\tau}=\bar{\lambda}=\rho_{p}^{(\mu)}. On the other hand, from μ(z)=μ~(wz)\mu(z)=\tilde{\mu}(w z) we must have ρp(μ)=wˉττp(μ~)(w)\rho_{p}^{(\mu)}=\bar{w}^{\tau} \tau_{p}^{(\tilde{\mu})}(w) (see [7]). Thus, we see that τp(μ~)(w)=1\tau_{p}^{(\tilde{\mu})}(w)=1

To prove the reciprocal of the statement let us assume that ww is such that τp(μ~)(w)=1\tau_{p}^{(\tilde{\mu})}(w)=1. Since μ(z)=μ~(wz)\mu(z)=\tilde{\mu}(w z) it follows that

ρp(μ)=wˉττp(μ~)(w)=wˉτ\rho_{p}^{(\mu)}=\bar{w}^{\tau} \tau_{p}^{(\tilde{\mu})}(w)=\bar{w}^{\tau}

Hence, by noticing that αn(μ)=wn+1αn(μ~),n≥0\alpha_{n}^{(\mu)}=w^{n+1} \alpha_{n}^{(\tilde{\mu})}, n \geq 0, we can use (3.11) to conclude that αn+p(μ)=λαn(μ),n≥0\alpha_{n+p}^{(\mu)}=\lambda \alpha_{n}^{(\mu)}, n \geq 0, with λˉ=ρp(μ)\bar{\lambda}=\rho_{p}^{(\mu)}.

Thus, from Theorem 3.2 it follows that μ\mu belongs to Np\{ν}N_{p} \backslash\{\nu\}.
Lemma 3.6 tells that given a measure μ~(μ~≠ν)\tilde{\mu}(\tilde{\mu} \neq \nu) with associated pp-periodic Verblunsky coefficients it is always possible to get exactly pp measures belonging to Np\{ν}N_{p} \backslash\{\nu\}. This result also shows how to construct the measures.

Notice that the condition τp(μ~)(w)=1\tau_{p}^{(\tilde{\mu})}(w)=1 is equivalent to ϕp(μ~;w)−ϕp∗(μ~;w)=0\phi_{p}(\tilde{\mu} ; w)-\phi_{p}^{*}(\tilde{\mu} ; w)=0. Thus, we see that τp(μ~)(w)=1\tau_{p}^{(\tilde{\mu})}(w)=1 if and only if ww is a possible pure point of μ~\tilde{\mu}. Hence, Lemma 3.6 says that to obtain a measure μ∈Np\{ν}\mu \in N_{p} \backslash\{\nu\} which is equivalent by rotation to μ~\tilde{\mu} we need to choose ww such that ww is a possible pure point of μ~\tilde{\mu}.

From Lemma 3.5 and Lemma 3.6 we get Theorem 1.2, which shows that the study of measures belonging to NpN_{p} is completely equivalent to that of measures in VpV_{p}.

The next results in this section deal with the pure points of measures in NpN_{p}.
Theorem 3.7. Let μ∈Np\mu \in N_{p}. Then, ww is a possible pure point of μ\mu if and only if the sequence {τn(μ)(w)}n=0∞\left\{\tau_{n}^{(\mu)}(w)\right\}_{n=0}^{\infty} given by (2.2) is periodic up to a phase and τn+p(μ)(w)=λˉτn(μ)(w),n≥0\tau_{n+p}^{(\mu)}(w)=\bar{\lambda} \tau_{n}^{(\mu)}(w), n \geq 0, with λˉ=ρp(μ)\bar{\lambda}=\rho_{p}^{(\mu)}.

Proof. Since the Lebesgue measure ν\nu does not have any pure point, without loss of generality, we can assume μ≠ν\mu \neq \nu.

Consider μ~(z)=μ(wˉλz)\tilde{\mu}(z)=\mu\left(\bar{w}_{\lambda} z\right) where wλw_{\lambda} is such that wλp=λw_{\lambda}^{p}=\lambda and λˉ=ρp(μ)\bar{\lambda}=\rho_{p}^{(\mu)}. By Lemma 3.5 we have that μ~∈Vp\{ν}\tilde{\mu} \in V_{p} \backslash\{\nu\}.

Notice that w~\widetilde{w} is a possible pure point of μ~\tilde{\mu} if and only if w=wˉλw~w=\bar{w}_{\lambda} \widetilde{w} is a possible pure point of μ\mu. On the other hand, w~\widetilde{w} is a possible pure point of μ~\tilde{\mu} if and only if τp(μ~)(w~)=1\tau_{p}^{(\tilde{\mu})}(\tilde{w})=1.

If ϕp(μ~;z)\phi_{p}(\tilde{\mu} ; z) and ϕp(μ;z)\phi_{p}(\mu ; z) are the associated respective orthogonal polynomials with degree pp, we also have ϕp(μ~;z)=wλpϕp(μ;wˉλz)\phi_{p}(\tilde{\mu} ; z)=w_{\lambda}^{p} \phi_{p}\left(\mu ; \bar{w}_{\lambda} z\right) and ϕp∗(μ~;z)=ϕp∗(μ;wˉλz)\phi_{p}^{*}(\tilde{\mu} ; z)=\phi_{p}^{*}\left(\mu ; \bar{w}_{\lambda} z\right). Hence, one can easily see that τp(μ~)(w~)=1\tau_{p}^{(\tilde{\mu})}(\tilde{w})=1 is equivalent to

wλˉpϕp(μ;wˉλw^)ϕp∗(μ;wˉλw^)=1w_{\bar{\lambda}}^{p} \frac{\phi_{p}\left(\mu ; \bar{w}_{\lambda} \hat{w}\right)}{\phi_{p}^{*}\left(\mu ; \bar{w}_{\lambda} \hat{w}\right)}=1

Furthermore, since (3.12) is equivalent to τp(μ)(w)=λˉ\tau_{p}^{(\mu)}(w)=\bar{\lambda}, we conclude that ww is a possible pure point of μ\mu if and only if τp(μ)(w)=λˉ\tau_{p}^{(\mu)}(w)=\bar{\lambda}, with λˉ=ρp(μ)\bar{\lambda}=\rho_{p}^{(\mu)}.

Now, we will prove that the condition τp(μ)(w)=λˉ\tau_{p}^{(\mu)}(w)=\bar{\lambda} holds if and only if the sequence {τn(μ)(w)}n=0∞\left\{\tau_{n}^{(\mu)}(w)\right\}_{n=0}^{\infty} is such that τn+p(μ)(w)=λˉτn(μ)(w),n≥0\tau_{n+p}^{(\mu)}(w)=\bar{\lambda} \tau_{n}^{(\mu)}(w), n \geq 0.

Assume that ww satisfies τp(μ)(w)=λˉ\tau_{p}^{(\mu)}(w)=\bar{\lambda}. Since, τ0(μ)(w)=1\tau_{0}^{(\mu)}(w)=1, we clearly have τp(μ)(w)=λˉτ0(μ)(w)\tau_{p}^{(\mu)}(w)=\bar{\lambda} \tau_{0}^{(\mu)}(w). Now, suppose that

τn+p(μ)(w)=λˉτn(μ)(w),0≤n≤k\tau_{n+p}^{(\mu)}(w)=\bar{\lambda} \tau_{n}^{(\mu)}(w), \quad 0 \leq n \leq k

Then, by using (3.13), (2.2) and Theorem 3.2 we see that

τk+1+p(μ)(w)=wτk+p(μ)(w)−αˉk+p(μ)1−wτk+p(μ)(w)αk+p(μ)=wλˉτk(μ)(w)−λˉαˉk(μ)1−wλˉτk(μ)(w)λαk(μ)=λˉτk+1(μ)(w)\tau_{k+1+p}^{(\mu)}(w)=\frac{w \tau_{k+p}^{(\mu)}(w)-\bar{\alpha}_{k+p}^{(\mu)}}{1-w \tau_{k+p}^{(\mu)}(w) \alpha_{k+p}^{(\mu)}}=\frac{w \bar{\lambda} \tau_{k}^{(\mu)}(w)-\bar{\lambda} \bar{\alpha}_{k}^{(\mu)}}{1-w \bar{\lambda} \tau_{k}^{(\mu)}(w) \lambda \alpha_{k}^{(\mu)}}=\bar{\lambda} \tau_{k+1}^{(\mu)}(w)

Hence, by mathematical induction, it follows that τn+p(μ)(w)=λˉτn(μ)(w),n≥0\tau_{n+p}^{(\mu)}(w)=\bar{\lambda} \tau_{n}^{(\mu)}(w), n \geq 0, with λˉ=ρp(μ)\bar{\lambda}=\rho_{p}^{(\mu)}.

Reciprocally, if τn+p(μ)(w)=λˉτn(μ)(w),n≥0\tau_{n+p}^{(\mu)}(w)=\bar{\lambda} \tau_{n}^{(\mu)}(w), n \geq 0, then by setting n=0n=0 we obtain τp(μ)(w)=λˉ\tau_{p}^{(\mu)}(w)=\bar{\lambda}. This completes the proof.

Theorem 3.8. Let μ∈Np\mu \in N_{p} and {α0(μ)}n=0∞\left\{\alpha_{0}^{(\mu)}\right\}_{n=0}^{\infty} its associated sequence of Verblunsky coefficients. In addition, let w∈Tw \in \mathbb{T} such that ww is a possible pure point of μ\mu. Then, ww is a pure point of μ\mu if and only if

∏j=1p∣1−wτj−1(μ)(w)αj−1(μ)∣2<∏j=1p[1−∣αj−1(μ)∣2]\prod_{j=1}^{p}\left|1-w \tau_{j-1}^{(\mu)}(w) \alpha_{j-1}^{(\mu)}\right|^{2}<\prod_{j=1}^{p}\left[1-\left|\alpha_{j-1}^{(\mu)}\right|^{2}\right]

Moreover, if ww is a pure point of μ\mu, then the mass at this point is given by

μ({w})=γγ+ζ\mu(\{w\})=\frac{\gamma}{\gamma+\zeta}

where ζ=∑n=1p∏j=1n∣1−wτj−1(μ)(w)αj−1(μ)∣21−∣αj−1(μ)∣2\zeta=\sum_{n=1}^{p} \prod_{j=1}^{n} \frac{\left|1-w \tau_{j-1}^{(\mu)}(w) \alpha_{j-1}^{(\mu)}\right|^{2}}{1-\left|\alpha_{j-1}^{(\mu)}\right|^{2}} and γ=1−∏j=1p∣1−wτj−1(μ)(w)αj−1(μ)∣21−∣αj−1(μ)∣2\gamma=1-\prod_{j=1}^{p} \frac{\left|1-w \tau_{j-1}^{(\mu)}(w) \alpha_{j-1}^{(\mu)}\right|^{2}}{1-\left|\alpha_{j-1}^{(\mu)}\right|^{2}}.
Proof. For j≥1j \geq 1, let qj=∣1−wτj−1(μ)(w)αj−1(μ)∣21−∣αj−1(μ)∣2q_{j}=\frac{\left|1-w \tau_{j-1}^{(\mu)}(w) \alpha_{j-1}^{(\mu)}\right|^{2}}{1-\left|\alpha_{j-1}^{(\mu)}\right|^{2}}. By Theorem 2.2 we know that ww is a pure point of μ\mu, if and only if, the infinite sum s(w)=∑n=1∞∏j=1nqjs(w)=\sum_{n=1}^{\infty} \prod_{j=1}^{n} q_{j} is convergent.

On the other hand, since μ∈Np\mu \in N_{p}, from Theorem 3.2 we have αj+p(μ)=λαj(μ),j≥0\alpha_{j+p}^{(\mu)}=\lambda \alpha_{j}^{(\mu)}, j \geq 0. Hence, by observing that ∣λ∣=1|\lambda|=1, from Theorem 3.7 we conclude that qj+p=qj,j≥1q_{j+p}=q_{j}, j \geq 1.

Thus, if q=∏j=1pqjq=\prod_{j=1}^{p} q_{j}, we can write s(w)s(w) as

s(w)=q1(∑n=0∞qn)+q1q2(∑n=0∞qn)+⋯+q1q2⋯qp(∑n=0∞qn)s(w)=q_{1}\left(\sum_{n=0}^{\infty} q^{n}\right)+q_{1} q_{2}\left(\sum_{n=0}^{\infty} q^{n}\right)+\cdots+q_{1} q_{2} \cdots q_{p}\left(\sum_{n=0}^{\infty} q^{n}\right)

Observe that s(w)s(w) is convergent if and only if ∣q∣<1|q|<1. Thus, the first part of the statement follows.

Furthermore, if ∣q∣<1|q|<1 using (3.14), we have

s(w)=(11−q)(∑n=1ρ∏j=1nqj)=ζγs(w)=\left(\frac{1}{1-q}\right)\left(\sum_{n=1}^{\rho} \prod_{j=1}^{n} q_{j}\right)=\frac{\zeta}{\gamma}

Finally, by Theorem 2.2 and (3.15), we get

μ({w})=11+s(w)=γγ+ζ\mu(\{w\})=\frac{1}{1+s(w)}=\frac{\gamma}{\gamma+\zeta}

Remark 3.4. It can be proved that Theorem 3.7 and Theorem 3.8 also hold in the general case where μ\mu is any probability measure with periodic up to a phase associated Verblunsky coefficients and λ\lambda is any complex number with ∣λ∣=1|\lambda|=1. Indeed, the hypothesis μ∈Np\mu \in N_{p} should be replaced by μ∈Up\mu \in U_{p}. Thus, the Verblunsky coefficients associated with the measure μ\mu satisfy αn+p(μ)=λαn(μ),n≥0\alpha_{n+p}^{(\mu)}=\lambda \alpha_{n}^{(\mu)}, n \geq 0, for λ∈T\lambda \in \mathbb{T} and p∈Np \in \mathbb{N}. Therefore, in the proof of Theorem 3.7 consider μ~(z)=μ(wˉλz)\tilde{\mu}(z)=\mu\left(\bar{w}_{\lambda} z\right), where wλw_{\lambda} is such that wλp=λw_{\lambda}^{p}=\lambda, with λ∈T\lambda \in \mathbb{T}. Since μ~(z)=μ(wˉλz)\tilde{\mu}(z)=\mu\left(\bar{w}_{\lambda} z\right) it is known that

αj(μ~)=wˉλj+1αj(μ),j≥0\alpha_{j}^{(\tilde{\mu})}=\bar{w}_{\lambda}^{j+1} \alpha_{j}^{(\mu)}, \quad j \geq 0

Consequently, since αn+p(μ)=λαn(μ),n≥0\alpha_{n+p}^{(\mu)}=\lambda \alpha_{n}^{(\mu)}, n \geq 0, we obtain

αn+p(μ~)=wˉλn+p+1λαn(μ)=wˉλn+1αn(μ)=αn(μ~),n≥0\alpha_{n+p}^{(\tilde{\mu})}=\bar{w}_{\lambda}^{n+p+1} \lambda \alpha_{n}^{(\mu)}=\bar{w}_{\lambda}^{n+1} \alpha_{n}^{(\mu)}=\alpha_{n}^{(\tilde{\mu})}, \quad n \geq 0

Hence, μ~∈Vp\{ν}\tilde{\mu} \in V_{p} \backslash\{\nu\} and the proof of Theorem 3.7 follows analogously. Moreover, notice that the proof of Theorem 3.8 is analogous when we replace the hypothesis μ∈Np\mu \in N_{p} by μ∈Up\mu \in U_{p}, since in this case we also have αn+p(μ)=λαn(μ),n≥0\alpha_{n+p}^{(\mu)}=\lambda \alpha_{n}^{(\mu)}, n \geq 0, with λ∈T\lambda \in \mathbb{T} (i.e., ∣λ∣=1|\lambda|=1 ).

4. Associated para-orthogonal polynomials

In (1.2) we have defined the space NpN_{p} in terms of the periodicity of the associated pair {{cn(μ)}n=1∞\left\{\left\{c_{n}^{(\mu)}\right\}_{n=1}^{\infty}\right., {mn(μ)}n=1∞}\left.\left\{m_{n}^{(\mu)}\right\}_{n=1}^{\infty}\right\} with {mn(μ)}n=0∞\left\{m_{n}^{(\mu)}\right\}_{n=0}^{\infty} being the minimal parameter sequence of the associated positive chain sequence {dn(μ)}n=1∞\left\{d_{n}^{(\mu)}\right\}_{n=1}^{\infty} as in Theorem 2.1.

Now, given a measure μ\mu in NpN_{p}, we investigate what happens with the associated pair {{cn(μ)}n=1∞\left\{\left\{c_{n}^{(\mu)}\right\}_{n=1}^{\infty}\right., {dn(μ)}n=1∞}\left.\left\{d_{n}^{(\mu)}\right\}_{n=1}^{\infty}\right\}

It is known (see, for example, [6]) that if {an}n=1∞\left\{a_{n}\right\}_{n=1}^{\infty} is a positive chain sequence, then {a1,n}n=1∞={an+1}n=1∞\left\{a_{1, n}\right\}_{n=1}^{\infty}=\left\{a_{n+1}\right\}_{n=1}^{\infty} is also a positive chain sequence. Furthermore, if {Mn}n=0∞\left\{M_{n}\right\}_{n=0}^{\infty} is the maximal parameter sequence for {an}n=1∞\left\{a_{n}\right\}_{n=1}^{\infty} then {M1,n}n=0∞={Mn+1}n=0∞\left\{M_{1, n}\right\}_{n=0}^{\infty}=\left\{M_{n+1}\right\}_{n=0}^{\infty} is the maximal parameter sequence of {a1,n}n=1∞\left\{a_{1, n}\right\}_{n=1}^{\infty}.

We say that {an}n=1∞\left\{a_{n}\right\}_{n=1}^{\infty} is a single parameter positive chain sequence (SPPCS) if its minimal parameter sequence {mn}n=0∞\left\{m_{n}\right\}_{n=0}^{\infty} coincide with its maximal parameter sequence {Mn}n=0∞\left\{M_{n}\right\}_{n=0}^{\infty}, or equivalently, if m0=M0=0m_{0}=M_{0}=0. Otherwise, we say that {an}n=1∞\left\{a_{n}\right\}_{n=1}^{\infty} is a non-single parameter positive chain sequence.

Thus, using the pp-periodicity of the associated pair {{cn(μ)}n=1∞,{mn(μ)}n=1∞}\left\{\left\{c_{n}^{(\mu)}\right\}_{n=1}^{\infty},\left\{m_{n}^{(\mu)}\right\}_{n=1}^{\infty}\right\}, we can state the following.
Lemma 4.1. Let μ\mu be a probability measure in NpN_{p} and {{cn(μ)}n=1∞,{dn(μ)}n=1∞}\left\{\left\{c_{n}^{(\mu)}\right\}_{n=1}^{\infty},\left\{d_{n}^{(\mu)}\right\}_{n=1}^{\infty}\right\} as in Theorem 2.1. Then {{cn(μ)}n=1∞,{dn+1(μ)}n=1∞}\left\{\left\{c_{n}^{(\mu)}\right\}_{n=1}^{\infty},\left\{d_{n+1}^{(\mu)}\right\}_{n=1}^{\infty}\right\} is a pair of pp-periodic real sequences with {dn+1(μ)}n=1∞\left\{d_{n+1}^{(\mu)}\right\}_{n=1}^{\infty} a non-single parameter positive chain sequence.

The next lemma is a kind of reciprocal of Lemma 4.1. It says that every probability measure μ∈P(T)\mu \in \mathcal{P}(\mathbb{T}) associated with a pair of real sequences {{cn(μ)}n=1∞,{dn(μ)}n=1∞}\left\{\left\{c_{n}^{(\mu)}\right\}_{n=1}^{\infty},\left\{d_{n}^{(\mu)}\right\}_{n=1}^{\infty}\right\} with {dn+1(μ)}n=1∞\left\{d_{n+1}^{(\mu)}\right\}_{n=1}^{\infty} a pp-periodic non-single

parameter positive chain sequence and {cn(μ)}n=1∞\left\{c_{n}^{(\mu)}\right\}_{n=1}^{\infty} also pp-periodic is a slight modification of some specific measure in NpN_{p}.

Lemma 4.2. Let μ∈P(T)\mu \in \mathcal{P}(\mathbb{T}) associated with a pair of real sequences {{cn(μ)}n=1∞,{dn(μ)}n=1∞}\left\{\left\{c_{n}^{(\mu)}\right\}_{n=1}^{\infty},\left\{d_{n}^{(\mu)}\right\}_{n=1}^{\infty}\right\} as in Theorem 2.1. Moreover, suppose that {dn+1(μ)}n=1∞\left\{d_{n+1}^{(\mu)}\right\}_{n=1}^{\infty} and {cn(μ)}n=1∞\left\{c_{n}^{(\mu)}\right\}_{n=1}^{\infty} are p-periodic sequences. Then, there exists 0≤ϵ<10 \leq \epsilon<1 and μ(0;.)∈Np\mu(0 ;.) \in N_{p}, with mass zero at z=1z=1 such that

∫Tℓ(z)dμ(z)=(1−ϵ)∫Tℓ(z)dμ(0;z)+ϵℓ(1)\int_{\mathbb{T}} \ell(z) d \mu(z)=(1-\epsilon) \int_{\mathbb{T}} \ell(z) d \mu(0 ; z)+\epsilon \ell(1)

for any Laurent polynomial ℓ\ell.
Proof. Let ϵ\epsilon be the mass of the measure μ\mu at z=1z=1. Let us consider the family of measures μ(δ;),0≤δ<1\mu(\delta ;), 0 \leq \delta<1, built from μ\mu and given by

∫Tℓ(z)dμ(δ;z)=1−δ1−ϵ∫Tℓ(z)dμ(z)+δ−ϵ1−ϵℓ(1)\int_{\mathbb{T}} \ell(z) d \mu(\delta ; z)=\frac{1-\delta}{1-\epsilon} \int_{\mathbb{T}} \ell(z) d \mu(z)+\frac{\delta-\epsilon}{1-\epsilon} \ell(1)

For each 0≤δ<10 \leq \delta<1, let {{cn(δ)}n=1∞,{dn(δ)}n=1∞}\left\{\left\{c_{n}(\delta)\right\}_{n=1}^{\infty},\left\{d_{n}(\delta)\right\}_{n=1}^{\infty}\right\} be the associated pair of the measure μ(δ;\mu(\delta ;.)asin) as in Theorem 2.1 and {mn(δ)}n=0∞\left\{m_{n}(\delta)\right\}_{n=0}^{\infty} the minimal parameter sequence for {dn(δ)}n=1∞\left\{d_{n}(\delta)\right\}_{n=1}^{\infty}. From results established in [7], for any 0≤δ<10 \leq \delta<1, we have

cn(δ)=cn(μ) and dn+1(δ)=dn+1(μ),n≥1c_{n}(\delta)=c_{n}^{(\mu)} \quad \text { and } \quad d_{n+1}(\delta)=d_{n+1}^{(\mu)}, \quad n \geq 1

Furthermore, if {Mn(μ)}n=0∞\left\{M_{n}^{(\mu)}\right\}_{n=0}^{\infty} is the maximal parameter sequence for {dn(μ)}n=1∞\left\{d_{n}^{(\mu)}\right\}_{n=1}^{\infty} then we have d1(δ)=d_{1}(\delta)= (1−δ)M1(μ)(1-\delta) M_{1}^{(\mu)}.

Since {dn+1(μ)}n=1∞\left\{d_{n+1}^{(\mu)}\right\}_{n=1}^{\infty} is a pp-periodic positive chain sequence it follows (see, for example, [6]) that {Mn+1(μ)}n=0∞\left\{M_{n+1}^{(\mu)}\right\}_{n=0}^{\infty} is the pp-periodic maximal parameter sequence for {dn+1(μ)}n=1∞\left\{d_{n+1}^{(\mu)}\right\}_{n=1}^{\infty}.

Hence, if we consider the measure μ(0;\mu(0 ;.)wehaveδ=0) we have \delta=0 and, consequently, from (4.2) {dn(0)}n=1∞\left\{d_{n}(0)\right\}_{n=1}^{\infty} is such that mn(0)=Mn(μ),n≥1m_{n}(0)=M_{n}^{(\mu)}, n \geq 1. Thus, it follows that {mn(0)}n=1∞\left\{m_{n}(0)\right\}_{n=1}^{\infty} is pp-periodic. On the other hand, again from (4.2) and by the pp-periodicity of the sequence {cn(μ)}n=1∞\left\{c_{n}^{(\mu)}\right\}_{n=1}^{\infty} we have

cn+p(0)=cn+p(μ)=cn(μ)=cn(0),n≥1c_{n+p}(0)=c_{n+p}^{(\mu)}=c_{n}^{(\mu)}=c_{n}(0), \quad n \geq 1

Thus, μ(0;\mu(0 ;.)belongstoNp) belongs to N_{p} and the result follows from (4.1) with δ=0\delta=0.
Lemma 4.1 and Lemma 4.2 show that by studying the NpN_{p} spaces we are, in fact, studying that measures in P(T)\mathcal{P}(\mathbb{T}) which are associated (as in Theorem 2.1) to a pair of real sequences {{cn}n=1∞,{dn}n=1∞}\left\{\left\{c_{n}\right\}_{n=1}^{\infty},\left\{d_{n}\right\}_{n=1}^{\infty}\right\} where {cn}n=1∞\left\{c_{n}\right\}_{n=1}^{\infty} and {dn+1}n=1∞\left\{d_{n+1}\right\}_{n=1}^{\infty} are pp-periodic sequences.

In [7] it was shown that given a measure μ∈P(T)\mu \in \mathcal{P}(\mathbb{T}) it is always possible to consider an associated sequence of polynomials, {Rn(z)}n=0∞\left\{R_{n}(z)\right\}_{n=0}^{\infty}, which is given by the recurrence formula

Rn+1(z)=[(1+icn+1)z+(1−icn+1)]Rn(z)−4dn+1zRn−1(z),n≥1R_{n+1}(z)=\left[\left(1+i c_{n+1}\right) z+\left(1-i c_{n+1}\right)\right] R_{n}(z)-4 d_{n+1} z R_{n-1}(z), \quad n \geq 1

with R0(z)=1R_{0}(z)=1 and R1(z)=(1+ic1)z+(1−ic1)R_{1}(z)=\left(1+i c_{1}\right) z+\left(1-i c_{1}\right), where

Rn(z)=∏j=1n[1−ρj−1αj−1]∏j=1n[1−R‾e(ρj−1αj−1)]zϕn(μ;z)−ρnϕn∗(μ;z)z−1R_{n}(z)=\frac{\prod_{j=1}^{n}\left[1-\rho_{j-1} \alpha_{j-1}\right]}{\prod_{j=1}^{n}\left[1-\overline{\mathcal{R}} e\left(\rho_{j-1} \alpha_{j-1}\right)\right]} \frac{z \phi_{n}(\mu ; z)-\rho_{n} \phi_{n}^{*}(\mu ; z)}{z-1}

Please cite this article in press as: C.F. Bracciali et al., Verblunsky coefficients related with periodic real sequences and associated measures on the unit circle, J. Math. Anal. Appl. (2016), http://dx.doi.org/10.1016/j.jmaa.2016.08.009

with ρn=ϕn(μ;1)/ϕn∗(μ;1),n≥0\rho_{n}=\phi_{n}(\mu ; 1) / \phi_{n}^{*}(\mu ; 1), n \geq 0. Notice that the sequence {(z−1)Rn(z)}n=0∞\left\{(z-1) R_{n}(z)\right\}_{n=0}^{\infty} is a sequence of paraorthogonal polynomials on the unit circle (POPUC). For more details on para-orthogonal polynomials we refer to [15][15].

Furthermore, in [5] it was shown how to recover, starting from the recurrence formula (4.3), the associated measure, the Verblunsky coefficients and the respective moments.

As a consequence of Lemma 4.1, Lemma 4.2 and the results established in [5,7][5,7] we can state the following.
Theorem 4.3. Given a measure μ∈Np\mu \in N_{p}, there exists an associated sequence of polynomials {Rn(z)}n=0∞\left\{R_{n}(z)\right\}_{n=0}^{\infty} satisfying (4.3) with {{cn}n=1∞,{dn+1}n=1∞}\left\{\left\{c_{n}\right\}_{n=1}^{\infty},\left\{d_{n+1}\right\}_{n=1}^{\infty}\right\} a pair of p-periodic real sequences and {dn+1}n=1∞\left\{d_{n+1}\right\}_{n=1}^{\infty} a non-single parameter positive chain sequence. Reciprocally, given a sequence of polynomials satisfying (4.3) with {{cn}n=1∞,{dn+1}n=1∞}\left\{\left\{c_{n}\right\}_{n=1}^{\infty},\left\{d_{n+1}\right\}_{n=1}^{\infty}\right\} a pair of p-periodic real sequences and {dn+1}n=1∞\left\{d_{n+1}\right\}_{n=1}^{\infty} a non-single parameter positive chain sequence, there exists an associated measure μ∈Np\mu \in N_{p}.

Proof. If μ∈Np\mu \in N_{p} and {{cn}n=1∞,{dn}n=1∞}\left\{\left\{c_{n}\right\}_{n=1}^{\infty},\left\{d_{n}\right\}_{n=1}^{\infty}\right\} is the associated pair as in Theorem 2.1, then by the results established in [7] there exists a sequence of polynomials {Rn(z)}n=0∞\left\{R_{n}(z)\right\}_{n=0}^{\infty} satisfying (4.3). Now, the result follows from Lemma 4.1.

Reciprocally, let {Rn(z)}n=0∞\left\{R_{n}(z)\right\}_{n=0}^{\infty} be a sequence of polynomials satisfying (4.3) with {{cn}n=1∞,{dn+1}n=1∞}\left\{\left\{c_{n}\right\}_{n=1}^{\infty},\left\{d_{n+1}\right\}_{n=1}^{\infty}\right\} a pair of pp-periodic real sequences and {dn+1}n=1∞\left\{d_{n+1}\right\}_{n=1}^{\infty} a non-single parameter positive chain sequence. Since {dn+1}n=1∞\left\{d_{n+1}\right\}_{n=1}^{\infty} is a non-single parameter positive chain sequence, we can obtain a new chain sequence {dn}n=1∞\left\{d_{n}\right\}_{n=1}^{\infty} by setting d1=(1−ϵ)M1d_{1}=(1-\epsilon) M_{1}, with 0≤ϵ<10 \leq \epsilon<1. Here, M1M_{1} denotes the first term of the maximal parameter sequence {Mn+1}n=0∞\left\{M_{n+1}\right\}_{n=0}^{\infty} for {dn+1}n=1∞\left\{d_{n+1}\right\}_{n=1}^{\infty}.

Thus, by the results established in [5], starting from the sequence {Rn(z)}n=0∞\left\{R_{n}(z)\right\}_{n=0}^{\infty} we can recover a unique probability measure on the unit circle, say μ^\hat{\mu}, associated with the pair {{cn}n=1∞,{dn}n=1∞}\left\{\left\{c_{n}\right\}_{n=1}^{\infty},\left\{d_{n}\right\}_{n=1}^{\infty}\right\}.

Now, using the measure μ^\hat{\mu} and Lemma 4.2, we can obtain an associated measure μ=μ^(0;.)∈Np\mu=\hat{\mu}(0 ;.) \in N_{p} with mass zero at z=1z=1.

The next result shows how one can use the zeros of the associated POPUC to determine the possible pure points of measures in NpN_{p}.

Theorem 4.4. Let μ∈Np\mu \in N_{p} and {Rn(z)}n=0∞\left\{R_{n}(z)\right\}_{n=0}^{\infty} be the associated sequence of polynomials satisfying the recurrence formula (4.3). Then, ww is a possible pure point of μ\mu if and only if ww satisfies the equation (z−1)Rp−1(z)=0(z-1) R_{p-1}(z)=0.

Proof. From the proof of Theorem 3.7, ww is a possible pure point of μ\mu if and only if τp(μ)(w)=λˉ\tau_{p}^{(\mu)}(w)=\bar{\lambda}, where λˉ=ρp(μ)\bar{\lambda}=\rho_{p}^{(\mu)}.

Furthermore, from (2.2), λˉ=τp(μ)(w)\bar{\lambda}=\tau_{p}^{(\mu)}(w) is equivalent to ϕp(μ;w)−λˉϕp∗(μ;w)=0\phi_{p}(\mu ; w)-\bar{\lambda} \phi_{p}^{*}(\mu ; w)=0, where ϕp(μ;z)\phi_{p}(\mu ; z) is the associated orthogonal polynomial with degree pp and ϕp∗(μ;z)\phi_{p}^{*}(\mu ; z) its reciprocal.

On the other hand (see [5]) we know that

ϕp(μ;z)=1∏k=1p(1+ick(μ))[Rp(z)−2(1−mp(μ))Rp−1(z)]\phi_{p}(\mu ; z)=\frac{1}{\prod_{k=1}^{p}\left(1+i c_{k}^{(\mu)}\right)}\left[R_{p}(z)-2\left(1-m_{p}^{(\mu)}\right) R_{p-1}(z)\right]

and, since znRn(1/zˉ)‾=Rn(z),n≥1z^{n} \overline{R_{n}(1 / \bar{z})}=R_{n}(z), n \geq 1, it follows that

ϕp∗(μ;z)=1∏k=1p(1−ick(μ))[Rp(z)−2(1−mp(μ))zRp−1(z)]\phi_{p}^{*}(\mu ; z)=\frac{1}{\prod_{k=1}^{p}\left(1-i c_{k}^{(\mu)}\right)}\left[R_{p}(z)-2\left(1-m_{p}^{(\mu)}\right) z R_{p-1}(z)\right]

Hence, since λˉ=ρp(μ)=∏k=1p(1−ick(μ))/(1+ick(μ))\bar{\lambda}=\rho_{p}^{(\mu)}=\prod_{k=1}^{p}\left(1-i c_{k}^{(\mu)}\right) /\left(1+i c_{k}^{(\mu)}\right), from (4.4) and (4.5) it follows that

ϕp(μ;w)−λˉϕp∗(μ;w)=0 if and only if (w−1)Rp−1(w)=0\phi_{p}(\mu ; w)-\bar{\lambda} \phi_{p}^{*}(\mu ; w)=0 \quad \text { if and only if } \quad(w-1) R_{p-1}(w)=0

This completes the proof.

4.1. A special case

Now, we present a case in which the zeros of Rn(z),n≥1R_{n}(z), n \geq 1, can be given in terms of the coefficients that appear in the recurrence formula (4.3).

Let cn=c,n≥1c_{n}=c, n \geq 1, and {dn+1}n=1∞\left\{d_{n+1}\right\}_{n=1}^{\infty} be a positive chain sequence. In this special case, the recurrence formula (4.3) can be given as

Rn+1(z)=[(1+ic)z+(1−ic)]Rn(z)−4dn+1zRn−1(z),n≥1R_{n+1}(z)=[(1+i c) z+(1-i c)] R_{n}(z)-4 d_{n+1} z R_{n-1}(z), \quad n \geq 1

with R0(z)=1R_{0}(z)=1 and R1(z)=(1+ic)z+(1−ic)R_{1}(z)=(1+i c) z+(1-i c).
To study the zeros of Rn(z),n≥1R_{n}(z), n \geq 1, is convenient to study the zeros of associated functions Wn(x)\mathcal{W}_{n}(x) defined on [−1,1][-1,1] by

Wn(x)=2−ne−inθ/2Rn(eiθ),n≥0\mathcal{W}_{n}(x)=2^{-n} e^{-i n \theta / 2} R_{n}\left(e^{i \theta}\right), \quad n \geq 0

with x=cos⁡(θ/2)x=\cos (\theta / 2). It was shown (see [2,9][2,9] ) that these functions satisfy the following recurrence relation

Wn+1(x)=(x−cn+11−x2)Wn(x)−dn+1Wn−1(x),n≥1\mathcal{W}_{n+1}(x)=\left(x-c_{n+1} \sqrt{1-x^{2}}\right) \mathcal{W}_{n}(x)-d_{n+1} \mathcal{W}_{n-1}(x), \quad n \geq 1

with W0(x)=1\mathcal{W}_{0}(x)=1 and W1(x)=x−c11−x2\mathcal{W}_{1}(x)=x-c_{1} \sqrt{1-x^{2}}.
Moreover, for any n≥1,Wn(x)n \geq 1, \mathcal{W}_{n}(x) has exactly nn distinct zeros xn,j=cos⁡(θn,j/2),j=1,2…,nx_{n, j}=\cos \left(\theta_{n, j} / 2\right), j=1,2 \ldots, n, in (−1,1)(-1,1) and they satisfy the following interlacing property

−1<xn+1,n+1<xn,n<xn+1,n<⋯<xn,1<xn+1,1<1,n≥1-1<x_{n+1, n+1}<x_{n, n}<x_{n+1, n}<\cdots<x_{n, 1}<x_{n+1,1}<1, \quad n \geq 1

In the case when cn=c,n≥1c_{n}=c, n \geq 1, from (4.7) we have

(x−c1−x2)Wn(x)=Wn+1(x)+dn+1Wn−1(x),n≥1\left(x-c \sqrt{1-x^{2}}\right) \mathcal{W}_{n}(x)=\mathcal{W}_{n+1}(x)+d_{n+1} \mathcal{W}_{n-1}(x), \quad n \geq 1

with W0(x)=1\mathcal{W}_{0}(x)=1 and W1(x)=x−c1−x2\mathcal{W}_{1}(x)=x-c \sqrt{1-x^{2}}.
Then, from (4.9), for each fixed n≥1n \geq 1,

(x−c1−x2)(W0(x)W1(x)W2(x)⋮Wn−2(x)Wn−1(x))=(010⋯00d201⋱⋮⋮0d30⋱00⋮⋱⋱⋱100⋯0dn−1010⋯00dn0)(W0(x)W1(x)W2(x)⋮Wn−2(x)Wn−1(x))+(000⋮0Wn(x))\left(x-c \sqrt{1-x^{2}}\right)\left(\begin{array}{c} \mathcal{W}_{0}(x) \\ \mathcal{W}_{1}(x) \\ \mathcal{W}_{2}(x) \\ \vdots \\ \mathcal{W}_{n-2}(x) \\ \mathcal{W}_{n-1}(x) \end{array}\right)=\left(\begin{array}{cccccc} 0 & 1 & 0 & \cdots & 0 & 0 \\ d_{2} & 0 & 1 & \ddots & \vdots & \vdots \\ 0 & d_{3} & 0 & \ddots & 0 & 0 \\ \vdots & \ddots & \ddots & \ddots & 1 & 0 \\ 0 & \cdots & 0 & d_{n-1} & 0 & 1 \\ 0 & \cdots & 0 & 0 & d_{n} & 0 \end{array}\right)\left(\begin{array}{c} \mathcal{W}_{0}(x) \\ \mathcal{W}_{1}(x) \\ \mathcal{W}_{2}(x) \\ \vdots \\ \mathcal{W}_{n-2}(x) \\ \mathcal{W}_{n-1}(x) \end{array}\right)+\left(\begin{array}{c} 0 \\ 0 \\ 0 \\ \vdots \\ 0 \\ \mathcal{W}_{n}(x) \end{array}\right)

Hence, from the above equality, it follows that the zeros xn,j(c)x_{n, j}^{(c)}, of Wn(x),n≥1\mathcal{W}_{n}(x), n \geq 1, are such that

xn,j(c)−c1−[xn,j(c)]2=λn,j,j=1,2,…,nx_{n, j}^{(c)}-c \sqrt{1-\left[x_{n, j}^{(c)}\right]^{2}}=\lambda_{n, j}, \quad j=1,2, \ldots, n

Please cite this article in press as: C.F. Bracciali et al., Verblunsky coefficients related with periodic real sequences and associated measures on the unit circle, J. Math. Anal. Appl. (2016), http://dx.doi.org/10.1016/j.jmaa.2016.08.009

where λn,j\lambda_{n, j} are the eigenvalues of the associated matrix A(d2,d3,…,dn)A\left(d_{2}, d_{3}, \ldots, d_{n}\right) given by

A(d2,d3,…,dn)=(010⋯00d201⋱⋮⋮0d30⋱00⋮⋱⋱⋱100⋯0dn−1010⋯00dn0)A\left(d_{2}, d_{3}, \ldots, d_{n}\right)=\left(\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 & 0 \\ d_{2} & 0 & 1 & \ddots & \vdots & \vdots \\ 0 & d_{3} & 0 & \ddots & 0 & 0 \\ \vdots & \ddots & \ddots & \ddots & 1 & 0 \\ 0 & \cdots & 0 & d_{n-1} & 0 & 1 \\ 0 & \cdots & 0 & 0 & d_{n} & 0 \end{array}\right)

Remark 4.1. When n=1n=1, the matrix defined in (4.11) reduces to the zero matrix.
Notice that, by setting c=0c=0 in (4.10), we obtain xn,j(0)=λn,j,1≤j≤nx_{n, j}^{(0)}=\lambda_{n, j}, 1 \leq j \leq n, which shows that −1<λn,j<1-1<\lambda_{n, j}<1. Moreover, from the interlancing property (4.8) for xn,j(0)=λn,j,1≤j≤nx_{n, j}^{(0)}=\lambda_{n, j}, 1 \leq j \leq n, it follows that the matrix A(d2,d3,…,dn)A\left(d_{2}, d_{3}, \ldots, d_{n}\right) has nn distinct eigenvalues. On the other hand, by solving (4.10) we immediately conclude that

xn,j(c)=λn,j+c1−λn,j2+c21+c2,j=1,2,…,nx_{n, j}^{(c)}=\frac{\lambda_{n, j}+c \sqrt{1-\lambda_{n, j}^{2}+c^{2}}}{1+c^{2}}, \quad j=1,2, \ldots, n

Thus, from (4.12), we can state the following.
Theorem 4.5. Let the sequences {Rn(z)}n=0∞\left\{R_{n}(z)\right\}_{n=0}^{\infty} and {Wn(x)}n=0∞\left\{\mathcal{W}_{n}(x)\right\}_{n=0}^{\infty} be given, respectively, by (4.6) and (4.9). Then, for each n≥1n \geq 1, the zeros xn,j(c)x_{n, j}^{(c)} of Wn(x)\mathcal{W}_{n}(x) are given by (4.12) and the zeros of Rn(z)R_{n}(z) are given by zn,j(c)=eiθn,j(c)z_{n, j}^{(c)}=e^{i \theta_{n, j}^{(c)}}, where θn,j(c)=2arccos⁡xn,j(c),j=1,2,…,n\theta_{n, j}^{(c)}=2 \arccos x_{n, j}^{(c)}, j=1,2, \ldots, n.

If we consider the measure μ∈Np\mu \in N_{p}, with associated pair {{cn}n=1∞,{dn}n=1∞}\left\{\left\{c_{n}\right\}_{n=1}^{\infty},\left\{d_{n}\right\}_{n=1}^{\infty}\right\} such that cn=c,n≥1c_{n}=c, n \geq 1, as an immediate consequence of Theorem 4.4 and Theorem 4.5 we have the following.

Corollary 4.5.1. Let μ∈Np,p≥2\mu \in N_{p}, p \geq 2, be the measure with associated pair {{cn}n=1∞,{dn}n=1∞}\left\{\left\{c_{n}\right\}_{n=1}^{\infty},\left\{d_{n}\right\}_{n=1}^{\infty}\right\} as in Theorem 2.1. Moreover, suppose that cn=c,n≥1c_{n}=c, n \geq 1. Then, ww is a possible pure point of μ\mu if and only if

w∈{1,eiθp−1,1(c),eiθp−1,2(c),…,eiθp−1,p−1(c)}w \in\left\{1, e^{i \theta_{p-1,1}^{(c)}}, e^{i \theta_{p-1,2}^{(c)}}, \ldots, e^{i \theta_{p-1, p-1}^{(c)}}\right\}

where θp−1,j(c)=2arccos⁡xp−1,j(c)\theta_{p-1, j}^{(c)}=2 \arccos x_{p-1, j}^{(c)} and xp−1,j(c)x_{p-1, j}^{(c)} is given by

xp−1,j(c)=λp−1,j+c1−λp−1,j2+c21+c2,j=1,2,…,p−1x_{p-1, j}^{(c)}=\frac{\lambda_{p-1, j}+c \sqrt{1-\lambda_{p-1, j}^{2}+c^{2}}}{1+c^{2}}, \quad j=1,2, \ldots, p-1

with λp−1,j\lambda_{p-1, j} an eigenvalue of the associated matrix A(d2,d3,…,dp−1)A\left(d_{2}, d_{3}, \ldots, d_{p-1}\right) as in (4.11).
Remark 4.2. Notice that for p=1p=1 and μ∈Np\mu \in N_{p}, then from Theorem 4.4 the only possible pure point of μ\mu is w=1w=1.

Corollary 4.5.2. Let μ~∈Np\tilde{\mu} \in N_{p} ( pp even) be the measure with associated pair {{c~n}n=1∞,{d~n}n=1∞}\left\{\left\{\tilde{c}_{n}\right\}_{n=1}^{\infty},\left\{\tilde{d}_{n}\right\}_{n=1}^{\infty}\right\} as in Theorem 2.1 and {m~n}n=0∞\left\{\tilde{m}_{n}\right\}_{n=0}^{\infty} the minimal parameter sequence for {d~n}n=1∞\left\{\tilde{d}_{n}\right\}_{n=1}^{\infty}. Moreover, suppose that c~n=\tilde{c}_{n}= (−1)nc,n≥1(-1)^{n} c, n \geq 1. Then, ww is a possible pure point of μ~\tilde{\mu} if and only if

w∈{β,eiθ^p−1,1,eiθ^p−1,2,…,eiθ^p−1,p−1}w \in\left\{\beta, e^{i \hat{\theta}_{p-1,1}}, e^{i \hat{\theta}_{p-1,2}}, \ldots, e^{i \hat{\theta}_{p-1, p-1}}\right\}

where β=−1+ic1−ic,θ^p−1,j=arg⁡β+2arccos⁡x^p−1,j(c)\beta=-\frac{1+i c}{1-i c}, \hat{\theta}_{p-1, j}=\arg \beta+2 \arccos \hat{x}_{p-1, j}^{(c)}, and x^p−1,j(c)\hat{x}_{p-1, j}^{(c)} is given by

x^p−1,j(c)=λ^p−1,j+c1−λ^p−1,j2+c21+c2,j=1,2,…,p−1\hat{x}_{p-1, j}^{(c)}=\frac{\hat{\lambda}_{p-1, j}+c \sqrt{1-\hat{\lambda}_{p-1, j}^{2}+c^{2}}}{1+c^{2}}, \quad j=1,2, \ldots, p-1

with λ^p−1,j\hat{\lambda}_{p-1, j} an eigenvalue of the associated matrix A(d^2,d^3,…,d^p−1)A\left(\hat{d}_{2}, \hat{d}_{3}, \ldots, \hat{d}_{p-1}\right). Here, A(d^2,d^3,…,d^p−1)A\left(\hat{d}_{2}, \hat{d}_{3}, \ldots, \hat{d}_{p-1}\right) is given as in (4.11) with d^2n=m^2nm^2n−1\hat{d}_{2 n}=\hat{m}_{2 n} \hat{m}_{2 n-1} and d^2n+1=(1−m^2n)(1−m^2n+1),n=1,2,…,(p−2)/2\hat{d}_{2 n+1}=\left(1-\hat{m}_{2 n}\right)\left(1-\hat{m}_{2 n+1}\right), n=1,2, \ldots,(p-2) / 2.

Proof. For p=2,Ap=2, A is the zero matrix, the only associated eigenvalue is zero and the result clearly holds.
Suppose p≥4p \geq 4. Furthermore, consider the measure μ^(z)=μ^(βz)\hat{\mu}(z)=\hat{\mu}(\beta z) with its associated pair of sequences {{c^n}n=1∞,{d^n}n=1∞}\left\{\left\{\hat{c}_{n}\right\}_{n=1}^{\infty},\left\{\hat{d}_{n}\right\}_{n=1}^{\infty}\right\} as in Theorem 2.1. It is known (see [4]) that

c^n=c,m^2n=m^2n and m^2n−1=1−m^2n−1,n≥1\hat{c}_{n}=c, \quad \hat{m}_{2 n}=\hat{m}_{2 n} \quad \text { and } \quad \hat{m}_{2 n-1}=1-\hat{m}_{2 n-1}, \quad n \geq 1

Hence, since

d^n=(1−m^n−1)m^n and d^n=(1−m^n−1)m^n,n≥1\hat{d}_{n}=\left(1-\hat{m}_{n-1}\right) \hat{m}_{n} \quad \text { and } \quad \hat{d}_{n}=\left(1-\hat{m}_{n-1}\right) \hat{m}_{n}, \quad n \geq 1

it is not hard to see that,

d^2n=m^2nm^2n−1 and d^2n+1=(1−m^2n)(1−m^2n+1),n≥1\hat{d}_{2 n}=\hat{m}_{2 n} \hat{m}_{2 n-1} \quad \text { and } \quad \hat{d}_{2 n+1}=\left(1-\hat{m}_{2 n}\right)\left(1-\hat{m}_{2 n+1}\right), \quad n \geq 1

Now, since μ^∈Np\hat{\mu} \in N_{p} the result follows from Corollary 4.5.1.

5. The limit periodic case

In Section 3 we saw that every measure μ∈Np\mu \in N_{p} is such that σess(μ)=∪Bj\sigma_{e s s}(\mu)=\cup \mathcal{B}_{j}, where each Bj\mathcal{B}_{j} is a closed subset of T\mathbb{T}. In other words, every measure with associated pair {{cn}n=1∞,{mn}n=1∞}\left\{\left\{c_{n}\right\}_{n=1}^{\infty},\left\{m_{n}\right\}_{n=1}^{\infty}\right\} of pp-periodic sequences has its essential support consisting of at most pp closed disjoint arcs on the unit circle.

The main purpose of this section is to study what happens with the essential support of a measure which is associated with a pair {{cn}n=1∞,{mn}n=1∞}\left\{\left\{c_{n}\right\}_{n=1}^{\infty},\left\{m_{n}\right\}_{n=1}^{\infty}\right\} of limit pp-periodic sequences. We say that a sequence of complex numbers {bn}n=1∞\left\{b_{n}\right\}_{n=1}^{\infty} is limit pp-periodic if the following holds

lim⁡n→∞bnp+k=lk,k=1,2,…,p\lim _{n \rightarrow \infty} b_{n p+k}=l_{k}, \quad k=1,2, \ldots, p

with lkl_{k} belonging to the extended complex plane.
To obtain the results of this section we use some results established in [1] where the authors have considered ratio asymptotics for orthogonal polynomials on the unit circle.

Let us begin by introducing some standard notations. Let μ∈P(T)\mu \in \mathcal{P}(\mathbb{T}) and {αn}n=0∞\left\{\alpha_{n}\right\}_{n=0}^{\infty} be its associated sequence of Verblunsky coefficients. We say that μ∈MT(L1,…,Lp;A1,…,Ap)\mu \in \mathcal{M}_{\mathbb{T}}\left(L_{1}, \ldots, L_{p} ; A_{1}, \ldots, A_{p}\right) if the following conditions hold

lim⁡n→∞∣αnp+k∣=Lk and lim⁡n→∞αˉnp+kαˉnp+k−1=Ak,k=1,2,…,p\lim _{n \rightarrow \infty}\left|\alpha_{n p+k}\right|=L_{k} \quad \text { and } \quad \lim _{n \rightarrow \infty} \frac{\bar{\alpha}_{n p+k}}{\bar{\alpha}_{n p+k-1}}=A_{k}, \quad k=1,2, \ldots, p

In [1] it was shown that in the case when p=1p=1 and L1∈(0,1]L_{1} \in(0,1], all measures in MT(L1;A1)\mathcal{M}_{\mathbb{T}}\left(L_{1} ; A_{1}\right) have the same essential support and, furthermore, for any measure μ∈MT(L1;A1)\mu \in \mathcal{M}_{\mathbb{T}}\left(L_{1} ; A_{1}\right) holds

σess(μ)={z∈T:θ0≤arg⁡(z)−arg⁡(A1)≤2π−θ0}\sigma_{e s s}(\mu)=\left\{z \in \mathbb{T}: \theta_{0} \leq \arg (z)-\arg \left(A_{1}\right) \leq 2 \pi-\theta_{0}\right\}

where sin⁡(θ0/2):=L1\sin \left(\theta_{0} / 2\right):=L_{1}. In this case, one can say that the sequence {αn}n=0∞\left\{\alpha_{n}\right\}_{n=0}^{\infty} obeys the López condition (see [19]).

For p≥2p \geq 2, also in [1] it was shown that, with the condition Lk∈(0,1),k=1,2,…,pL_{k} \in(0,1), k=1,2, \ldots, p, all measures μ∈MT(L1,…,Lp;A1,…,Ap)\mu \in \mathcal{M}_{\mathbb{T}}\left(L_{1}, \ldots, L_{p} ; A_{1}, \ldots, A_{p}\right) have the same essential support and this essential support is the union of at most pp disjoint closed arcs on T\mathbb{T}.

The following theorem provide us information about the essential support of measures such that their associated pairs {{cn(μ)}n=1∞,{mn(μ)}n=1∞}\left\{\left\{c_{n}^{(\mu)}\right\}_{n=1}^{\infty},\left\{m_{n}^{(\mu)}\right\}_{n=1}^{\infty}\right\} are limit pp-periodic with p=1p=1.

Theorem 5.1. Let μ∈P(T)\mu \in \mathcal{P}(\mathbb{T}) be the measure associated with the pair {{cn(μ)}n=1∞,{mn(μ)}n=1∞}\left\{\left\{c_{n}^{(\mu)}\right\}_{n=1}^{\infty},\left\{m_{n}^{(\mu)}\right\}_{n=1}^{\infty}\right\}. Suppose that lim⁡n→∞cn(μ)=c,lim⁡n→∞mn(μ)=m\lim _{n \rightarrow \infty} c_{n}^{(\mu)}=c, \lim _{n \rightarrow \infty} m_{n}^{(\mu)}=m and d=(1−m)md=(1-m) m. Then, the following holds.
(i) If 1−4d+c2=01-4 d+c^{2}=0 then σess(μ)=T\sigma_{e s s}(\mu)=\mathbb{T}.
(ii) If 1−4d+c2≠01-4 d+c^{2} \neq 0 and c∈Rc \in \mathbb{R} then σess(μ)={z∈T:θ0≤arg⁡(z)−arg⁡(w)≤2π−θ0}\sigma_{e s s}(\mu)=\left\{z \in \mathbb{T}: \theta_{0} \leq \arg (z)-\arg (w) \leq 2 \pi-\theta_{0}\right\}, where w=1−ic1+icw=\frac{1-i c}{1+i c} and θ0=2arcsin⁡(1−4d+c21+c2)\theta_{0}=2 \arcsin \left(\sqrt{\frac{1-4 d+c^{2}}{1+c^{2}}}\right). In particular, when m=0m=0 we have that μ\mu is a pure point measure and σess(μ)={−w}\sigma_{e s s}(\mu)=\{-w\}.
(iii) If c=±∞c= \pm \infty then μ\mu is a pure point measure and σess(μ)={1}\sigma_{e s s}(\mu)=\{1\}.

Proof. Since lim⁡n→∞cn(μ)=c,lim⁡n→∞mn(μ)=m\lim _{n \rightarrow \infty} c_{n}^{(\mu)}=c, \lim _{n \rightarrow \infty} m_{n}^{(\mu)}=m and d=(1−m)md=(1-m) m, from (2.1) we have

lim⁡n→∞∣αn+1(μ)∣=1−4d+c21+c2\lim _{n \rightarrow \infty}\left|\alpha_{n+1}^{(\mu)}\right|=\sqrt{\frac{1-4 d+c^{2}}{1+c^{2}}}

On the other hand, again from (2.1)

lim⁡n→∞αˉn+1(μ)αˉn(μ)=lim⁡n→∞[1−icn+1(μ)1+icn+1(μ)][1−2mn+2(μ)+icn+2(μ)1+icn+2(μ)][1+icn+1(μ)1−2mn+1(μ)+icn+1(μ)]\lim _{n \rightarrow \infty} \frac{\bar{\alpha}_{n+1}^{(\mu)}}{\bar{\alpha}_{n}^{(\mu)}}=\lim _{n \rightarrow \infty}\left[\frac{1-i c_{n+1}^{(\mu)}}{1+i c_{n+1}^{(\mu)}}\right]\left[\frac{1-2 m_{n+2}^{(\mu)}+i c_{n+2}^{(\mu)}}{1+i c_{n+2}^{(\mu)}}\right]\left[\frac{1+i c_{n+1}^{(\mu)}}{1-2 m_{n+1}^{(\mu)}+i c_{n+1}^{(\mu)}}\right]

Now, if 1−4d+c2=01-4 d+c^{2}=0 from (5.3) we have lim⁡n→∞∣αn+1(μ)∣=0\lim _{n \rightarrow \infty}\left|\alpha_{n+1}^{(\mu)}\right|=0 and, consequently, statement (i) holds (see [18, Theorem 4.3.8]).

If 1−4d+c2≠01-4 d+c^{2} \neq 0, from (5.3) we have lim⁡n→∞∣αn+1(μ)∣=L1\lim _{n \rightarrow \infty}\left|\alpha_{n+1}^{(\mu)}\right|=L_{1}, where L1=1−4d+c21+c2L_{1}=\sqrt{\frac{1-4 d+c^{2}}{1+c^{2}}} is such that L1∈(0,1]L_{1} \in(0,1].
Furthermore, 1−4d+c2≠01-4 d+c^{2} \neq 0 implies d≠1/4d \neq 1 / 4 or c≠0c \neq 0. Thus, from (5.4) we conclude that

lim⁡n→∞αˉn+1(μ)αˉn(μ)=1−ic1+ic=w\lim _{n \rightarrow \infty} \frac{\bar{\alpha}_{n+1}^{(\mu)}}{\bar{\alpha}_{n}^{(\mu)}}=\frac{1-i c}{1+i c}=w

Hence, if we set A1=wA_{1}=w in (5.5), then the condition (5.1) is satisfied. Thus, by (5.2), we have

σess(μ)={z∈T:θ0≤arg⁡(z)−arg⁡(w)≤2π−θ0}\sigma_{e s s}(\mu)=\left\{z \in \mathbb{T}: \theta_{0} \leq \arg (z)-\arg (w) \leq 2 \pi-\theta_{0}\right\}

Moreover, if d=0d=0 one can observe that θ0=π\theta_{0}=\pi, and, consequently, σess(μ)={−w}\sigma_{e s s}(\mu)=\{-w\}. This completes the proof of (ii).

To show (iii) it is sufficient to observe that if c=±∞c= \pm \infty then in (5.3) we have L1=1L_{1}=1 and in (5.5) we have A1=−1A_{1}=-1. Hence, from (5.2), the result follows.

Remark 5.1. Notice that, from (1.1), the number dd mentioned in Theorem 5.1 is such that d=lim⁡n→∞dn(μ)d=\lim _{n \rightarrow \infty} d_{n}^{(\mu)}, where {dn(μ)}n=1∞\left\{d_{n}^{(\mu)}\right\}_{n=1}^{\infty} is the associated positive chain sequence. Thus, from well known results on chain sequences,

we have d≤1/4d \leq 1 / 4 and, consequently, one can see that the condition 1−4d+c2=01-4 d+c^{2}=0 reduces to c=0c=0 and d=1/4d=1 / 4.

The next results deal with the case where p≥2p \geq 2.
Lemma 5.2. Let p≥2p \geq 2 and Lk∈(0,1),k=1,2,…,pL_{k} \in(0,1), k=1,2, \ldots, p. Then,

Np∩MT(L1,…,Lp;A1,…,Ap)≠∅N_{p} \cap \mathcal{M}_{\mathbb{T}}\left(L_{1}, \ldots, L_{p} ; A_{1}, \ldots, A_{p}\right) \neq \emptyset

if and only if there exist numbers c1,…,cp∈Rc_{1}, \ldots, c_{p} \in \mathbb{R} and m1,…,mp∈(0,1)m_{1}, \ldots, m_{p} \in(0,1) such that

Ak=[1−ick1+ick+1][1−2mk+1+ick+11−2mk+ick] and Lk=(1−2mk+1)2+ck+121+ck+12A_{k}=\left[\frac{1-i c_{k}}{1+i c_{k+1}}\right]\left[\frac{1-2 m_{k+1}+i c_{k+1}}{1-2 m_{k}+i c_{k}}\right] \quad \text { and } \quad L_{k}=\sqrt{\frac{\left(1-2 m_{k+1}\right)^{2}+c_{k+1}^{2}}{1+c_{k+1}^{2}}}

for k=1,2,…,pk=1,2, \ldots, p, with cp+1:=c1c_{p+1}:=c_{1} and mp+1:=m1m_{p+1}:=m_{1}.
Proof. Suppose that there exists a measure μ∈Np∩MT(L1,…,Lp;A1,…,Ap)\mu \in N_{p} \cap \mathcal{M}_{\mathbb{T}}\left(L_{1}, \ldots, L_{p} ; A_{1}, \ldots, A_{p}\right). Let {αn(μ)}n=0∞\left\{\alpha_{n}^{(\mu)}\right\}_{n=0}^{\infty} be the associated sequence of Verblunsky coefficients and {{cn(μ)}n=1∞,{mn(μ)}n=1∞}\left\{\left\{c_{n}^{(\mu)}\right\}_{n=1}^{\infty},\left\{m_{n}^{(\mu)}\right\}_{n=1}^{\infty}\right\} the associated pair of real sequences. Since μ∈Np\mu \in N_{p} we have

cnp+k(μ)=ck(μ) and mnp+k(μ)=mk(μ),n≥0,k=1,2,…,pc_{n p+k}^{(\mu)}=c_{k}^{(\mu)} \quad \text { and } \quad m_{n p+k}^{(\mu)}=m_{k}^{(\mu)}, \quad n \geq 0, \quad k=1,2, \ldots, p

On the other hand, since μ∈MT(L1,…,Lp;A1,…,Ap)\mu \in \mathcal{M}_{\mathbb{T}}\left(L_{1}, \ldots, L_{p} ; A_{1}, \ldots, A_{p}\right), from (5.1) it follows that

lim⁡n→∞∣αnp+k(μ)∣=Lk and lim⁡n→∞αˉnp+k(μ)αˉnp+k−1(μ)=Ak,k=1,2,…,p\lim _{n \rightarrow \infty}\left|\alpha_{n p+k}^{(\mu)}\right|=L_{k} \quad \text { and } \quad \lim _{n \rightarrow \infty} \frac{\bar{\alpha}_{n p+k}^{(\mu)}}{\bar{\alpha}_{n p+k-1}^{(\mu)}}=A_{k}, \quad k=1,2, \ldots, p

Hence, from (2.1), (5.7) and (5.8) we conclude that

Ak=[1−ick(μ)1+ick+1(μ)][1−2mk+1(μ)+ick+1(μ)1−2mk(μ)+ick(μ)] and Lk=[1−2mk+1(μ)]2+[ck+1(μ)]21+[ck+1(μ)]2A_{k}=\left[\frac{1-i c_{k}^{(\mu)}}{1+i c_{k+1}^{(\mu)}}\right]\left[\frac{1-2 m_{k+1}^{(\mu)}+i c_{k+1}^{(\mu)}}{1-2 m_{k}^{(\mu)}+i c_{k}^{(\mu)}}\right] \quad \text { and } \quad L_{k}=\sqrt{\frac{\left[1-2 m_{k+1}^{(\mu)}\right]^{2}+\left[c_{k+1}^{(\mu)}\right]^{2}}{1+\left[c_{k+1}^{(\mu)}\right]^{2}}}

k=1,2,…,pk=1,2, \ldots, p.
Thus, the result follows by setting ck=ck(μ)c_{k}=c_{k}^{(\mu)} and mk=mk(μ),k=1,2,…,pm_{k}=m_{k}^{(\mu)}, k=1,2, \ldots, p.
Reciprocally, if there exist numbers c1,…,cp∈Rc_{1}, \ldots, c_{p} \in \mathbb{R} and m1,…,mp∈(0,1)m_{1}, \ldots, m_{p} \in(0,1) such that (5.6) holds, from Theorem 2.1 we can construct a measure μ∈Np\mu \in N_{p} by setting

cnp+k(μ)=ck and mnp+k(μ)=mk,n≥0,k=1,2,…,pc_{n p+k}^{(\mu)}=c_{k} \quad \text { and } \quad m_{n p+k}^{(\mu)}=m_{k}, \quad n \geq 0, \quad k=1,2, \ldots, p

Clearly, the constructed measure μ\mu also belongs to MT(L1,…,Lp;A1,…,Ap)\mathcal{M}_{\mathbb{T}}\left(L_{1}, \ldots, L_{p} ; A_{1}, \ldots, A_{p}\right).
Theorem 5.3. Let μ∈P(T)\mu \in \mathcal{P}(\mathbb{T}) be a probability measure and {{cn(μ)}n=1∞,{mn(μ)}n=1∞}\left\{\left\{c_{n}^{(\mu)}\right\}_{n=1}^{\infty},\left\{m_{n}^{(\mu)}\right\}_{n=1}^{\infty}\right\} its associated pair of real sequences. Suppose that

lim⁡n→∞cnp+k(μ)=ck,lim⁡n→∞mnp+k(μ)=mk,k=1,2,…,p\lim _{n \rightarrow \infty} c_{n p+k}^{(\mu)}=c_{k}, \quad \lim _{n \rightarrow \infty} m_{n p+k}^{(\mu)}=m_{k}, \quad k=1,2, \ldots, p

with p≥2,ck∈R,mk∈(0,1)p \geq 2, c_{k} \in \mathbb{R}, m_{k} \in(0,1) and (1−2mk)2+ck2>0\left(1-2 m_{k}\right)^{2}+c_{k}^{2}>0. Moreover, let μ^∈Np\hat{\mu} \in N_{p} be a probability measure with associated pair {{cn(μ^)}n=1∞,{mn(μ^)}n=1∞}\left\{\left\{c_{n}^{(\hat{\mu})}\right\}_{n=1}^{\infty},\left\{m_{n}^{(\hat{\mu})}\right\}_{n=1}^{\infty}\right\} satisfying

cnp+k(μ^)=ck and mnp+k(μ^)=mk,n≥0,k=1,2,…,pc_{n p+k}^{(\hat{\mu})}=c_{k} \quad \text { and } \quad m_{n p+k}^{(\hat{\mu})}=m_{k}, \quad n \geq 0, \quad k=1,2, \ldots, p

Then, the measures μ\mu and μ^\hat{\mu} have the same essential support.
Proof. Firstly, from (2.1) and (5.9) we clearly have that μ∈MT(L1,…,Lp;A1,…,Ap)\mu \in \mathcal{M}_{\mathbb{T}}\left(L_{1}, \ldots, L_{p} ; A_{1}, \ldots, A_{p}\right), where

Ak=[1−ick1+ick+1][1−2mk+1+ick+11−2mk+ick] and Lk=(1−2mk+1)2+ck+121+ck+12,k=1,2,…,pA_{k}=\left[\frac{1-i c_{k}}{1+i c_{k+1}}\right]\left[\frac{1-2 m_{k+1}+i c_{k+1}}{1-2 m_{k}+i c_{k}}\right] \quad \text { and } \quad L_{k}=\sqrt{\frac{\left(1-2 m_{k+1}\right)^{2}+c_{k+1}^{2}}{1+c_{k+1}^{2}}}, \quad k=1,2, \ldots, p

with cp+1=c1c_{p+1}=c_{1} and mp+1=m1m_{p+1}=m_{1}.
Hence, by Lemma 5.2 it follows that Np∩MT(L1,…,Lp;A1,…,Ap)≠∅N_{p} \cap \mathcal{M}_{\mathbb{T}}\left(L_{1}, \ldots, L_{p} ; A_{1}, \ldots, A_{p}\right) \neq \emptyset. Moreover, from (5.10) we also have that μ^∈MT(L1,…,Lp;A1,…,Ap)\hat{\mu} \in \mathcal{M}_{\mathbb{T}}\left(L_{1}, \ldots, L_{p} ; A_{1}, \ldots, A_{p}\right).

Since every measure in MT(L1,…,Lp;A1,…,Ap)\mathcal{M}_{\mathbb{T}}\left(L_{1}, \ldots, L_{p} ; A_{1}, \ldots, A_{p}\right) has the same essential support, we conclude that σess(μ)=σess(μ^)\sigma_{e s s}(\mu)=\sigma_{e s s}(\hat{\mu}).

6. Examples

Using the following examples we discuss the results obtained in the previous sections.

Example 1. We consider the measure μ~\tilde{\mu} associated with the OPUC {ϕn(α)}n=0∞\left\{\phi_{n}^{(\alpha)}\right\}_{n=0}^{\infty} known as Geronimus polynomials (see [12-14] and [18, p. 83]). The respective Verblunsky coefficients are

αn(μ~)=−ϕn+1(α)‾(0)=α,n≥0\alpha_{n}^{(\tilde{\mu})}=-\overline{\phi_{n+1}^{(\alpha)}}(0)=\alpha, \quad n \geq 0

From [12] it follows that μ~\tilde{\mu}, as a probability measure, is such that

∫02πℓ(eiθ)dμ~(eiθ)=∫θ∣α∣2π−θ∣α∣ℓ(eiθ)cos⁡2(θ∣α∣/2)−cos⁡2(θ/2)2π∣1+α∣sin⁡((θ−ϑα)/2)dθ\int_{0}^{2 \pi} \ell\left(e^{i \theta}\right) d \tilde{\mu}\left(e^{i \theta}\right)=\int_{\theta_{|\alpha|}}^{2 \pi-\theta_{|\alpha|}} \ell\left(e^{i \theta}\right) \frac{\sqrt{\cos ^{2}\left(\theta_{|\alpha|} / 2\right)-\cos ^{2}(\theta / 2)}}{2 \pi|1+\alpha| \sin \left(\left(\theta-\vartheta_{\alpha}\right) / 2\right)} d \theta

when Re(α)+∣α∣2≤0\mathcal{R} e(\alpha)+|\alpha|^{2} \leq 0 and

∫02πℓ(eiθ)dμ~(eiθ)=∫θ∣α∣2π−θ∣α∣ℓ(eiθ)cos⁡2(θ∣α∣/2)−cos⁡2(θ/2)2π∣1+α∣sin⁡((θ−ϑα)/2)dθ+δαℓ(eiϑα)\int_{0}^{2 \pi} \ell\left(e^{i \theta}\right) d \tilde{\mu}\left(e^{i \theta}\right)=\int_{\theta_{|\alpha|}}^{2 \pi-\theta_{|\alpha|}} \ell\left(e^{i \theta}\right) \frac{\sqrt{\cos ^{2}\left(\theta_{|\alpha|} / 2\right)-\cos ^{2}(\theta / 2)}}{2 \pi|1+\alpha| \sin \left(\left(\theta-\vartheta_{\alpha}\right) / 2\right)} d \theta+\delta_{\alpha} \ell\left(e^{i \vartheta_{\alpha}}\right)

when Re(α)+∣α∣2>0\mathcal{R} e(\alpha)+|\alpha|^{2}>0. Here, ℓ\ell is any Laurent polynomial and the values of ϑα,θ∣α∣\vartheta_{\alpha}, \theta_{|\alpha|} and δα\delta_{\alpha} are given by

eiϑα=wα=1+αˉ1+α,θ∣α∣=2arcsin⁡(∣α∣) and δα=2(Re(α)+∣α∣2)∣1+α∣2e^{i \vartheta_{\alpha}}=w_{\alpha}=\frac{1+\bar{\alpha}}{1+\alpha}, \quad \theta_{|\alpha|}=2 \arcsin (|\alpha|) \quad \text { and } \quad \delta_{\alpha}=\frac{2\left(\mathcal{R} e(\alpha)+|\alpha|^{2}\right)}{|1+\alpha|^{2}}

Assuming ϑα\vartheta_{\alpha} to be such that −π<ϑα<π-\pi<\vartheta_{\alpha}<\pi, let us consider the rotated measure μ~(wαz)\tilde{\mu}\left(w_{\alpha} z\right) which we simply denote by μ\mu. That is,

μ(z)=μ~(wαz)\mu(z)=\tilde{\mu}\left(w_{\alpha} z\right)

Notice that μ~∈V1\tilde{\mu} \in V_{1}. Furthermore (see [8])

cn(μ~)=−4Im(α)∣α∣(1−∣α∣2)n−1(∣α∣−Re(α))(1+∣α∣)2n−1+(∣α∣+Re(α))(1−∣α∣)2n−1,n≥1c_{n}^{(\tilde{\mu})}=\frac{-4 \mathcal{I} m(\alpha)|\alpha|\left(1-|\alpha|^{2}\right)^{n-1}}{(|\alpha|-\mathcal{R} e(\alpha))(1+|\alpha|)^{2 n-1}+(|\alpha|+\mathcal{R} e(\alpha))(1-|\alpha|)^{2 n-1}}, \quad n \geq 1

and

mn(μ~)=12(∣α∣−Re(α))(1+∣α∣)2n+(∣α∣+Re(α))(1−∣α∣)2n(∣α∣−Re(α))(1+∣α∣)2n−1+(∣α∣+Re(α))(1−∣α∣)2n−1,n≥1m_{n}^{(\tilde{\mu})}=\frac{1}{2} \frac{(|\alpha|-\mathcal{R} e(\alpha))(1+|\alpha|)^{2 n}+(|\alpha|+\mathcal{R} e(\alpha))(1-|\alpha|)^{2 n}}{\left(|\alpha|-\mathcal{R} e(\alpha)\right)(1+|\alpha|)^{2 n-1}+(|\alpha|+\mathcal{R} e(\alpha))(1-|\alpha|)^{2 n-1}}, \quad n \geq 1

Hence, (6.1) and (6.2) show us that μ~∈V1\N1\tilde{\mu} \in V_{1} \backslash N_{1} whenever α∉R\alpha \notin \mathbb{R}.
To find the values of {cn(μ)}n=1∞\left\{c_{n}^{(\mu)}\right\}_{n=1}^{\infty} and {mn(μ)}n=1∞\left\{m_{n}^{(\mu)}\right\}_{n=1}^{\infty} we use Theorem 2.1. Observe that αn(μ)=wαn+1α,n≥0\alpha_{n}^{(\mu)}=w_{\alpha}^{n+1} \alpha, n \geq 0. Thus, from Theorem 2.1, we easily verify that ρn(μ)=wα−n,n≥0\rho_{n}^{(\mu)}=w_{\alpha}^{-n}, n \geq 0. From this,

cn(μ)=−Im(α)1+Re(α) and mn(μ)=121−∣α∣2[1+Re(α)],n≥1c_{n}^{(\mu)}=\frac{-\mathcal{I} m(\alpha)}{1+\mathcal{R} e(\alpha)} \quad \text { and } \quad m_{n}^{(\mu)}=\frac{1}{2} \frac{1-|\alpha|^{2}}{[1+\mathcal{R} e(\alpha)]}, \quad n \geq 1

Therefore, by (6.3) we conclude that μ∈N1\mu \in N_{1}. This is what it is expected by Lemma 3.6. Moreover, if α∉R\alpha \notin \mathbb{R} we have that μ∈N1\V1\mu \in N_{1} \backslash V_{1}.

From Theorem 4.4, w=1w=1 is the only possible pure point of μ\mu. Moreover, from Theorem 3.8, w=1w=1 is a pure point of μ\mu if and only if ∣1−α0(μ)∣2<1−∣α0(μ)∣2\left|1-\alpha_{0}^{(\mu)}\right|^{2}<1-\left|\alpha_{0}^{(\mu)}\right|^{2}. This last condition is clearly equivalent to Re(α)+∣α∣2>0\mathcal{R} e(\alpha)+|\alpha|^{2}>0, as expected.

Again from Theorem 3.8, the size of the mass at w=1w=1 is μ({1})=γ/(γ+ζ)\mu(\{1\})=\gamma /(\gamma+\zeta), with ζ=∣1−α0(μ)∣2/[1−\zeta=\left|1-\alpha_{0}^{(\mu)}\right|^{2} /[1- ∣α0(μ)∣2]\left.\left|\alpha_{0}^{(\mu)}\right|^{2}\right] and γ=1−ζ\gamma=1-\zeta.

Hence, by observing that

ζ=1−∣α∣21+2Re(α)+∣α∣2\zeta=\frac{1-|\alpha|^{2}}{1+2 \mathcal{R} e(\alpha)+|\alpha|^{2}}

we obtain

μ({1})=2(Re(α)+∣α∣2)∣1+α∣2=δα\mu(\{1\})=\frac{2\left(\mathcal{R} e(\alpha)+|\alpha|^{2}\right)}{|1+\alpha|^{2}}=\delta_{\alpha}

as it should be, since μ(z)=μ~(wαz)\mu(z)=\tilde{\mu}\left(w_{\alpha} z\right).
Notice also that, from (6.1) and (6.2),

lim⁡n→∞cn(μ~)=0 and lim⁡n→∞mn(μ~)=12(1+∣α∣)\lim _{n \rightarrow \infty} c_{n}^{(\tilde{\mu})}=0 \quad \text { and } \quad \lim _{n \rightarrow \infty} m_{n}^{(\tilde{\mu})}=\frac{1}{2}(1+|\alpha|)

Thus, from part ii) of Theorem 5.1, we obtain

σess(μ~)={z∈T:θ0≤arg⁡(z)≤2π−θ0}\sigma_{e s s}(\tilde{\mu})=\left\{z \in \mathbb{T}: \theta_{0} \leq \arg (z) \leq 2 \pi-\theta_{0}\right\}

where θ0=2arcsin⁡(∣α∣)\theta_{0}=2 \arcsin (|\alpha|), according with the known results for Geronimus polynomials and associated measures.

Example 2. Let μ(z;b1,b2,c)\mu\left(z ; b_{1}, b_{2}, c\right) be the probability measure associated with the pair of real sequences {cn(μ)}n=1∞\left\{c_{n}^{(\mu)}\right\}_{n=1}^{\infty} and {dn(μ)}n=1∞\left\{d_{n}^{(\mu)}\right\}_{n=1}^{\infty} given by

cn(μ)=c and dn(μ)=(1−mn−1(μ))mn(μ),n≥1c_{n}^{(\mu)}=c \quad \text { and } \quad d_{n}^{(\mu)}=\left(1-m_{n-1}^{(\mu)}\right) m_{n}^{(\mu)}, \quad n \geq 1

where c∈Rc \in \mathbb{R} and the real sequence {mn(μ)}n=0∞\left\{m_{n}^{(\mu)}\right\}_{n=0}^{\infty} is such that m0(μ)=0m_{0}^{(\mu)}=0,

m2n−1(μ)=1−b12 and m2n(μ)=1−b22,n≥1m_{2 n-1}^{(\mu)}=\frac{1-b_{1}}{2} \quad \text { and } \quad m_{2 n}^{(\mu)}=\frac{1-b_{2}}{2}, \quad n \geq 1

with b1,b2∈R,∣b1∣<1b_{1}, b_{2} \in \mathbb{R},\left|b_{1}\right|<1 and ∣b2∣<1\left|b_{2}\right|<1.
If c≠0c \neq 0, one can observe that μ∈N2\V2\mu \in N_{2} \backslash V_{2}. Furthermore, αn+2(μ)=λαn(μ),n≥0\alpha_{n+2}^{(\mu)}=\lambda \alpha_{n}^{(\mu)}, n \geq 0, with λˉ=(1−ic1+ic)2=ρ2(μ)\bar{\lambda}=\left(\frac{1-i c}{1+i c}\right)^{2}=\rho_{2}^{(\mu)}, according to Theorem 3.2.

The solutions for the equation w2=λw^{2}=\lambda are

wλ,1=(1+ic1−ic) and wλ,2=−(1+ic1−ic)w_{\lambda, 1}=\left(\frac{1+i c}{1-i c}\right) \quad \text { and } \quad w_{\lambda, 2}=-\left(\frac{1+i c}{1-i c}\right)

By Lemma 3.5, there exist exactly two measures in V2V_{2} which are equivalent by rotation to μ\mu.
The first one is the measure μ~(z;b1,b2,c)=μ(wˉλ,1z;b1,b2,c)\tilde{\mu}\left(z ; b_{1}, b_{2}, c\right)=\mu\left(\bar{w}_{\lambda, 1} z ; b_{1}, b_{2}, c\right). It is not difficult to see that the Verblunsky coefficients associated with μ~\tilde{\mu} satisfy

α2n(μ~)=b1−ic1+ic and α2n+1(μ~)=b2−ic1+ic,n≥0\alpha_{2 n}^{(\tilde{\mu})}=\frac{b_{1}-i c}{1+i c} \quad \text { and } \quad \alpha_{2 n+1}^{(\tilde{\mu})}=\frac{b_{2}-i c}{1+i c}, \quad n \geq 0

which confirm that μ~∈V2\tilde{\mu} \in V_{2}.
The second one is the measure μ~(z;b1,b2,c)=μ(wˉλ,2z;b1,b2,c)\tilde{\mu}\left(z ; b_{1}, b_{2}, c\right)=\mu\left(\bar{w}_{\lambda, 2} z ; b_{1}, b_{2}, c\right). For this measure it is known (see [4]) that

cn(μ~)=(−1)nc,m2n(μ~)=m2n(μ) and m2n−1(μ~)=1−m2n−1(μ),n≥1c_{n}^{(\tilde{\mu})}=(-1)^{n} c, \quad m_{2 n}^{(\tilde{\mu})}=m_{2 n}^{(\mu)} \quad \text { and } \quad m_{2 n-1}^{(\tilde{\mu})}=1-m_{2 n-1}^{(\mu)}, \quad n \geq 1

Hence, from (2.1) we conclude that μ^∈V2\hat{\mu} \in V_{2} and the sequence of corresponding Verblunsky coefficients is such that

α2n(μ^)=−b1+ic1+ic and α2n+1(μ^)=b2−ic1+ic,n≥0\alpha_{2 n}^{(\hat{\mu})}=\frac{-b_{1}+i c}{1+i c} \quad \text { and } \quad \alpha_{2 n+1}^{(\hat{\mu})}=\frac{b_{2}-i c}{1+i c}, \quad n \geq 0

Furthermore, from Theorem 3.4, we have

V2∩N2={μ~(z;b1,b2,c):c∈R and b1,b2∈(−1,1)}V_{2} \cap N_{2}=\left\{\tilde{\mu}\left(z ; b_{1}, b_{2}, c\right): c \in \mathbb{R} \text { and } b_{1}, b_{2} \in(-1,1)\right\}

See [4] for the complete characterization of the space V2∩N2V_{2} \cap N_{2}.
Using the fact that μ~(z;b1,b2,c)=μ(wˉλ,2z;b1,b2,c)\tilde{\mu}\left(z ; b_{1}, b_{2}, c\right)=\mu\left(\bar{w}_{\lambda, 2} z ; b_{1}, b_{2}, c\right), we can derive the weight function w(θ)w(\theta) associated with the measure μ\mu.

If we denote the weight function of μ^\hat{\mu} by w^(θ)\hat{w}(\theta), then it is known (see [4]) that

w^(θ)=(1−b12)(1−b22)−[(1+c2)cos⁡θ−b1b2−c2]2∣(1+b2)∣sin⁡θ+c(1−cos⁡θ)∥\hat{w}(\theta)=\frac{\sqrt{\left(1-b_{1}^{2}\right)\left(1-b_{2}^{2}\right)-\left[\left(1+c^{2}\right) \cos \theta-b_{1} b_{2}-c^{2}\right]^{2}}}{\left|\left(1+b_{2}\right)\right| \sin \theta+c(1-\cos \theta) \|}

Moreover, the bands B^1\hat{B}_{1} and B^2\hat{B}_{2} for the measure μ~\tilde{\mu} are determined by the points hatzj+=eihatthetaj+andandhatzj−=\hat{z}_{j}^{+}=e^{i \hat{\theta}_{j}^{+}}andand \hat{z}_{j}^{-}=hatzj+=eihatthetaj+andandhatzj−= eiθ^j−,j∈{1,2}e^{i \hat{\theta}_{j}^{-}}, j \in\{1,2\}, with

θ^1+=arccos⁡((1−b12)1/2(1−b22)1/2+c2+b1b21+c2) and θ^2+=2π−θ^1+,θ^1−=arccos⁡(c2−(1−b12)1/2(1−b22)1/2+b1b21+c2) and θ^2−=2π−θ^1−.\begin{aligned} & \hat{\theta}_{1}^{+}=\arccos \left(\frac{\left(1-b_{1}^{2}\right)^{1 / 2}\left(1-b_{2}^{2}\right)^{1 / 2}+c^{2}+b_{1} b_{2}}{1+c^{2}}\right) \quad \text { and } \quad \hat{\theta}_{2}^{+}=2 \pi-\hat{\theta}_{1}^{+}, \\ & \hat{\theta}_{1}^{-}=\arccos \left(\frac{c^{2}-\left(1-b_{1}^{2}\right)^{1 / 2}\left(1-b_{2}^{2}\right)^{1 / 2}+b_{1} b_{2}}{1+c^{2}}\right) \quad \text { and } \quad \hat{\theta}_{2}^{-}=2 \pi-\hat{\theta}_{1}^{-} . \end{aligned}

Please cite this article in press as: C.F. Bracciali et al., Verblunsky coefficients related with periodic real sequences and associated measures on the unit circle, J. Math. Anal. Appl. (2016), http://dx.doi.org/10.1016/j.jmaa.2016.08.009

Thus, if η=arg⁡(wλ,2)\eta=\arg \left(w_{\lambda, 2}\right), we immediately conclude that

w(θ)=w^(θ+η)=(1−b12)(1−b22)−[(1+c2)cos⁡(θ+η)−b1b2−c2]2∣(1+b2){sin⁡(θ+η)+c[1−cos⁡(θ+η)]}∣w(\theta)=\hat{w}(\theta+\eta)=\frac{\sqrt{\left(1-b_{1}^{2}\right)\left(1-b_{2}^{2}\right)-\left[\left(1+c^{2}\right) \cos (\theta+\eta)-b_{1} b_{2}-c^{2}\right]^{2}}}{\left|\left(1+b_{2}\right)\{\sin (\theta+\eta)+c[1-\cos (\theta+\eta)]\}\right|}

The associated bands B1\mathcal{B}_{1} and B2\mathcal{B}_{2} (as in Corollary 1.2.1) for the measure μ\mu are determined by the points zj+=eithetaj+z_{j}^{+}=e^{i \theta_{j}^{+}}zj+=eithetaj+and zj−=eiθj−,j∈{1,2}z_{j}^{-}=e^{i \theta_{j}^{-}}, j \in\{1,2\}, with

θ1+=θ^1+−η,θ2+=2π−θ1+,θ1−=θ^1−−η and θ2−=2π−θ1−\theta_{1}^{+}=\hat{\theta}_{1}^{+}-\eta, \quad \theta_{2}^{+}=2 \pi-\theta_{1}^{+}, \quad \theta_{1}^{-}=\hat{\theta}_{1}^{-}-\eta \quad \text { and } \quad \theta_{2}^{-}=2 \pi-\theta_{1}^{-}

From Theorem 4.4 (see also Corollary 4.5.1) the possible pure points of μ\mu are w1=1w_{1}=1 and w2=wˉλ,2w_{2}=\bar{w}_{\lambda, 2}. It is not difficult to see that

τn+2(μ)(w1)=λˉτn(μ)(w1) and τn+2(μ)(w2)=λˉτn(μ)(w2),n≥0\tau_{n+2}^{(\mu)}\left(w_{1}\right)=\bar{\lambda} \tau_{n}^{(\mu)}\left(w_{1}\right) \quad \text { and } \quad \tau_{n+2}^{(\mu)}\left(w_{2}\right)=\bar{\lambda} \tau_{n}^{(\mu)}\left(w_{2}\right), \quad n \geq 0

according with Theorem 3.7 .
Furthermore, from Theorem 3.8 one can see that w1w_{1} is a pure point of μ\mu if and only if b1+b2>0b_{1}+b_{2}>0. Moreover, if

ζ1=∑n=12∏j=1n∣1−ρj−1(μ)αj−1(μ)∣21−∣αj−1(μ)∣2 and γ1=1−∏j=12∣1−ρj−1(μ)αj−1(μ)∣21−∣αj−1(μ)∣2\zeta_{1}=\sum_{n=1}^{2} \prod_{j=1}^{n} \frac{\left|1-\rho_{j-1}^{(\mu)}\alpha_{j-1}^{(\mu)}\right|^{2}}{1-\left|\alpha_{j-1}^{(\mu)}\right|^{2}} \quad \text { and } \quad \gamma_{1}=1-\prod_{j=1}^{2} \frac{\left|1-\rho_{j-1}^{(\mu)}\alpha_{j-1}^{(\mu)}\right|^{2}}{1-\left|\alpha_{j-1}^{(\mu)}\right|^{2}}

again by Theorem 3.8 we obtain

μ({w1})=γ1γ1+ζ1=b1+b21+b2\mu\left(\left\{w_{1}\right\}\right)=\frac{\gamma_{1}}{\gamma_{1}+\zeta_{1}}=\frac{b_{1}+b_{2}}{1+b_{2}}

Similarly, w2w_{2} is a pure point of μ\mu if and only if b2−b1>0b_{2}-b_{1}>0. Moreover, if

ζ2=∑n=12∏j=1n∣1−w2τj−1(μ)(w2)αj−1(μ)∣21−∣αj−1(μ)∣2 and γ2=1−∏j=12∣1−w2τj−1(μ)(w2)αj−1(μ)∣21−∣αj−1(μ)∣2\zeta_{2}=\sum_{n=1}^{2} \prod_{j=1}^{n} \frac{\left|1-w_{2} \tau_{j-1}^{(\mu)}\left(w_{2}\right) \alpha_{j-1}^{(\mu)}\right|^{2}}{1-\left|\alpha_{j-1}^{(\mu)}\right|^{2}} \quad \text { and } \quad \gamma_{2}=1-\prod_{j=1}^{2} \frac{\left|1-w_{2} \tau_{j-1}^{(\mu)}\left(w_{2}\right) \alpha_{j-1}^{(\mu)}\right|^{2}}{1-\left|\alpha_{j-1}^{(\mu)}\right|^{2}}

we obtain

μ({w2})=γ2γ2+ζ2=b2−b11+b2\mu\left(\left\{w_{2}\right\}\right)=\frac{\gamma_{2}}{\gamma_{2}+\zeta_{2}}=\frac{b_{2}-b_{1}}{1+b_{2}}

These results on the pure points of μ\mu are expected and could be obtained from the results established in [4] since μ(z;b1,b2,c)=μ^(wλ,2z;b1,b2,c)\mu\left(z ; b_{1}, b_{2}, c\right)=\hat{\mu}\left(w_{\lambda, 2} z ; b_{1}, b_{2}, c\right).

Example 3. Let ψ(κ),0≤κ<1\psi^{(\kappa)}, 0 \leq \kappa<1, be the nontrivial probability measures given by

∫Cℓ(z)dψ(κ)(z)=(1−κ)∫Cℓ(z)12πizdz+κℓ(i)\int_{\mathbb{C}} \ell(z) d \psi^{(\kappa)}(z)=(1-\kappa) \int_{\mathbb{C}} \ell(z) \frac{1}{2 \pi i z} d z+\kappa \ell(i)

for any Laurent polynomial ℓ\ell.
Associated with each of the measures ψ(κ),0≤κ<1\psi^{(\kappa)}, 0 \leq \kappa<1, there exists a unique pair of real sequences {{cn(ψ(κ))}n=1∞,{mn(ψ(κ))}n=1∞}\left\{\left\{c_{n}^{\left(\psi^{(\kappa)}\right)}\right\}_{n=1}^{\infty},\left\{m_{n}^{\left(\psi^{(\kappa)}\right)}\right\}_{n=1}^{\infty}\right\}, with

c4s+1(ψ(κ))=κ4sκ+1,m4s+1(ψ(κ))=12(4sκ+1)2+κ2(4sκ+1)2c4s+2(ψ(κ))=−2κ2[(4s+1)κ+1]2,m4s+2(ψ(κ))=12(4sκ+1)[[(4s+2)κ+1]2+κ2][(4s+1)κ+1]3c4s+3(ψ(κ))=−κ(4s+2)κ+1,m4s+3(ψ(κ))=12[(4s+2)κ+1]2−κ2[(4s+2)κ+1]2c4s+4(ψ(κ))=0,m4s+4(ψ(κ))=12(4s+2)κ+1(4s+3)κ+1\begin{array}{ll} c_{4 s+1}^{\left(\psi^{(\kappa)}\right)}=\frac{\kappa}{4 s \kappa+1}, & m_{4 s+1}^{\left(\psi^{(\kappa)}\right)}=\frac{1}{2} \frac{(4 s \kappa+1)^{2}+\kappa^{2}}{(4 s \kappa+1)^{2}} \\ c_{4 s+2}^{\left(\psi^{(\kappa)}\right)}=\frac{-2 \kappa^{2}}{[(4 s+1) \kappa+1]^{2}}, & m_{4 s+2}^{\left(\psi^{(\kappa)}\right)}=\frac{1}{2} \frac{(4 s \kappa+1)\left[[(4 s+2) \kappa+1]^{2}+\kappa^{2}\right]}{[(4 s+1) \kappa+1]^{3}} \\ c_{4 s+3}^{\left(\psi^{(\kappa)}\right)}=\frac{-\kappa}{(4 s+2) \kappa+1}, & m_{4 s+3}^{\left(\psi^{(\kappa)}\right)}=\frac{1}{2} \frac{[(4 s+2) \kappa+1]^{2}-\kappa^{2}}{[(4 s+2) \kappa+1]^{2}} \\ c_{4 s+4}^{\left(\psi^{(\kappa)}\right)}=0, & m_{4 s+4}^{\left(\psi^{(\kappa)}\right)}=\frac{1}{2} \frac{(4 s+2) \kappa+1}{(4 s+3) \kappa+1} \end{array}

for s≥0s \geq 0 (see [3]).
Clearly, the sequences {cn(ψ(κ))}n=1∞\left\{c_{n}^{\left(\psi^{(\kappa)}\right)}\right\}_{n=1}^{\infty} and {mn(ψ(κ))}n=1∞\left\{m_{n}^{\left(\psi^{(\kappa)}\right)}\right\}_{n=1}^{\infty} are such that

lim⁡n→∞cn(ψ(κ))=0 and lim⁡n→∞mn(ψ(κ))=12\lim _{n \rightarrow \infty} c_{n}^{\left(\psi^{(\kappa)}\right)}=0 \quad \text { and } \quad \lim _{n \rightarrow \infty} m_{n}^{\left(\psi^{(\kappa)}\right)}=\frac{1}{2}

Therefore, by part i) of Theorem 5.1 we have that σess(ψ(κ))=T\sigma_{e s s}\left(\psi^{(\kappa)}\right)=\mathbb{T}, as expected, since the measures ψ(κ)\psi^{(\kappa)}, 0≤κ<10 \leq \kappa<1, are simple modifications of the Lebesgue measure dν(z)=(2πiz)−1dzd \nu(z)=(2 \pi i z)^{-1} d z, which include mass point at z=iz=i.

Example 4. In order to give an application of Theorem 5.3 we consider, in this example, the family of probability measures μ(δ;z),0≤δ<1\mu(\delta ; z), 0 \leq \delta<1, with associated pair {{cn(δ)}n=1∞,{dn(δ)}n=1∞}\left\{\left\{c_{n}(\delta)\right\}_{n=1}^{\infty},\left\{d_{n}(\delta)\right\}_{n=1}^{\infty}\right\} satisfying

cn(δ)=(−1)nc and mn(δ)=121+(n−2)δ1+(n−1)δ,n≥1c_{n}(\delta)=(-1)^{n} c \quad \text { and } \quad m_{n}(\delta)=\frac{1}{2} \frac{1+(n-2) \delta}{1+(n-1) \delta}, \quad n \geq 1

where c∈Rc \in \mathbb{R} and {mn(δ)}n=0∞\left\{m_{n}(\delta)\right\}_{n=0}^{\infty} is the minimal parameter sequence for {dn(δ)}n=1∞\left\{d_{n}(\delta)\right\}_{n=1}^{\infty}.
Notice that, from (6.5), we have

lim⁡n→∞c2n+k(δ)=(−1)kc and lim⁡n→∞m2n+k(δ)=12,k=1,2\lim _{n \rightarrow \infty} c_{2 n+k}(\delta)=(-1)^{k} c \quad \text { and } \quad \lim _{n \rightarrow \infty} m_{2 n+k}(\delta)=\frac{1}{2}, \quad k=1,2

Hence, for c≠0c \neq 0, we can consider the measure μ^(z;0,0,c)\hat{\mu}(z ; 0,0, c) as in Example 2 and apply Theorem 5.3 to conclude, from (6.6), that σess(μ(δ;.))=σess(μ^)\sigma_{e s s}(\mu(\delta ;.))=\sigma_{e s s}(\hat{\mu}).

On the other hand, from [4], it is known that σess(μ^)=C1∪C2\sigma_{e s s}(\hat{\mu})=\mathcal{C}_{1} \cup \mathcal{C}_{2}, where

C1={z=eiθ:0≤θ≤arccos⁡(c2−1c2+1)} and C2={z=eiθ:2π−arccos⁡(c2−1c2+1)≤θ≤2π}\mathcal{C}_{1}=\left\{z=e^{i \theta}: 0 \leq \theta \leq \arccos \left(\frac{c^{2}-1}{c^{2}+1}\right)\right\} \quad \text { and } \quad \mathcal{C}_{2}=\left\{z=e^{i \theta}: 2 \pi-\arccos \left(\frac{c^{2}-1}{c^{2}+1}\right) \leq \theta \leq 2 \pi\right\}

Thus, for 0≤δ<10 \leq \delta<1 and c≠0c \neq 0, we obtain σess(μ(δ;.))=C1∪C2\sigma_{e s s}(\mu(\delta ;.))=\mathcal{C}_{1} \cup \mathcal{C}_{2}. Furthermore, if c=0c=0, by applying part i) of Theorem 5.1 we clearly have that σess(μ(δ;.))=T\sigma_{e s s}(\mu(\delta ;.))=\mathbb{T}.

The results obtained can be confirmed by observing that

∫Tℓ(z)dμ(δ;z)=(1−δ)∫Tℓ(z)dμ^(z;0,0,c)+δℓ(1)\int_{\mathbb{T}} \ell(z) d \mu(\delta ; z)=(1-\delta) \int_{\mathbb{T}} \ell(z) d \hat{\mu}(z ; 0,0, c)+\delta \ell(1)

for any Laurent polynomial ℓ\ell.

Acknowledgments

The authors would like to thank the referees for their valuable and constructive comments which helped to improve the manuscript.

The authors deeply thank Professor A. Martínez-Finkelshtein for his many valuable advices with respect to this work.

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