A CMV connection between orthogonal polynomials on the unit circle and the real line

M. Derevyagin, L. Vinet and A. Zhedanov introduced in Constr. Approx. 36 (2012) 513-535 a new connection between orthogonal polynomials on the unit circle and the real line. It maps any real CMV matrix into a Jacobi one depending on a real parameter $\lambda$. These authors prove that this map yields a natural link between the Jacobi polynomials on the unit circle and the little and big $-1$ Jacobi polynomials on the real line. They also provide explicit expressions for the measure and orthogonal polynomials associated with the Jacobi matrix in terms of those related to the CMV matrix, but only for the value $\lambda=1$ which simplifies the connection -basic DVZ connection-. However, similar explicit expressions for an arbitrary value of $\lambda$ -- (general) DVZ connection -- are missing. This is the main problem overcome in this paper. This work introduces a new approach to the DVZ connection which formulates it as a two-dimensional eigenproblem by using known properties of CMV matrices. This allows us to go further, providing explicit relations between the measures and orthogonal polynomials for the general DVZ connection. It turns out that this connection maps a measure on the unit circle into a rational perturbation of an even measure supported on two symmetric intervals of the real line, which reduce to a single interval for the basic DVZ connection, while the perturbation becomes a degree one polynomial. Some instances of the DVZ connection are shown to give new one-parameter families of orthogonal polynomials on the real line.


Introduction
Any sequence p n of orthonormal polynomials with respect to a measure on the real line (OPRL) is characterized by a Jacobi matrix, encoding the corresponding three term recurrence relation, J ppxq " xppxq, p " pp 0 , p 1 , p 2 , . . . q t .
Without loss of generality, we can suppose the measure normalized so that p 0 " 1. When the measure is supported on the unit circle T " tz P C : |z| " 1u, another kind of recurrence relation characterizes the corresponding sequences of orthogonal polynomials (OPUC). In the case of monic OPUC φ n , it has the form [11,12,16,17] φ n`1 pzq " zφ n pzq´α n φnpzq, α n P C, |α n | ă 1, where φnpzq " z n φ n p1{zq is known as the reversed polynomial of φ n and the parameters α n are called Verblunsky coefficients. Introducing the complementary parameters ρ n " a 1´|α n | 2 , the orthonormal OPUC ϕ n with positive leading coefficients become ϕ n pzq " κ n φ n pzq " κ n p´α n´1`¨¨¨`z n q, κ n " pρ 0 ρ 1¨¨¨ρn´1 q´1.
They provide a simple matrix representation of the recurrence relation (see [3,16,18]), which reads as Cχpzq " zχpzq, χ " pχ 0 , χ 1 , χ 2 , . . . q t , in terms of the so called CMV matrix C -the unitary analogue of a Jacobi matrix [14,16]-, a five-diagonal unitary matrix which factorizes as C " ML, with We refer to this as the Θ-factorization of a CMV matrix. Analogously to the case of the real line, we will assume that χ 0 " ϕ 0 " 1 by normalizing the measure. Since this paper revolves around a new connection between OPUC and OPRL, it is worth commenting on known ones. The paradigm of such connections, due to Szegő [11,16,17], starts with a measure µ on T which is symmetric under conjugation, which means that the Verblunsky coefficients are real or, in other words, the related OPUC ϕ n have real coefficients. The measure σ induced on r´2, 2s by the mapping z Þ Ñ x " z`z´1 is called its Szegő projection, which we denote by σ " Szpµq. The related OPRL are given by [16,17] p n pxq " z´n ϕ 2n pzq`ϕ2 n pzq a 2p1´α 2n´1 q , x " z`z´1.
Here and in what follows we use the convention α´1 "´1. A more recent OPUC-OPRL connection goes back to works of Delsarte and Genin in the framework of the split Levinson algorithm [6,7,8]. They realized that p n pxq " pz 1{2 q´n ϕ n pzq`ϕnpzq a 2p1´α n´1 q , x " z 1{2`z´1{2 , constitute an OPRL sequence with respect to an even measure on r´2, 2s. We will refer to this as the DG connection between OPUC and OPRL. Although the relations between the measures and orthogonal polynomials are simple for the above two connections, the situation is somewhat different regarding the CMV and Jacobi matrices: for both connections, the relations expressing the Jacobi parameters in terms of the Verblunsky coefficients are non-trivial to invert [6,16,17]. This changes completely with the new connection (see Proposition 4.1), directly defined by a very simple relation between CMV and Jacobi matrices (see [9] and Section 2). We refer to this by the surname initials of its discoverers, Derevyagin, Vinet and Zhedanov: the DVZ connection.
DVZ has other properties which distinguish it from Szegő and DG. First, for each OPUC instance with real coefficients, instead of a single OPRL sequence, it gives a family of OPRL depending on an arbitrary real parameter λ. Besides, up to λ "˘1, the DVZ measure on the real line is supported on two disjoint symmetric intervals, although it is not even, in contrast to the DG measure on r´2, 2s. The price to pay for these differences is a slightly more involved relation between the OPUC and OPRL linked by DVZ, which includes a less trivial relation between the unit circle and real line variables than for Szegő and DG.
The rest of the paper is devoted to the DVZ connection, which is detailed in the following way: Section 2 describes two especially simple cases of the DVZ connection, corresponding λ "˘1. The aim is to introduce a new eigenproblem translation of DVZ in the easiest possible setting. Section 3 reconciles this new approach with the original one in [9], in which the basic DVZ connection for λ " 1 arises as a composition of DG and a Christoffel transformation on the real line [1,19], i.e. the multiplication of a measure on the real line by a real polynomial. The main results of the paper are in Section 4 where, extending the previous eigenmachinery to the general DVZ connection given by an arbitrary λ P R, we obtain explicit expressions for the corresponding measures and orthogonal polynomials on the real line in terms of the unit circle counterparts. This is shown to provide new examples of one-parameter OPRL families in Section 5.

A new approach to the basic DVZ connection
Derevyagin, Vinet and Zhedanov established in [9] a new connection between OPUC and OPRL, closely related to the DG connection. The starting point for this was the Θ-factorization C " ML of a CMV matrix, given by the tridiagonal factors (4). If the Verblunsky coefficients are real, a Jacobi matrix K can be built up out of the symmetric unitary factors L and M, In [9], Derevyagin, Vinet and Zhedanov identified the measure and OPRL related to K in terms of the measure and OPUC associated with C. Such an identification appeared surprisingly when combining the DG connection with a Christoffel transformation on the real line. This defines what we will call the basic DVZ connection (a more general one will come later on), which maps OPUC with real coefficients into OPRL with respect to a measure supported on r´2, 2s. We will refer to K as the basic DVZ transform of C. Since L 2 " M 2 " I for α n P R and |α n | ă 1, the Jacobi matrix (7) satisfies where : denotes the adjoint of a matrix. This identity encodes the relation between the basic DVZ connection and the Szegő projection -whose Jacobi matrix is hidden in C`C : [13,16]-, and suggests also a link with symmetrization processes on r´2, 2s -adressed by the mapping x Þ Ñ x 2´2 (see Section 3), which is represented by the operation K 2´2 I on the Jacobi matrix K-. Actually, we will show that the basic DVZ connection follows by a concatenation of a symmetrization process [15] and a Christoffel transformation [2,5,10] on the unit circle, followed by the Szegő projection. Equivalently, we can implement first the Szegő mapping, and then perform the real line version of the symmetrization (see Section 3) and Christoffel [1,19]. In this section we will present the basic DVZ connection in a more natural and concise way, directly looking for the relation between the OLPUC related to C and the OPRL associated with the basic DVZ transform K, and then using this to uncover the relation between the corresponding orthogonal polynomials and orthogonality measures. The interpretation in terms of symmetrizations, Szegő and Christoffel will follow as a byproduct of these results.
The OPRL q n corresponding to the Jacobi matrix K are the solutions of the formal eigenvalue equation Kqpxq " xqpxq, q " pq 0 , q 1 , q 2 , . . . q t , q 0 " 1.
We will identify these OPRL by using some relations, often not fully exploited, between the OLPUC χ n related to C and the factors L, M [3], Bearing in mind that L and M are involutions for α n P p´1, 1q, the above relations can be equivalently written as Lχ˚pzq " z´1χpzq, Mχpzq " χ˚pzq.

Therefore,
Kχpzq " pz`1qχ˚pzq, Kχ˚pzq " pz´1`1qχpzq, which show that, for each fixed z P C˚" Czt0u, the linear subspace spantχpzq, χ˚pzqu is invariant for K. This implies that K has formal eigenvectors upzqχpzq`vpzqχ˚pzq given by the eigenvalue equation The eigenvalues x are the solutions of for a choice of the square root z 1{2 (here and in what follows we understand that z n{2 " pz 1{2 q n , n P Z, for that choice of the square root). The corresponding eigenvectors of K are spanned by We conclude that the OPRL q n are given by where we have used (3). This coincides with the expression given in [9]. The orthogonality measure ν on the real line for q n follows from that one µ on the unit circle for χ n . To obtain this relation, let us parametrize the unit circle as T " te iθ : θ P r0, 2πqu and consider µ as a measure on r0, 2πs, where the mass at 1 P T, if any, is distributed equally between θ " 0 and θ " 2π so that µ remains symmetric under the transformation θ Þ Ñ 2π´θ, which represents the conjugation of e iθ . Then, the restriction of x " z 1{2`z´1{2 to the unit circle leads to the map which is a one-to-one transformation between r0, 2πs and r´2, 2s with inverse θpxq " 2 arccos x 2 , x P r´2, 2s.
The symmetric extension of µ to r0, 2πs is crucial to identify the orthogonality measure of q n , and guarantees that the relation xp2π´θq "´xpθq makes sense for any angle θ in the interval r0, 2πs where µ is supported. The orthogonality measure of q n pxq " X n pzq{X 0 pzq arises from the observation that the components X n of the formal eigenvector (8) are orthogonal with respect to µ. Indeed, Due to the orthonormality of χ n and χ n˚w ith respect to µ, the first term in the right hand side is 2I. On the other hand, taking into account that the OLPUC have real coefficients due to the symmetry of the measure under conjugation, the second term vanishes because its integrand simply changes sign under the transformation θ Þ Ñ 2π´θ which leaves the measure µ invariant. Therefore, Xpe iθ qXpe iθ q : dµpθq " 2I, which proves that the orthogonality measure of q n is a result also present in [9].
Since θp´xq " 2π´θpxq, the symmetry of µ under θ Þ Ñ 2π´θ makes dµpθpxqq symmetric under x Þ Ñ´x. However, the orthogonality measure ν does not preserve finally this symmetry due to the additional factorˇˇX 0 pe iθpxq{2 qˇˇ2 " 2`x that comes from the relation dνpxq " 1 2ˇX 0 pe iθpxq{2 qˇˇ2 dµpθpxqq. The above discussion may be extended to the tridiagonal matrix which is the conjugated of a Jacobi matrix by a diagonal sign matrix, Therefore, the formal eigenvalue equation K´q´pxq " xq´pxq, q´" pq0 , q1 , q2 , . . . q t , q0 " 1, defines a sequence qń of OPRL whose leading coefficients have a sign given by the corresponding diagonal coefficient of Π. A direct translation of the previous reasoning to this case shows that K´has formal eigenvectors x "´ipz 1{2´z´1{2 q.
As for the orthogonality measure ν´of qń , in this case a convenient parametrization of the unit circle is T " te iθ : θ P p´π, πsu. Accordingly, we consider the measure µ of χ n as a measure on r´π, πs, with any possible mass at´1 P T distributed equally between θ "´π and θ " π to preserve the symmetry of µ under the transformation θ Þ Ñ´θ representing the conjugation of e iθ . Then, on the unit circle, x "´ipz 1{2`z´1{2 q becomes which maps r´π, πs one-to-one onto r´2, 2s, and has the inverse The same kind of argument as in the previous case proves that Bearing in mind that θp´xq "´θp´xq, the measure ν´fails to be symmetric with respect to x Þ Ñ´x only due to the factor 2´x. The next section will present a closer look at the relations between the basic DVZ connection and the well known Szegő projection between the unit circle and the real line. This will make symmetrization processes and Christoffel transformations to enter into the game.

DVZ, Christoffel, Szegő and symmetrizations
Originally, in [9], the basic DVZ connection arises as a combination of well known transformations which surprisingly links the OPUC related to a CMV matrix C " ML and the OPRL associated to the Jacobi matrix K " L`M. We will see that such an interpretation follows directly from our more direct approach. For this purpose, let us rewrite the DVZ relation (9) between OPUC and OPRL as This relation suggests introducing the symmetrized OPUC [15], characterized by the Verblunsky coefficientsα 2k " 0 andα 2k`1 " α k . They are orthonormal with respect to a measure which is is invariant for the mapping θ Þ Ñ θ`π -i.e., symmetric under the change of sign of e iθ -, namely, where µ is the measure on r0, 2πs which makes ϕ n orthonormal, so that dµp2θq and dµp2θ´2πq are supported on r0, πs and rπ, 2πs respectively. Since dµpθq " 2dμpθ{2q, we can express the OPRL (9) and the measure (12) for the basic DVZ transform as Given a measure µ on T which is symmetric under conjugation, its Szegő projection is the measure σ " Szpµq induced on r´2, 2s by the mapping z Þ Ñ x " z`z´1, which is explicitly given by dσpxq " 2dµparccospx{2qq. The above expression for the measure ν shows that the basic DVZ connection is a combination of three transformations: symmetrization µ Þ Ñμ, Szegő projectionμ Þ Ñσ " Szpμq and the Christoffel transformation dσpxq Þ Ñ 1 2 p2`xq dσpxq. Let us have a closer look at the effect of these transformations on the orthogonal polynomials. The OPRL associated with the Szegő projection σ " Szpµq have the form [16,17] On the other hand, due to the symmetry e iθ Þ Ñ´e iθ ofμ, its Szegő projection σ " Szpμq becomes an even measure, i.e. symmetric under x Þ Ñ´x, thus its OPRL have the form [4, Chapter I] p 2n pxq " P n px 2 q,p 2n`1 pxq " x r P n px 2 q, with P n and r P n polynomials with orthonormality measures dmpxq and x dmpxq supported on R`. Bearing in mind (17), an expression similar to (19) for the OPRLp n related to the the Szegő projectionσ " Szpμq shows that which coincides with (6) once we substitute z by z 1{2 . This shows that the DG connection follows just by combining the symmetrization and the Szegő projection, thus the DG measure on r´2, 2s is given by dσpxq " 2dμparccospx{2qq " dµp2 arccospx{2qq. Besides,p 2n pxq "p 2n pz`z´1q " p n pz 2`z´2 q " p n px 2´2 q.
Therefore, P n pxq " p n px´2q, dmpxq " dσpx´2q and r P n pxq "p n px´2q, wherep n are orthonormal with respect to dσpxq " px`2q dσpxq. Hence, so thatp n pz 2`z´2 q " Combining this with (17) we find that Bearing in mind thatα 2n " 0, this identifies q n , as it is given in (18), with the OPRL related to the measure 1 2 px`2q dσpxq, in agreement with the previous interpretation of DVZ as a composition of the symmetrization, Szegő and Christoffel transformations.
Using the following notation, The above diagram has more than the three alluded transformations whose combination gives the basic DVZ connection. Let us comment on them. First, every Christoffel transformation dσpxq Þ Ñ ℘pxq dσpxq between measures on r´2, 2s is lifted to a similar one dµpθq Þ Ñ ℘pe iθ`e´iθ q dµpθq between the measures on T from which they come as Szegő projections. This explains the CMV matrix p D in the right upper corner: its measure 1 2 |z`1| 2 dμpzq is the Christoffel transform on the unit circle (see [2,5,10] for this notion) whose Szegő projection is the measure ν of the basic DVZ transform K.
Also, the Szegő projection of the symmetrization process on T yields a transformation which maps any measure σ on r´2, 2s into an even measureσ on the same interval. Since z Þ Ñ z 2 reads as x Þ Ñ x 2´2 if x " z`z´1, we conclude that where ξ stands for the one-to-one mapping between r0, 2s and r´2, 2s induced by

The general DVZ connection
In this section we intend to extend the basic DVZ connection to arbitrary real linear combinations of the factors L, M of a CMV matrix C " ML, i.e. a linear pencil which we denote by K λ 0 ,λ 1 is a Jacobi matrix when λ k ą 0, otherwise it is related to a true Jacobi matrix J λ 0 ,λ 1 by conjugation with a diagonal sign matrix, As in the case of the basic DVZ connection, our aim is to find explicit relations between the measures and orthogonal polynomials associated with C and J λ 0 ,λ 1 .
Prior to the discussion of these relations we will clarify the OPRL targets of this connection with OPUC. In other words, which Jacobi matrices may be expressed as J λ 0 ,λ 1 " J λ 0 ,λ 1 ppα n q ně0 q for some sequence pα n q ně0 of real Verblunsky coefficients? This question is answered by the following proposition.
Proposition 4.1. A Jacobi matrix J given by (1) has the form J λ 0 ,λ 1 ppα n q ně0 q for some λ k P Rzt0u and a sequence pα n q ně0 in p´1, 1q iff the complex numbers z n " ř n j"0 b j`i a n satisfy the following conditions for n ě 0: (i) All the points z n with even index n lie in a single circumference C 0 centered at a point in Rzt0u.
(ii) All the points z n with odd index n lie in the circumference C 1 with the same center as C 0 and passing through the origin.
If this is the case, λ 1 is the common center of C 0 and C 1 , |λ 0 | is the radius of C 0 and, for any sign of λ 0 , so that the mapping pλ 0 , λ 1 , pα n q ně0 q Þ Ñ J λ 0 ,λ 1 ppα n q ně0 q is one-to-one up to the identity J λ 0 ,λ 1 ppα n q ně0 q " J´λ 0 ,λ 1 ppp´1q n`1 α n q ně0 q.
From these conditions we find (23) and which proves (i), (ii), as well as the relations between λ k and the circumferences C k . The equality (24) follows directly from the explicit form of J λ 0 ,λ 1 ppα n q ně0 q. Suppose now that (i) and (ii) are true, which means that (26) holds for some λ k P Rzt0u, where the sign of λ 0 may be arbitrarily chosen. Introducing ρ n " a n {|λ ǫpnq | and defining α n by (23), the relation (26) becomes α 2 n`ρ 2 n " 1. We conclude that pα n q ně0 is a sequence in p´1, 1q such that ρ n " a 1´a 2 n and (25) is satisfied, i.e. J " J λ 0 ,λ 1 ppα n q ně0 q. Regarding this equality, according to this discussion, the only freedom in λ k and α n is the choice of the sign of λ 0 , whose alteration only changes the sign of the parameters α n with even index n. This proves that the map pλ 0 , λ 1 , pα n q ně0 q Þ Ñ J λ 0 ,λ 1 ppα n q ně0 q fails to be one-to-one only due to (24).
The previous proposition states that the Jacobi matrices arising from the present general version of DVZ are generated by selecting a couple of concentric circumferences -C 0 and C 1 -with center on Rzt0u, one of them -C 1 -passing though the origin. The choice of a sequence z n P C ǫpnq with positive imaginary part determines the corresponding Jacobi parameters a n " Imz n and b n " Repz n´zn´1 q.

Besides,
where Π is the diagonal sign matrix given in (14). Thus, the OPRL sequence p pλq " pp pλq 0 , p pλq 0 , . . . q t given by J λ p pλq pxq " xp pλq pxq, p pλq 0 " 1, is related by p pλq " Πq pλq with the solutions of Hence, the polynomials q pλq n and p pλq n differ in at most a sign, so q pλq is also an OPRL sequence with respect to the measure ν λ . Due to these reasons, in what follows we will consider only linear combinations of CMV factors with the form K λ for an arbitrary λ P Rzt0u , referring to the solutions q pλq n of (27) as the corresponding OPRL. We will refer to the problem of finding the relations between the orthogonal polynomials and measures related to a CMV matrix C " ML and the tridiagonal matrix K λ " L`λM as the (general) DVZ connection between OPUC and OPRL.
This general version of DVZ was already considered in [9]. Nevertheless, unlike the case of the basic DVZ connection, [9] does not provide any closed formula for the relations between the orthogonal polynomials and measures associated with a CMV matrix C " ML and its (general) DVZ transform, This will change with the present approach, whose simplicity leads to completely explicit expressions for such general DVZ relations between orthogonal polynomials and measures.

OPRL for the general DVZ connection
We are looking for the solutions of the formal eigenvalue equation which yield an OPRL sequence q pλq n with positive leading coefficients for λ ą 0, and having a sign given by the diagonal entries of Π in (14) if λ ă 0. The same method used previously for λ "˘1 provides the relation between the OPRL q pλq n and the OLPUC χ n of C for any value of λ P Rzt0u. Since The eigenvalues x are the solutions of the corresponding eigenvectors being spanned by This suggests the identification q pλq n pxq " X n pzq X 0 pzq " xχ n pzq`pz`λqχ n˚p zq x`z`λ , z " zpxq, where z " zpxq is a complex mapping satisfying (30). The validity of this result only needs to check that the above expression yields a well defined function of x because then (29) follows from the eigenvector property of Xpzq. However, (30) determines x as function of z only up to a sign which could depend arbitrarily on z. Also, although z Þ Ñ 1`λ 2`λ pz`z´1q maps Czt0u onto C, there are in general two values of z giving the same value of x 2 which are inverse of each other. Therefore, all that can be said is that (30) establishes a one-to-one correspondence between subsets tz, z´1u Ă Czt0u and subsets tx,´xu Ă C. As a consequence, there are infinitely many functions z " zpxq well defined on the whole complex plane and satisfying (30), so that (32) becomes a true function of x when substituting z by zpxq. The fact that, for any of these choices, q pλq n pxq satisfies (29), not only identifies these functions as the OPRL related to K λ , but consequently shows that q pλq n pxq do not depend on the particular choice of zpxq. A more direct proof of this result is given by the following theorem, which is of practical interest since it provides an alternative expression of q pλq n pxq which turns out to be more useful to identify its dependence on x in particular situations.
we conclude that both determinants, when divided by z´z´1, become real linear combinations of U j pz`z´1q, which thus have the form Q n pz`z´1q and r Q n pz`z´1q for some real polynomials Q n and r Q n . The relations (3) lead to This implies that deg q pλq n " n because z`z´1 " λ´1x 2´p λ`λ´1q due to (30). Introducing the full expansion χ n pzq " ř k c pnq j z j into the determinants defining Q n and r Q n we obtain where we assume (34) extended to every j P Z. The relations given in the theorem follow by noticing that U´j "´U j´2 . Finally, the relation q pλq pxq " Xpzq{X 0 pzq shows that it is a formal eigenvector of K λ with eigenvalue x and such that q

Orthogonality measure for the general DVZ connection
To obtain the orthogonality measure ν λ of the OPRL q pλq n we will introduce the measures µ`and µ´induced by µ on the upper and lower arcs of the unit circle, T`" te iθ : θ P r0, πsu and T´" te iθ : θ P r´π, 0su, with any eventual mass point of µ at˘1 equally distributed between µ`and µ´. Then, we can rewrite any integral with respect to µ as ż Besides, the symmetry of µ under conjugation means that dµ`pzq " dµ´pz´1q, so that ż The orthogonality measure ν λ will arise from the orthogonality of the components of the eigenvector (31) with respect to µ, a result given in the next proposition. To make it more precise and prove this orthogonality we define the following map on T xpe iθ q " ? 1`λ 2`2 λ cos θ, which satisfies (30) and xpz´1q " xpzq. When restricted to T˘, (35) provides a homeomorphism between T˘and the interval r|1´|λ||, 1`|λ|s, with an inverse mapping given by zpxq " e˘i θpxq , where θpxq " arccos t λ pxq 2 , t λ pxq " λ´1x 2´p λ`λ´1q.
Now we can formulate the precise meaning of the orthogonality property for the eigenvector (31). Proof. Since xpzq satisfies (30), the above sum of integrals can be rewritten as ż T pχpzqχpzq :`χ˚p zqχ˚pzq : q dµpzq ż T`ˆz`λ xpzq χ˚pzqχpzq :`z´1`λ xpzq χpzqχ˚pzq :˙d µ`pzq ż T´ˆz`λ xpzq χ˚pzqχpzq :`z´1`λ xpzq χpzqχ˚pzq :˙d µ´pzq The OLPUC χ n have real coefficients due to the symmetry of µ under conjugation. As a consequence, the integrands of the last two summands are invariant under conjugation of z. Since this operation exchanges µ´and µ`, the last two terms cancel each other. On the other hand, the orthogonality of χ n and χ n˚w ith respect to µ implies that the first integral is 2I.
The orthogonality measure of q pλq n arises as a consequence of the previous result. Except for the cases λ "˘1, it is a measure supported on two disjoint symmetric intervals.
Theorem 4.4. If µ is a measure on T which is symmetric under conjugation, then the OPRL q pλq n pxq of the corresponding general DVZ transform K λ are orthonormal with respect to the measure where θ˘pxq is the homeomorphism between Eλ and r0, πs given by the restriction of θpxq in (36) to Eλ , and µ˘is the measure induced by µ on T˘with the mass points of µ at˘1 equally distributed between µ˘.

Proof. With the notation of Proposition 4.3, we have
X˘pe˘i θpxq q " q pλq n p˘xq X0 pe˘i θpxq q.
Therefore, using the homeomorphisms z " e˘i θ`pxq between Eλ and T˘, the result of such a proposition may be rewritten as ż Eλ q pλq n pxq q pλq n pxq : |X0 pe iθpxq q| 2 dµ`pe iθ`pxq q ż Eλ q pλq n p´xqq pλq n p´xq : |X0 pe´i θpxq q| 2 dµ´pe´i θ`pxq q " 2I. Besides, Bearing in mind that z " zpxq satisfies (30), we find that The theorem follows by inserting this equality into (37) and performing in the second integral of (37) the change of variables x Þ Ñ´x, taking into account that it maps Eλ into Eλ and θ´pxq " θ`p´xq.
We should highlight that, in the expression for ν λ given by the previous theorem, dµ`pe iθ`pxq q is a measure on Eλ while dµ´pe´i θ´pxq q is a measure on Eλ , so that the sum of both is even, i.e. symmetric under the reflection x Þ Ñ´x. Nevertheless, the measure ν λ is not even due to the additional factor which, by the way, is non-negative on E λ . Bearing in mind that dµ`pzq " dµ´pz´1q, we can also express the measure ν λ using only the measure µ`on the upper arc T`, i.e.
Since θ˘pxq " θpxq for x P Eλ , it is tempting to rewrite the above as However, (36) is not one-to-one between E λ and T`because θp´xq " θpxq, thus (40) should be considered simply as a symbolic representation of (39).
Although zpxq fails to be one-to-one between E λ and T because the four edges of E λ are mapped into˘1, this difficulty is overcome due to the factor px`λq 2´1 which cancels any possible mass point at the two edges˘1´λ, making unnecessary to cover them with the mapping zpxq. Therefore, (41) makes perfect sense considering zpxq as a one-to-one map between E λ zt1´λ,´1´λu and T.
The connection between ν λ and µ can be made more explicit for the absolutely continuous and pure point components. Suppose that a measure µ on T has the form dµpe iθ q " wpθq dθ`ÿ k m k pδpθ´θ k q`δpθ`θ k qq dθ, θ k P r0, πs, where wp´θq " wpθq and the mass points appear in symmetric pairs due to the invariance of µ under conjugation. The above expression assumes for η " 0, π the artificial splitting of any mass point m δpθ´ηq dθ as pm{2qpδpθ´ηq`δpθ`ηqq dθ, something which allows us to identify dµ˘pe iθ q " wpθq dθ`ÿ k m k δpθ¯θ k q dθ.
On the other hand, Finally, we conclude that The explicitness of this expression for ν λ makes it of particular interest for the analysis of specific examples. Among the possible mass points of µ, those located at˘1 deserve special attention since they are the only ones which are artificially splitted to obtain ν λ . If µ has a mass m at˘1, then µ˘have a mass m{2 at the same point which, according to the previous results, leads to the following masses for ν λ at the edges of E λ : In agreement with our previous observation, the edges˘1´λ P E λ are free of mass points for the measure ν λ whenever λ ‰˘1. In the case λ "˘1, the only point of E˘1 " r´2, 2s which cannot support a mass point of ν˘1 is¯2.

Examples
In this section we will apply the general DVZ connection to some well known families of OPUC -whose details can be found for instance in [16, Sect. 1.6]-, which will provide new explicit examples of OPRL. Theorem 4.2 yields a representation of these OPRL in terms of the Chebyshev polynomials U n on r´2, 2s (see (33)) evaluated on the function t λ pxq defined in (36). For convenience, in what follows we will use the abbreviationÛ n " U n pt λ pxqq, which is a λ-dependent polynomial of degree 2n in x.

Bernstein-Szegő polynomials
As an introductory example, we will analyze the general DVZ connection when the OPUC are the degree one real Bernstein-Szegő polynomials, i.e. those defined by the Verblunsky coefficients α n " α δ n,0 , α P p´1, 1q.
Using the results of Theorem 4.2 we can express the related OPRL as On the other hand, if θ " θpxq is given by (36), then thus, according to (42), the orthogonality measure of q pλq n reads as x P E λ .

Lebesgue measure with a mass point
Let us consider now the probability measure obtained by inserting a mass point at z " 1 into the Lebesgue measure on the unit circle, i.e.
The DVZ transform is the Jacobi matrix K λ in (28) built out of the above coefficients α n , while Theorem 4.2 provides the corresponding OPRL q To get these expressions we have taken into account that 1`α n´1 " α n´1 {α n . The measure ν λ which makes q pλq n orthonormal follows from (42) and (43), which lead to dν λ pxq " 1´m 2π|λ| d px`λq 2´1 1´px´λq 2 dx`m δpx´1´λq dx, x P E λ .

Second kind polynomials of the previous example
The alluded second kind polynomials are those with Verblunsky coefficients α n "´1 n`m´1 , m P p0, 1q.
Therefore, ρ n and κ n are the same as in the previous example, while the Carathéodory function F of the orthogonality measure is the inverse of that one for the preceding example, i.e.
The above results seem to be new for m ‰ 1{2, adding an item to the list of OPUC examples given in [16,Sect. 1.6], where this case is discussed only for m " 1{2. Furthermore, via DVZ connection, they provide a new OPRL family with Jacobi matrix K λ as in (28), which differs from that one arising in Example 5.2 only in the main diagonal (up to the first one, the diagonal coefficients of K λ are the opposite of those in the analogous matrix for Example 5.2). According to Theorem 4.2 and using that 1`α n´1 " α n´1 {α n´2 , we find that the corresponding OPRL are α n´2 px´λ´m α n´2 α 2n´2 qÛ k´1 4mα n´1 px´λ´1q ř k´2 j"0 pj`1qÛ j ‰ , n " 2k, κ n " px´λ´α n´1 qÛ ḱ α n´1 α n´2 p1`2mp1`pn´1qpx´λqqqα n´2 qÛ k´1 4mα n´1 px´λ´1q ř k´2 j"0 pj`1qÛ j ‰ , n " 2k`1.