Orthogonal polynomials on the unit circle: Verblunsky coefficients with some restrictions imposed on a pair of related real sequences
Cleonice F. Bracciali, Jairo S. Silva, A. Sri Ranga, Daniel O. Veronese
aa r X i v : . [ m a t h . C A ] A ug Orthogonal polynomials on the unit circle: Verblunsky coefficients withsome restrictions imposed on a pair of related real sequences ∗ Cleonice F. Bracciali a , Jairo S. Silva b, † , A. Sri Ranga a , Daniel O. Veronese c a Departamento de Matem´atica Aplicada, IBILCE,UNESP - Universidade Estadual Paulista, 15054-000, S˜ao Jos´e do Rio Preto, SP, Brazil. b Depto de Matem´atica, Universidade Federal do Maranh˜ao, 65080-805, S˜ao Lu´ıs, MA, Braziland P´os-Gradua¸c˜ao em Matem´atica, IBILCE,UNESP - Universidade Estadual Paulista, 15054-000, S˜ao Jos´e do Rio Preto, SP, Brazil. c ICTE, Universidade Federal do Triˆangulo Mineiro, 38064-200, Uberaba, MG.
Abstract
It was shown recently that associated with a pair of real sequences {{ c n } ∞ n =1 , { d n } ∞ n =1 } , with { d n } ∞ n =1 a positive chain sequence, there exists a unique nontrivial prob-ability measure µ on the unit circle. The Verblunsky coefficients { α n } ∞ n =0 associated withthe orthogonal polynomials with respect to µ are given by the relation α n − = τ n − (cid:20) − m n − ic n − ic n (cid:21) , n ≥ , where τ = 1, τ n = Q nk =1 (1 − ic k ) / (1 + ic k ), n ≥ { m n } ∞ n =0 is the minimal parametersequence of { d n } ∞ n =1 . In this manuscript we consider this relation and its consequences byimposing some restrictions of sign and periodicity on the sequences { c n } ∞ n =1 and { m n } ∞ n =1 .When the sequence { c n } ∞ n =1 is of alternating sign, we use information about the zeros ofassociated para-orthogonal polynomials to show that there is a gap in the support of themeasure in the neighbourhood of z = −
1. Furthermore, we show that it is possible to ge-nerate periodic Verblunsky coefficients by choosing periodic sequences { c n } ∞ n =1 and { m n } ∞ n =1 with the additional restriction c n = − c n − , n ≥ . We also give some results on periodicVerblunsky coefficients from the point of view of positive chain sequences. An example isprovided to illustrate the results obtained.
Keywords: Para-orthogonal polynomials, Probability measures, Periodic Verblunsky coefficients,Chain sequences, Alternating sign sequence.2010 Mathematics Subject Classification: 42C05, 33C47.
Orthogonal polynomials on the unit circle (OPUC) have been commonly known as Szeg˝o poly-nomials in honor of G´abor Szeg˝o who introduced them in the first half of the 20th century.Because of their applications in quadrature rules, signal processing, operator and spectral theo-ry and many other topics, these polynomials have received a lot of attention in recent years (see, ∗ 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 CAPESof Brazil. † [email protected] (corresponding author). or example, [2, 4, 7, 12, 13, 16, 17, 21]). For many years a first hand text for an introduction tothese polynomials has been the classical book [20] of Szeg˝o. However, for recent and more up todate texts on this subject we refer to the two volumes of Simon [18, 19]. For further interestingreading on this subject we refer to Chapter 8 of Ismail’s recent book [11].Given a nontrivial probability measure µ ( z ) = µ ( e iθ ) on the unit circle T = { z = e iθ : 0 ≤ θ ≤ π } , the associated sequence of OPUC { φ n } are those with the property Z T ¯ z j φ n ( z ) dµ ( z ) = Z π e − ijθ φ n ( e iθ ) dµ ( e iθ ) = 0 , ≤ j ≤ n − , n ≥ . Letting κ − n = k φ n k = R T | φ n ( z ) | dµ ( z ), the orthonormal polynomials on the unit circle are ϕ n ( z ) = κ n φ n ( z ), n ≥ φ n ( z ), n ≥
0, considered as monic polynomials, satisfy the so called forwardand backward recurrence relations, respectively, φ n ( z ) = zφ n − ( z ) − α n − φ ∗ n − ( z ) ,φ n ( z ) = (1 − | α n − | ) zφ n − ( z ) − α n − φ ∗ n ( z ) , n ≥ , (1.1)where α n − = − φ n (0) and φ ∗ n ( z ) = z n φ n (1 / ¯ z ) denotes the reversed (reciprocal) polynomialof φ n ( z ). The numbers α n , in recent years, have been referred to as Verblunsky coefficients.It is known that these coefficients are such that | α n | < n ≥
0. Moreover, the OPUC andthe associated measure are completely determined from these coefficients (see for example [18],Theorem 1.7.11). A very nice and short constructive proof of this last statement can be foundin [9].It was shown in [6] that given any nontrivial probability measure on the unit circle, thencorresponding to this measure there exists a pair of real sequences { c n } ∞ n =1 and { d n } ∞ n =1 , where { d n } ∞ n =1 is also a positive chain sequence. In Theorem 2.1 we have given the complete informationregarding this statement and its reciprocal. To be precise, the sequences { c n } ∞ n =1 and { d n } ∞ n =1 are the coefficients of the three term recurrence formula R n +1 ( z ) = [(1 + ic n +1 ) z + (1 − ic n +1 )] R n ( z ) − d n +1 zR n − ( z ) , n ≥ , (1.2)with R ( z ) = 1 and R ( z ) = (1 + ic ) z + (1 − ic ) , where R n ( z ) = Q nj =1 (cid:2) − τ j − α j − (cid:3)Q nj =1 (cid:2) − R e ( τ j − α j − ) (cid:3) zφ n ( z ) − τ n φ ∗ n ( z ) z − , with τ n = φ n (1) /φ ∗ n (1), n ≥ { c n } ∞ n =1 and { d n } ∞ n =1 , it is possible to recover the associated probabilitymeasure using certain rational functions that follow from the recurrence formula (1.2). In [3],using standard arguments involving continued fractions, series expansions at infinity and atthe origin, and Helly’s Selection Theorem, the associated measure µ is given as a limit of asubsequence of discrete measures ψ n ( e iθ ) whose pure points (those different from z = 1) areexactly the zeros of R n ( z ). Results given in [3] enable us to give information about the supportof the measure µ by analysing the zeros z n,j = e iθ n,j , j = 1 , . . . , n , of R n ( z ) , or, equivalently,by analysing the zeros of the functions W n ( x ), given by W n ( x ) = 2 − n e − inθ/ R n ( e iθ ) , n ≥ , (1.3)where x = cos( θ/ {W n } ∞ n =0 satisfy the three term recurrenceformula (see [1, 8]) W n +1 ( x ) = (cid:16) x − c n +1 p − x (cid:17) W n ( x ) − d n +1 W n − ( x ) , n ≥ , (1.4)2ith W ( x ) = 1 and W ( x ) = x − c √ − x .For any n ≥ W n ( x ) has exactly n distinct zeros x n,j = cos( θ n,j / j = 1 , . . . , n , in( − , R n ( z )and R n +1 ( z ) is by proving the interlacing property − < x n +1 ,n +1 < x n,n < x n +1 ,n < · · · < x n, < x n +1 , < , n ≥ , (1.5)for the zeros of W n ( x ) and W n +1 ( x ) using the three term recurrence formula (1.4).The aim of this manuscript is to study sequences of Verblunsky coefficients where the relatedsequences { c n } ∞ n =1 and { m n } ∞ n =1 have restrictions of sign and periodicity. We show that, undercertain conditions, it is possible to estimate the support of the associated measure and to getperiodic Verblunsky coefficients. Furthermore, we discuss some geometric aspects related tothese restrictions.This manuscript is organized as follows. In Section 2 we give a summary of all requiredtheoretical results. Section 3 deals with the results concerning measures for which the associatedsequence { c n } ∞ n =1 has the alternating sign property, namely, c n = ( − n ˜ c n , for n ≥ , where˜ c n is a positive (or negative) sequence of real numbers. In Section 4 relations with periodicVerblunsky coefficients are considered. Finally, in Section 5 we give an example to illustrate theresults obtained. In this section we present some results concerning nontrivial probability measures and positivechain sequences (for more details on chain sequences we refer to [5] and [22]). Furthermore,some results about periodic Verblunsky coefficients are presented.We begin with two theorems established in [6]. The first theorem provides a characterizationfor nontrivial probability measures in terms of two sequences { c n } ∞ n =1 and { d n } ∞ n =1 . Theorem 2.1 (a)
Given a nontrivial probability measure µ on the unit circle, then associatedwith it there exists an unique pair of real sequences {{ c n } ∞ n =1 , { d n } ∞ n =1 } , where { d n } ∞ n =1 is alsoa positive chain sequence. Specifically, if { α n } ∞ n =0 is the associated sequence of Verblunskycoefficients and if the sequence τ n is such that τ = 1 and τ n = τ n − − τ n − α n − − τ n − α n − , n ≥ , then m = 0 ,c n = − Im ( τ n − α n − )1 − Re ( τ n − α n − ) and m n = 12 | − τ n − α n − | [1 − Re ( τ n − α n − )] , n ≥ , where { m n } ∞ n =0 is the minimal parameter sequence of { d n } ∞ n =1 . Moreover, the maximal parametersequence { M n } ∞ n =0 of { d n } ∞ n =1 is such that M is the value of the jump in the measure at z = 1 . (b) Conversely, given a pair of real sequences {{ c n } ∞ n =1 , { d n } ∞ n =1 } , where { d n } ∞ n =1 is also apositive chain sequence then associated with this pair there exists an unique nontrivial probabil-ity measure µ supported on the unit circle. Specifically, if { m n } ∞ n =0 is the minimal parametersequence of { d n } ∞ n =1 , then τ = 1 ,τ n − α n − = 1 − m n − ic n − ic n and τ n = 1 − ic n ic n τ n − , n ≥ . (2.1) Moreover, the measure has a jump M at z = 1 , where { M n } ∞ n =0 is the maximal parametersequence of { d n } ∞ n =1 . Theorem 2.2
The probability measure µ has a pure point at w ( | w | = 1) if, and only if, ∞ X n =1 n Y j =1 | − wτ j − ( w ) α j − | − | α j − | = λ ( w ) < ∞ . Moreover, the size of the mass at the point z = w is equal to t = [1 + λ ( w )] − . Here, τ ( w ) = 1 and τ j +1 ( w ) = φ j +1 ( w ) φ ∗ j +1 ( w ) = wτ j ( w ) − α j − wτ j ( w ) α j , j ≥ . (2.2)Now we discuss a result obtained in [3], which leads to a relation between the zeros of thepolynomials R n ( z ) and the measure associated with the pair of sequences {{ c n } ∞ n =1 , { d n } ∞ n =1 } .Consider the new sequence of polynomials { Q n } satisfying Q n +1 ( z ) = [(1 + ic n +1 ) z + (1 − ic n +1 )] Q n ( z ) − d n +1 zQ n − ( z ) , n ≥ , with Q ( z ) = 0 and Q ( z ) = 2 d . Let z n,j = e iθ n,j be the zeros of R n ( z ) , λ n, = 1 − Q n (1) R n (1) and λ n,j = Q n ( z n,j )(1 − z n,j ) R ′ n ( z n,j ) , with j ∈ { , , ..., n } . Thus, as shown in [3], P nj =0 λ n,j = 1 and also λ n,j > , j = 0 , , ..., n. In addition, if we define the sequence of step-functions ψ n ( e iθ ) , n ≥ , on [0 , π ] by ψ n ( e iθ ) = , θ = 0 ,λ n, , < θ ≤ θ n, , P kj =0 λ n,j , θ n,k < θ ≤ θ n,k +1 , k = 1 , , ..., n − , , θ n,n < θ ≤ π (2.3)then by Helly Selection Theorem a subsequence of ψ n ( e iθ ) converges to the measure µ ( e iθ )associated with the pair {{ c n } ∞ n =1 , { d n } ∞ n =1 } as established in Theorem 2.1.As an immediate consequence of this result, we can state the following. Theorem 2.3
Let {{ c n } ∞ n =1 , { d n } ∞ n =1 } be a pair of real sequences with { d n } ∞ n =1 a positive chainsequence. Moreover, let R n ( z ) be the sequence of polynomials given by (1.2) and µ be the measureassociated with this pair of sequences. In addition, suppose that the zeros of R n ( z ) lie on anclosed arc B of the unit circle, for n ≥ . Then, the support of the measure µ lie within B ∪ { } . We now present a review of basic results on measures with periodic Verblunsky coefficients.For more details regarding these results we refer to [10, 14, 15, 19].Let { α n } ∞ n =0 be a p -periodic sequence ( α n + p = α n , n ≥
0) of Verblunsky coefficients as-sociated with the measure denoted by µ ( p ) (here, p is a fixed natural number). Consider thediscriminant function ∆( z ) = z − p/ Tr( T p ( z )) where T p ( z ) = A ( α p − , z ) . . . A ( α , z ) , (2.4) A ( α j , z ) = (1 − | α j | ) − / (cid:18) z − α j − α j z (cid:19) , j = 0 , . . . , p − , (2.5)and Tr( T p ( z )) denotes the trace of T p ( z ) . It is well known that all the p distinct solutions of the equation ∆( z ) = 2, which we denoteby z +1 , . . . , z + p , lie on the unit circle T . In the same way, the p distinct solutions of the equation4( z ) = −
2, denoted by z − , . . . z − p , also lie on T . Using these solutions it is possible to show thatthe unit circle can be decomposed into 2 p alternating sets G , B , G , . . . , B p with each gap, G j , open and each band, B j , closed. Moreover, each band B j is given by B j = { z ∈ T | arg( z σ j j ) ≤ arg( z ) ≤ arg( z − σ j j ) } with σ j = ( − j +1 , j = 1 , , . . . , p. Now we mention four fundamental results (see [19, Chapter 11]) which give a completelycharacterization of probability measures on the unit circle associated with periodic Verblunskycoefficients. The first result provides information about the absolutely continuous part and thesingular part of the measure.
Theorem 2.4
Let { α j } ∞ j =0 be a sequence of Verblunsky coefficients of period p and let dµ ( p ) = w ( θ ) dθ π + dµ ( p ) s be the associated probability measure. Then, if B , . . . , B p are the correspondingbands we have that ∪ B j is the essential support of the a.c. spectrum and dµ ( p ) s [ ∪ B j ] = ∅ . Moreover, in each disjoint open arc on T \ ∪ pj =1 B j , µ ( p ) has either no support or a single purepoint. The next theorem provides information about the associated weight function w ( θ ) . Theorem 2.5
Let { α j } ∞ j =0 be a sequence of Verblunsky coefficients of period p and let dµ ( p ) = w ( θ ) dθ π + dµ ( p ) s be the associated measure. Then, for e iθ ∈ ∪ B j ,w ( θ ) = p − ∆ ( e iθ )2 | Im( e − ipθ/ ) ϕ p ( e iθ ) | . (2.6) In particular, (i) On ∪ B intj , w ( θ ) > . (ii) At an edge of a band that is by a closed gap (a gap which is empty), w ( θ ) > . (iii) At an edge, θ , of a band that is by an open gap, w ( θ ) ∼ c ( θ − θ ) if ϕ ∗ p ( e iθ ) − ϕ p ( e iθ ) = 0 . (iv) At an edge, θ , of a band that is by an open gap, w ( θ ) ∼ c ( θ − θ ) − if ϕ ∗ p ( e iθ ) − ϕ p ( e iθ ) = 0 . Finally, the following two theorems lead to a complete characterization for the pure pointsof the measure.
Theorem 2.6
Let { α j } ∞ j =0 be a sequence of Verblunsky coefficients of period p. Then π ( z ) = ϕ ∗ p ( z ) − ϕ p ( z ) (2.7) has all its zeros in the set of gap closures, one in each gap closure. Theorem 2.7
Let { α j } ∞ j =0 be a sequence of Verblunsky coefficients of period p and let µ ( p ) theassociated measure. Let θ be a point in a gap closure where ϕ ∗ p ( e iθ ) − ϕ p ( e iθ ) = 0 . Then, either µ ( p ) has no pure point in the gap or else it has a pure point at z = e iθ . The results on periodic Verblunsky coefficients presented above will be used in sections 4and 5. 5
On measures associated with alternating sign sequences { c n } First, we provide two lemmas that will be useful to derive the subsequent results.
Lemma 3.1
Let W n ( x ) satisfying (1.4) and R n ( z ) satisfying (1.2) . Then, the following state-ments are equivalent: (i) c n = ( − n c , n ≥ and c ∈ R ; (ii) For n ≥ , R n ( z ) has real coefficients and R n +1 ( z ) = [(1 − ic ) z + (1 + ic )] e R n ( z ) , where e R ( x ) = 1 and e R n ( z ) is also a polynomial with real coefficients; (iii) For n ≥ , W n ( x ) is an even polynomial of degree n and W n +1 ( x ) = (cid:0) x + c √ − x (cid:1) f W n ( x ) with f W ( x ) = 1 and f W n ( x ) an even polynomial of degree n . Proof. (i) ⇒ (ii) Since R ( z ) = 1 and R ( z ) = [(1 − ic ) z + (1 + ic )] e R ( z ) , with e R ( z ) = 1 , itfollows that the result holds for n = 0 . Furthermore, if (ii) holds for n = k ∈ N then, from thethree term recurrence relation (1.2), we obtain R k +1) ( z ) = [(1 + ic ) z + (1 − ic )] [(1 − ic ) z + (1 + ic )] e R k ( z ) − d k +2 zR k ( z )= (cid:2) (1 + c ) z + 2(1 − c ) z + (1 + c ) (cid:3) e R k ( z ) − d k +2 zR k ( z ) . Consequently, since we are assuming that e R k ( z ) and R k ( z ) are polynomials with real coeffi-cients, we conclude that R k +1) ( z ) also has real coefficients. Moreover, using again (1.2), wecan see that R k +1)+1 ( z ) = [(1 − ic ) z + (1 + ic )] R k +1) ( z ) − d k +3 z [(1 − ic ) z + (1 + ic )] e R k ( z )= [(1 − ic ) z + (1 + ic )] e R k +1) ( z ) , where e R k +1) ( z ) = R k +1) ( z ) − d k +3 z e R k ( z ) is also a polynomial with real coefficients, once R k +1) ( z ) and e R k ( z ) have real coefficients. Therefore, using mathematical induction, we con-clude that the statement (ii) holds for all n ≥ ⇒ (iii) By (ii), R n ( z ) has real coefficients for n ≥
0. Moreover, from the three termrecurrence relation (1.2) we have that R n ( z ) is a self-inversive polynomial, i.e., R ∗ n ( z ) = z n R n (1 / ¯ z ) = R n ( z ). Therefore, it follows (see [1, Lemma 2 . W n ( x ) is an even poly-nomial of degree 2 n in the variable x = cos θ/
2. Similarly, since e R n ( z ) is also a self-inversivepolynomial with real coefficients, we have that f W n ( x ) = (4 e iθ ) − n/ e R n ( e iθ ) is an even polyno-mial of degree 2 n . Then, since R n +1 ( z ) = [(1 − ic ) z + (1 + ic )] e R n ( z ) , from the relation (1.3)it follows that W n +1 ( x ) = ( x + c √ − x ) f W n ( x ) . (iii) ⇒ (i) Using the assumption (iii) and the three term recurrence relation (1.4), we have,for s ≥ W s ( x ) = ( x − c s p − x )( x + c p − x ) f W s − ( x ) − d s W s − ( x ) (3.1)and( x + c p − x ) f W s ( x ) = ( x − c s +1 p − x ) W s ( x ) − d s +1 ( x + c p − x ) f W s − ( x ) . (3.2)Hence, since W s ( x ), W s − ( x ), f W s ( x ) and f W s − ( x ) are even polynomials, we can use therelations (3.1) and (3.2) to conclude that c s = c and c s +1 = − c , for s ≥
1. Moreover, usingthe assumption (iii) and the definition of W ( x ) it is easy to see that c = − c. R n ( z ) satisfyingˆ R n +1 ( z ) = [(1 + i ˆ c n +1 ) z + (1 − i ˆ c n +1 )] ˆ R n ( z ) − d n +1 z ˆ R n − ( z ) , n ≥ , (3.3)with ˆ R ( z ) = 1 , ˆ R ( z ) = (1 + i ˆ c ) z + (1 − i ˆ c ) and ˆ c n = − c n . The following lemma gives the relation between the polynomials R n ( z ) and ˆ R n ( z ) . Lemma 3.2
Let R n ( z ) satisfying (1.2) and ˆ R n ( z ) satisfying (3.3) . Then, the following holds R n ( z ) = ˆ R n (¯ z ) , n = 0 , , , . . . . Proof.
The proof can be given by mathematical induction. Clearly, the result holds for n = 0and n = 1 . Suppose that the result holds for n = 0 , , . . . , k. Then, from the recurrence relations(1.2) and (3.3), we haveˆ R k +1 (¯ z ) = [(1 − ic k +1 )¯ z + (1 + ic k +1 )] ˆ R k (¯ z ) − d k +1 ¯ z ˆ R k − (¯ z )= [(1 − ic k +1 )¯ z + (1 + ic k +1 )] R k ( z ) − d k +1 ¯ zR k − ( z ) . Hence, the result follows by taking the complex conjugate on the above relation.Observe that the Lemma 3.2 provides also a relation between the zeros of the polynomials R n ( z ) and the zeros of ˆ R n ( z ) , namely, if z n,j is a zero of R n ( z ) then z n,j is a zero of ˆ R n ( z ) . Now we consider the problem of giving estimates for the support of measures whose sequences { c n } ∞ n =1 are of alternating sign. We start with the case c n = ( − n c , where c ∈ R . Let C and C be closed arcs on the unit circle given by C = { z ∈ T : 0 ≤ arg( z ) ≤ θ c } and C = { z ∈ T : 2 π − θ c ≤ arg( z ) ≤ π } , where θ c = arccos (cid:16) c − c +1 (cid:17) ∈ [0 , π ] . Then, we can state the following.
Theorem 3.3
Let µ be the probability measure on the unit circle associated with the pair of se-quences {{ c n } ∞ n =1 , { d n } ∞ n =1 } where c n = ( − n c, c ∈ R and { d n } ∞ n =1 is a positive chain sequence.Then, the support of µ lie within C ∪ C . Proof.
Without loss of generality we assume that c ≥ . Consider the polynomials R n ( z ) givenby (1.2). If we show that all zeros of R n ( z ) lie on C ∪ C , then from Theorem 2.3 we obtain thedesired result. To show this, we use the functions W n ( x ) defined in (1.4) which are associatedto the polynomials R n ( z ) . By Lemma 3.1 we have that W n +1 ( x ) = ( x + c √ − x ) f W n ( x ) with f W n ( x ) an evenpolynomial of degree 2 n. Moreover, W n ( x ) is also an even polynomial of degree 2 n. This meansthat − c √ c is always a zero of W n +1 ( x ) and the other 2 n zeros of these functions have asymmetry about the origin. Likewise, W n ( x ) being an even polynomial, all of their zeros aresymmetric with respect to the origin.Therefore, from the symmetry of the zeros observed above and taking into account theinterlacing property (1.5) it follows that all zeros of W n ( x ) lie in (cid:16) − , − c √ c i ∪ h c √ c , (cid:17) . Finally, if we denote the zeros of W n ( x ) by x n,j and the zeros of R n ( z ) by z n,j , then theyare related by x n,j = cos (cid:16) θ n,j (cid:17) where z n,j = e iθ n,j , j = 1 , , . . . , n. This shows that R n ( z ) hasall of its zeros on C ∪ C . Notice that Theorem 3.3 leads to an estimative for the support of the measure in the casewhere c n = ( − n ˜ c n and ˜ c n is a constant sequence. We use this initial estimative as motivationto obtain a more general result. 7 heorem 3.4 Let µ be the probability measure on the unit circle associated with the pair ofsequences {{ c n } ∞ n =1 , { d n } ∞ n =1 } where c n = ( − n ˜ c n , ˜ c n ≥ c > and { d n } ∞ n =1 is a positive chainsequence. Then, the support of µ lie within C ∪ C . Proof.
Firstly, notice that for ε with 0 < ε < c, we have ˜ c n ≥ c > c ε > , where c ε = c − ε .Since ˜ c n > c ε > , for x = − c ε √ c ε and x = c ε √ c ε , one can observe that, for n ≥ (cid:18) x − ˜ c n q − x (cid:19) = sign (cid:18) x − ˜ c n q − x (cid:19) = − (cid:18) x + ˜ c n q − x (cid:19) = sign (cid:18) x + ˜ c n q − x (cid:19) = 1 . (3.5)Now, we will show that all zeros of W n ( x ) lie in ( − , x ] ∪ [ x , − ˜ c √ c isthe only zero of W ( x ) in ( − ,
1) and that ˜ c > c ε , it follows that the result is valid for n = 1 . In addition, from (3.5), sign( W ( x j )) = 1 , j ∈ { , } . Hence, from the three term recurrence relation (1.4) for W ( x ) and from (3.4) we concludethat sign( W ( x j )) = − , j ∈ { , } . Suppose that there exists at least one zero of W ( x ) insidethe interval ( x , x ) . Then, since W ( x ) < W ( x ) < W ( x ) has twozeros in ( x , x ) . But this cannot happen because the only zero of W ( x ) is outside of ( x , x )and the two zeros of W ( x ) interlace with the zero of W ( x ). Thus, the result also holds for n = 2 . Again, from the recurrence relation for W ( x ) , sign( W ( x j )) = − , sign( W ( x j )) = 1 , j ∈{ , } , and by (3.5) it follows that sign( W ( x j )) = − , j ∈ { , } . Hence, using the interlacingproperty for the zeros of W ( x ) and W ( x ) , and the fact that there exist no zeros of W ( x ) in( x , x ) , it follows that W ( x ) cannot vanish in ( x , x ) . Continuing this procedure, by mathematical induction, one can easily see thatsign( W n ( x j )) = ( − ⌊ n/ ⌋ , j ∈ { , } , n = 0 , , , . . . and, by the same arguments used before, W n ( x ), n ≥
1, cannot vanish in ( x , x ) . Finally, by letting ε → , we see that W n ( x ) has all its zeros in (cid:16) − , − c √ c i ∪ h c √ c , (cid:17) or,equivalently, R n ( z ) , n ≥ , has all of its zeros on C ∪ C . Now, the result follows by Theorem2.3.
Corollary 3.4.1
Let µ be the probability measure on the unit circle associated with the pair ofsequences {{ c n } ∞ n =1 , { d n } ∞ n =1 } , where c n = ( − n ˜ c n , ˜ c n ≤ c < and { d n } ∞ n =1 is a positive chainsequence. Then, the support of µ lie on C ∪ C . Proof.
First, one can observe that − c n = ( − n ( − ˜ c n ) , with − ˜ c n ≥ − c > . Hence, if ˆ µ is theprobability measure associated to the pair {{− c n } ∞ n =1 , { d n } ∞ n =1 } , from Theorem 3.4 it followsthat ˆ R n ( z ) given by (3.3) has all zeros on C ∪ C and that ˆ µ has its support within C ∪ C . Nowthe result is an immediate consequence of Lemma 3.2.Now we consider the measure µ associated the the pair {{ c n } ∞ n =1 , { d n } ∞ n =1 } , where { c n } ∞ n =1 satisfy the condition c n = − c n − , n ≥ . Starting from µ we desire to get a new measure ˜ µ associated with the pair n { ˜ c n } ∞ n =1 , { ˜ d n } ∞ n =1 o ,where the sequence { ˜ c n } ∞ n =1 must satisfy the condition ˜ c n = ˜ c n − = c n , n ≥ . Let us consider the sequence of complex numbers { β n } ∞ n =1 given by β n = − (cid:18) ic n − ic n (cid:19) , n = 1 , , . . . . (3.6)8he next theorem shows how to get the required measure ˜ µ from the measure µ. Theorem 3.5
Let µ be the probability measure on the unit circle associated with the pair ofsequences {{ c n } ∞ n =1 , { d n } ∞ n =1 } where c n = − c n − , n ≥ . Let { β n } ∞ n =1 be the sequence of com-plex numbers defined by (3.6) . In addition, let ˜ µ be the measure associated with the sequence ofVerblunsky coefficients { ˜ α n } ∞ n =0 given by ˜ α n +1 = n +1 Y j =1 β j α n +1 and ˜ α n = n Y j =1 β j β n +1 α n , n = 0 , , , . . . , (3.7) where { α n } ∞ n =0 is the sequence of Verblunsky coefficients corresponding to µ. If n { ˜ c n } ∞ n =1 , { ˜ d n } ∞ n =1 o is the pair of sequences associated with the measure ˜ µ and if { ˜ m n } ∞ n =0 is the minimal parametersequence for { ˜ d n } ∞ n =1 , then the following holds ˜ c n = ˜ c n − = c n , ˜ m n − = 1 − m n − and ˜ m n = m n , n = 1 , , . . . . Proof.
Using the assumption c n = − c n − , we obtain τ n = 1 and τ n +1 = 1 + ic n +2 − ic n +2 , n = 0 , , . . . . (3.8)Hence, from (2.1) and (3.8), we have α n = 1 − m n +1 + ic n +2 ic n +2 and α n +1 = 1 − m n +2 − ic n +2 ic n +2 , n = 0 , , . . . . (3.9)Now let { ˆ m n } ∞ n =0 be the minimal parameter sequence for a positive chain sequence { ˆ d n } ∞ n =1 and { ˆ α n } ∞ n =0 the Verblunsky coefficients of a probability measure on the unit circle, ˆ µ , associatedwith the pair of real sequences n { ˆ c n } ∞ n =1 , { ˆ d n } ∞ n =1 o , whereˆ c n = ˆ c n − = c n , ˆ m n − = 1 − m n − and ˆ m n = m n , n = 1 , , . . . . (3.10)Using the relations (2.1), (3.6), (3.7), (3.9) and (3.10), one can see that for n = 0 , , . . . , ˆ α n +1 = (cid:18) i ˆ c n +1 − i ˆ c n +1 (cid:19) n Y k =1 i ˆ c k − i ˆ c k ! (cid:20) − m n +2 − i ˆ c n +2 − i ˆ c n +2 (cid:21) = (cid:18) ic n +2 − ic n +2 (cid:19) n Y j =1 (cid:18) ic j − ic j (cid:19) (cid:20) − m n +2 − ic n +2 ic n +2 (cid:21) = n +1 Y j =1 β j α n +1 = ˜ α n +1 . Similarly, using again (2.1), (3.6), (3.7), (3.9) and (3.10), we obtain for n = 0 , , . . . , ˆ α n = n Y k =1 i ˆ c k − i ˆ c k ! (cid:20) − m n +1 − i ˆ c n +1 − i ˆ c n +1 (cid:21) = n Y j =1 (cid:18) ic j − ic j (cid:19) (cid:20) − (cid:18) ic n +2 − ic n +2 (cid:19)(cid:21) (cid:20) − m n +1 + ic n +2 ic n +2 (cid:21) = n Y j =1 β j β n +1 α n = ˜ α n . Thus, ˜ α n = ˆ α n for n ≥ µ = ˆ µ . Hence, from the uniqueness of the pair n { ˜ c n } ∞ n =1 , { ˜ d n } ∞ n =1 o given by Theorem 2.1, we have ˜ m = ˆ m = 0 , ˜ c n = ˆ c n and ˜ m n = ˆ m n , n = 1 , , . . . , which completes the proof of the theorem. 9 orollary 3.5.1 Let µ be the probability measure on the unit circle associated with the pairof sequences {{ c n } ∞ n =1 , { d n } ∞ n =1 } , where c n = ( − n c, n ≥ and c ∈ R . In addition, let β = − (cid:16) ic − ic (cid:17) and ˜ µ ( z ) = µ ( βz ) the measure associated with the pair n { ˜ c n } ∞ n =1 , { ˜ d n } ∞ n =1 o . Then,for n ≥ , ˜ c n = c. Proof.
First, notice that if ˜ µ ( z ) = µ ( βz ) , the corresponding Verblunsky coefficients are relatedby ˜ α n = β n +1 α n , n ≥ c n = c, n ≥ . The first theorem in this section gives a characterization of measures with periodic Verblunskycoefficients in terms of the pair of real sequences {{ c n } ∞ n =1 , { d n } ∞ n =1 } , where { d n } ∞ n =1 is a positivechain sequence. Throughout in this section b n = 1 − m n , n ≥ , where { m n } ∞ n =0 is the minimalparameter sequence of { d n } ∞ n =1 . Theorem 4.1
Let µ be the probability measure on the unit circle associated with the pair ofsequences {{ c n } ∞ n =1 , { d n } ∞ n =1 } . Then, the measure µ has periodic Verblunsky coefficients { α n } ∞ n =0 of period p if, and only if, for n ≥ , n + p X j = n +1 arg (cid:18) ic j − ic j (cid:19) = arg (cid:18) b n +1 − ic n +1 − ic n +1 (cid:19) − arg (cid:18) b n + p +1 − ic n + p +1 − ic n + p +1 (cid:19) + 2 k n π, k n ∈ Z (4.1) and b n +1 + c n +1 c n +1 = b n + p +1 + c n + p +1 c n + p +1 . (4.2) Proof.
First one can observe, from (2.1), that for n ≥ α n + p = α n ⇔ τ n + p (cid:20) b n + p +1 − ic n + p +1 − ic n + p +1 (cid:21) = τ n (cid:20) b n +1 − ic n +1 − ic n +1 (cid:21) ⇔ n + p Y j = n +1 ic j − ic j (cid:20) b n + p +1 − ic n + p +1 − ic n + p +1 (cid:21) = (cid:20) b n +1 − ic n +1 − ic n +1 (cid:21) . Now the result follows by comparing, respectively, the modulus and the argument of thenumbers n + p Y j = n +1 ic j − ic j (cid:20) b n + p +1 − ic n + p +1 − ic n + p +1 (cid:21) and (cid:20) b n +1 − ic n +1 − ic n +1 (cid:21) , n ≥ . We say that µ is a symmetric measure if dµ ( z ) = − dµ (1 /z ) , z ∈ T . From results establishedin [3] one can observe that µ is symmetric if and only if c n = 0, n ≥
1, with { c n } ∞ n =1 given as inTheorem 2.1. Thus, as a consequence of Theorem 4.1, we have the following result. Corollary 4.1.1
Let µ be the probability measure on the unit circle associated with the pair ofsequences {{ c n } ∞ n =1 , { d n } ∞ n =1 } , where { c n } ∞ n =1 and { m n } ∞ n =1 are periodic sequences of period p. In addition, suppose that c n = − c n − , n ≥ . Then, (i) if p is even the measure µ has p − periodic sequence of Verblunsky coefficients; if p is odd, the measure µ is symmetric and has p − periodic sequence of Verblunsky coeffi-cients. Proof. (i) Clearly, we have that (4.1) and (4.2) hold. Hence the result follows by Theorem4.1.(ii) If p is odd, using the periodicity of c n and the assumption that c n = − c n − , we concludethat c n = 0, n ≥ . Hence, µ is symmetric. Moreover, since { m n } ∞ n =1 is a periodic sequence ofperiod p and c n = 0, n ≥
1, the conditions (4.1) and (4.2) of Theorem 4.1 can be easily verified.Consequently, the measure µ has p − periodic sequence of Verblunsky coefficients.The Corollary 4.1.1 shows that if we choose the sequence { c n } ∞ n =1 p − periodic ( p even) andsuch that c n = − c n − , then it is possible, by choosing { m n } ∞ n =1 also p − periodic, to get ameasure µ ( p ) whose Verblunsky coefficients are periodic with the same period. Notice that inthe case when c n = − c n − and c n > c n <
0) for n ≥ { c n } ∞ n =1 has thealternating sign property.The next theorem provides a geometric characterization for the choice of { c n } ∞ n =1 and { m n } ∞ n =1 considered above. Theorem 4.2
Let p be an even natural number and µ ( p ) be the probability measure associatedwith the pair {{ c n } ∞ n =1 , { d n } ∞ n =1 } . Then, the following statements are equivalent: (i)
The sequences { c n } ∞ n =1 and { m n } ∞ n =1 are p − periodic with c n = − c n − , n ≥ . (ii) The sequence of Verblunsky coefficients { α n } ∞ n =0 associated with the measure µ ( p ) is p − periodic. In addition, for k ∈ { , , . . . , p − } , the straight lines connecting α k to and α k +1 to − are parallel. Proof. (i) ⇒ (ii) From Corollary 4.1.1 it is immediate that { α n } ∞ n =0 is a periodic sequence withperiod p. On the other hand, by the assumption that c n = − c n − and by (2.1), for n ≥ , wehave α n = b n +1 + ic n +2 ic n +2 = 1 + λ n ( − − ic n +1 ) , where λ n = − b n +1 c n +1 . Similarly, for n ≥ α n +1 = b n +2 − ic n +2 ic n +2 = − λ n +1 ( − ic n +2 ) , where λ n +1 = − b n +2 c n +2 . Hence, for each k ∈ { , , . . . , p − } , one can see that α k ∈ r k , where r k is the straight linewith parametric equation given by r k ( t ) = 1 + t ( − − ic k +1 ) , t ∈ R . Similarly, for each k ∈ { , , . . . , p − } , one can see that α k +1 ∈ r k +1 , where r k +1 is thestraight line with parametric equation given by r k +1 ( t ) = 1 + t ( − ic k +2 ) , t ∈ R . Finally, since − − ic k +1 = − ic k +2 it follows that r k k r k +1 , for each k ∈ { , , . . . , p − } . (ii) ⇒ (i) Let α j = x j + iy j , j = 0 , , . . . , p − . If j = 2 k, k ∈ { , , . . . , p − } , we can write α k = 1 + λ k ( − − i ˜ c k +1 ) , (4.3)where λ k = 1 − ˜ b k +1 c k +1 , ˜ c k +1 = y k x k − b k +1 = 1 + ( x k − + y k x k − . (4.4)11ikewise, if j = 2 k + 1 , k ∈ { , , . . . , p − } , we can write α k +1 = − λ k +1 ( − i ˜ c k +2 ) , (4.5)where λ k +1 = − b k +2 c k +2 , ˜ c k +2 = − y k +1 x k +1 and ˜ b k +2 = − x k +1 ) + y k +1 x k +1 . (4.6)Hence, if we set ˜ b n = 1 − m n, from α n + p = α n , (4.4) and (4.6) one can see that˜ c n + p = ˜ c n and ˜ m n + p = ˜ m n , n = 1 , , . . . . (4.7)For each k ∈ { , , . . . , p − } , let r k be the straight line connecting α k to 1 and r k +1 thestraight line connecting α k +1 to − . Then, from (4.3), (4.5), (4.7) and since r k k r k +1 , k ∈{ , , . . . , p − } , it follows that ˜ c n +2 = − ˜ c n +1 , n = 0 , , . . . . (4.8)Hence, from (4.3) to (4.8) we have, for n ≥ ,α n = ˜ b n +1 + i ˜ c n +2 i ˜ c n +2 and α n +1 = ˜ b n +2 − i ˜ c n +2 i ˜ c n +2 . (4.9)Finally, using the formula (2.1) for α n and the relation (4.9) one can see, by mathematicalinduction, that for n ≥ , ˜ c n = c n and ˜ m n = m n . This completes the proof.Observe that Theorem 4.2 shows that to choose a periodic sequence { α n } of period p ( p even) with α j on certain parallel straight lines is equivalent to choosing the sequences { c n } and { m n } also p − periodic with the additional property c n +2 = − c n +1 , n ≥ . In Fig. 1 and Fig.2 we show some examples of possible choices for { c n } and { m n } . Figure 1:
Verblunsky coefficients associated tothe choice { c n } = ( − c, c, − c, c, . . . ) and { b n } =( b , b , b , b , . . . ), with c > . Figure 2:
Verblunsky coefficients associated tothe choice { c n } = ( − c , c , − c , c , − c , c , . . . )and { b n } = ( b , b , b , b , b , b , . . . ), with c < c > µ ( p ) whose associated Verblunsky coef-ficients are periodic. In [19] there is another approach to the same problem.We begin with a lemma that leads to a characterization of the possible pure points (that wedenote by w ) of the measure µ ( p ) in terms of the sequence { τ n ( w ) } defined in (2.2). Lemma 4.3
Let µ ( p ) be a probability measure on the unit circle with p − periodic Verblunskycoefficients. Then, w is a possible pure point of the measure µ ( p ) if, and only if, the sequence { τ n ( w ) } ∞ n =0 is periodic of period p. Proof.
By Theorem 2.6 and Theorem 2.7 we see that w is a possible pure point of µ ( p ) if,and only if, ϕ p ( w ) − ϕ ∗ p ( w ) = 0 . Notice that the condition ϕ p ( w ) − ϕ ∗ p ( w ) = 0 is equivalent to τ p ( w ) = 1 . Furthermore, using the periodicity of the sequence { α n } ∞ n =0 and the recurrence relation (2.2),we also see that τ p ( w ) = 1 is equivalent to the periodicity of the sequence { τ n ( w ) } ∞ n =0 . The next theorem provides a way to determinate all the pure points of the measure µ ( p ) andalso, to calculate the mass of each pure point. Theorem 4.4
Let µ ( p ) be a probability measure on the unit circle with p − periodic sequence { α n } ∞ n =0 of Verblunsky coefficients. In addition, suppose that w is a point on the unit circle suchthat ϕ p ( w ) − ϕ ∗ p ( w ) = 0 . Then, w is a pure point of µ ( p ) if, and only if, p Y j =1 | − wτ j − ( w ) α j − | < p Y j =1 (cid:2) − | α j − | (cid:3) . Moreover, if w is a pure point of µ ( p ) , then the mass at this point is given by µ ( p ) ( { w } ) = γγ + δ , where δ = p X n =1 n Y j =1 | − wτ j − ( w ) α j − | − | α j − | and γ = 1 − p Y j =1 | − wτ j − ( w ) α j − | − | α j − | . Proof.
For j = 1 , , . . . , let q j = | − wτ j − ( w ) α j − | −| α j − | . By Theorem 2.2 we know that w is a pure point if, and only if, the infinite sum λ ( w ) = P ∞ n =1 Q nj =1 q j is convergent.By Lemma 4.3 and by the periodicity of { α n } ∞ n =0 it follows that q j + p = q j , j ≥ . Thus, if q = p Y j =1 q j , we can write λ ( w ) as λ ( w ) = q ∞ X n =0 q n ! + q q ∞ X n =0 q n ! + · · · + q q · · · q p ∞ X n =0 q n ! . (4.10)Observe that λ ( w ) is convergent if, and only if, | q | < . Thus, the first part of the statementfollows.Furthermore, if | q | < λ ( w ) = (cid:18) − q (cid:19) p X n =1 n Y j =1 q j = δγ . (4.11)13inally, by Theorem 2.2 and (4.11), we get µ ( p ) ( { w } ) = 11 + λ ( w ) = γγ + δ . In this section we discuss, using the following example, the results obtained in the previoussections.Let the real sequences { c n } ∞ n =1 and { d n } ∞ n =1 be given by c n = ( − n c and d n = (1 − m n − ) m n , n ≥ , where c ∈ R and the real sequence { m n } ∞ n =0 is such that m = 0, m n − = 1 − b m n = 1 − b , n ≥ , with b , b ∈ R and | b | , | b | < c = 0 , { c n } ∞ n =1 has the alternating sign property and that { d n } ∞ n =1 is a positivechain sequence, with { m n } ∞ n =0 being its minimal parameter sequence. Moreover, { c n } ∞ n =1 and { m n } ∞ n =1 are periodic sequences of period 2.By Theorem 2.1, associated with the pair {{ c n } ∞ n =1 , { d n } ∞ n =1 } , there exists an unique prob-ability measure, say µ (2) , on the unit circle. Furthermore, from Corollary 4.1.1 follows that thesequence of Verblunsky coefficients of µ (2) is periodic with period 2 (in Fig. 1, it is illustratedthe position of these coefficients for the case c > { c n } ∞ n =1 one can also see that τ n = 1 and τ n +1 = 1 + ic − ic , n ≥ . (5.1)Thus, from (2.1) we have, for n ≥ α n = b + ic ic = ( b + c ) + ic (1 − b )1 + c and α n +1 = b − ic ic = ( b − c ) − ic (1 + b )1 + c . In this case, since p = 2 , we have ∆( z ) = z − Tr( T ( z )) . By (2.4) and (2.5) T ( z ) = (1 − | α | ) − / (1 − | α | ) − / (cid:18) z − α − α z (cid:19) (cid:18) z − α − α z (cid:19) . Hence, computing ∆( e iθ ) one can see that, for θ ∈ [0 , π ) , ∆( e iθ ) = 2 (cid:26) c [(1 − b )(1 − b )] / cos θ + b b − c [(1 − b )(1 − b )] / (cid:27) and, consequently q − ∆ ( e iθ ) = 2 s − (cid:20) (1 + c ) cos θ + b b − c (1 − b ) / (1 − b ) / (cid:21) . Furthermore, considering the normalized orthogonal polynomials ϕ ( z ) = κ φ ( z ) one canalso verify that ϕ ( z ) = 1(1 − b ) / (1 − b ) / (cid:8) (1 + c ) z + [( b b − b − c ) + ic ( b + 1)] z + [( c − b ) − ic ( b + 1)] (cid:9) θ ∈ [0 , π ) , Im( e − iθ ϕ ( e iθ )) = (1 + b )[sin θ + c (1 − cos θ )](1 − b ) / (1 − b ) / . Hence, from Theorem 2.5, the weight function w ( θ ) associated to µ (2) is such that w ( θ ) = p (1 − b )(1 − b ) − [(1 + c ) cos θ + b b − c ] | (1 + b )[sin θ + c (1 − cos θ )] | . Now we need to compute the bands B and B for the measure µ (2) . By solving the equation∆( e iθ ) = 2 we find the solutions θ +1 = arccos (1 − b ) / (1 − b ) / + c − b b c ! and θ +2 = 2 π − θ +1 . Likewise, by solving ∆( e iθ ) = − θ − = arccos c − (1 − b ) / (1 − b ) / − b b c ! and θ − = 2 π − θ − . Thus, each band B j is determined by the points z + j = e iθ + j and z − j = e iθ − j , j ∈ { , } . To determine the possible pure points of µ (2) , by Theorem 2.6 and Theorem 2.7 we need tosolve the equation ϕ ( z ) − ϕ ∗ ( z ) = 0 , whose solutions are w = 1 and w = c − c − i c c . Now looking at the bands B j and for the possible pure points w j , it is not hard to see thatthe measure µ (2) is always supported on C ∪ C , in accordance with Theorem 3.4.Finally, we give a complete characterization about the singular part of the measure µ (2) interms of the parameters b , b and c. Firstly, we analyze the point w = 1 . Notice that τ n ( w ) = τ n given by (5.1) is pe-riodic of period 2 , according to Lemma 4.3. From Theorem 4.4 one can see that w is apure point of µ (2) if, and only if, b + b > . Moreover, if δ = X n =1 n Y j =1 | − τ j − α j − | − | α j − | and γ = 1 − Y j =1 | − τ j − α j − | − | α j − | , again by Theorem 4.4 we obtain µ (2) ( { w } ) = γ γ + δ = b + b b . Now we consider the point w = c − c − i c c = − ic − ic . From Corollary 3.5.1, if ˜ µ ( z ) = µ ( w z ) , we have ˜ c n = c, n ≥ . Moreover,˜ τ n = n Y k =1 − i ˜ c k i ˜ c k = (cid:18) − ic ic (cid:19) n , n ≥ . On the other hand, it is known (see, for example, [6]) that ˜ τ n = w − n τ n ( w ) , n ≥ . Hence,one can see that τ n ( w ) = ( − n , n ≥ . Thus, it follows that τ n ( w ) is periodic of period 2 , according to Lemma 4.3.From Theorem 4.4, w is a pure point of µ (2) if, and only if, b − b > . Moreover, if δ = X n =1 n Y j =1 | − w τ j − ( w ) α j − | − | α j − | and γ = 1 − Y j =1 | − w τ j − ( w ) α j − | − | α j − | , we obtain µ (2) ( { w } ) = γ γ + δ = b − b b . , , Support of µ (2) in the case 0 < c < b > b > Figure 4:
Support of µ (2) in the case 0 < c < < b ≤ − b . Figure 5:
Support of µ (2) in the case 0 < c < < b ≤ b . Figure 6:
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