On perturbed orthogonal polynomials on the real line and the unit circle via Szegő's transformation
aa r X i v : . [ m a t h . C A ] M a y Noname manuscript No. (will be inserted by the editor)
On perturbed orthogonal polynomials on the real lineand the unit circle via Szeg˝o’s transformation
K. Castillo · F. Marcell´an · J. Rivero
Received: date / Accepted: date
Abstract
By using the Szeg˝o’s transformation we deduce new relations be-tween the recurrence coefficients for orthogonal polynomials on the real lineand the Verblunsky parameters of orthogonal polynomials on the unit circle.Moreover, we study the relation between the corresponding S -functions and C -functions. Keywords
Szeg˝o transformation · co-polynomials · spectral transformations · transfer matrices Mathematics Subject Classification (2000) · dµ be a non-trivial probability measure supported on I ⊆ R . The sequenceof polynomials { p n } n > where p n ( x ) = γ n x n + δ n x n − + (lower degree terms) , γ n > , K. CastilloCMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, PortugalE-mail: [email protected]; [email protected]. Marcell´anInstituto de Ciencias Matem´aticas (ICMAT) and Departamento de Matem´aticas, Universi-dad Carlos III de Madrid, 28911 Legan´es, Madrid, SpainE-mail: [email protected]. RiveroDepartamento de Matem´aticas, Universidad Carlos III de Madrid, 28911 Legan´es, Madrid,SpainE-mail: [email protected] K. Castillo et al. is said to be an orthonormal polynomial sequence with respect to dµ if Z I p n ( x ) p m dµ ( x ) = δ n,m , m ≥ . The corresponding monic orthogonal polynomials (with leading coefficientequal to 1) are P n ( x ) = p n ( x ) /γ n , see [4,17]. These polynomials satisfy thefollowing three-term recurrence relation P n +1 ( x ) = ( x − b n +1 ) P n ( x ) − d n P n − ( x ) , d n = 0 , d = 1 , n > , (1.1)where the recurrence coefficients are given by b n = δ n γ n − δ n +1 γ n +1 , d n = a n , a n = γ n − γ n > , n ≥ . Notice that the initial conditions P − ( x ) = 0 and P ( x ) = 1 hold. The three-term recurrence relation (1.1) is often represented in matrix form x P ( x ) = JP ( x ) , P = [ P , P , . . . ] T , where J is a semi-infinite tridiagonal matrix J = b d b d b d b . . .. . . . . . , which is called the monic Jacobi matrix [12].The Stieltjes or Cauchy transformation of the orthogonality measure dµ isdefined by S µ ( x ) = Z I dµ ( y ) x − y , x ∈ C \ I. It has a particular interest in the theory of orthogonal polynomials on the realline (OPRL, in short). S µ ( x ) admits the following series expansion S µ ( x ) = ∞ X k =0 u k x k +1 , where u k are the moments associated with dµ , i.e., u k = Z I x k dµ ( x ) . By a spectral transformation of the S -function S µ ( x ), we mean a new S -function associated with a measure d e µ , a modification of the original measure n perturbed orthogonal polynomials on the real line and the unit circle 3 dµ . We refer to pure rational spectral transformation as a transformation of S µ ( x ) given by S σ ( x ) ˙= A ( x ) S µ ( x ) , A ( x ) = (cid:20) a ( x ) b ( x ) c ( x ) d ( x ) (cid:21) , (1.2)where a ( x ), b ( x ), c ( x ), and d ( x ) are non-zero polynomials that provide a ’true’asymptotic behavior to (1.2) (see [20]). In (1.2), we adopt the notation ˙=introduced in [17], i.e., for the homography mapping f ( x ) = a ( x ) g ( x ) + b ( x ) c ( x ) g ( x ) + d ( x ) , a ( x ) d ( x ) − b ( x ) c ( x ) = 0 , we will write f ( x ) ˙= A ( x ) g ( x ) . dσ be a non-trivial probability measure supported on the unit circle T = { z ∈ C : | z | = 1 } parametrized by z = e iθ . There exists a unique sequence { φ n } n > of orthonormal polynomials φ n ( z ) = κ n z n + (lower degree terms) , κ n > , such that Z π − π φ n ( e iθ ) φ m ( e iθ ) dσ ( θ ) = δ n,m , m ≥ . The corresponding monic polynomials are defined by Φ n ( z ) = φ n ( z ) /κ n . Thesepolynomials satisfy the following recurrence relations (see [9,17,19]) Φ n +1 ( z ) = zΦ n ( z ) − α n Φ ∗ n ( z ) , n ≥ , (1.3) Φ ∗ n +1 ( z ) = Φ ∗ n ( z ) − α n zΦ n ( z ) , n ≥ , (1.4)with initial condition Φ ( z ) = 1. The polynomial Φ ∗ n ( z ) = z n Φ n ( z − ) isthe so-called reversed polynomial and the complex numbers { α n } n ≥ where α n = − Φ n +1 (0), are known as Verblunsky, Schur, Geronimus, or reflectionparameters. Let notice that | α n | <
1. If we replace in (1.3) the sequence { α n } n ≥ by {− α n } n ≥ , then we obtain the sequence of second kind polyno-mials { Ω n } n > .The Riesz-Herglotz transform of the measure dσ is given by F σ ( z ) = Z π − π e iθ + ze iθ − z dσ ( θ ) . Since F σ (0) = 1 and ℜ F σ ( z ) > D = { z ∈ C : | z | < } , F σ ( z ) is called a Carath´eodory function [10], or, simply, C -function. Let c k bethe k -th moment associated with the measure dσ , i.e, c k = Z π − π e − ikθ dσ ( θ ) . K. Castillo et al. F σ ( z ) can be written in terms of the moments { c n } n > as follows F ( z ) = 1 + 2 ∞ X k =1 c k z k . As for the real line case, by a spectral transformation of a C -function F σ ( z )we mean a new C -function associated with a measure dψ , a modification ofthe original measure dσ . We refer to pure rational spectral transformation asa transformation of F σ ( z ) given by F ψ ( z ) ˙= E ( z ) F σ ( z ) , E ( z ) = (cid:20) A ( z ) B ( z ) C ( z ) D ( z ) (cid:21) , (1.5)where A ( z ), B ( z ), C ( z ), and D ( z ) are non-zero polynomials that provide a’true’ behavior to (1.5) around the origin (see [2]).1.3 Szeg˝o transformation and Geronimus relationsLet us assume that the measure dµ is supported on the interval [ − , dσ such that dσ ( θ ) = 12 | dµ (cos θ ) | . In particular, if dµ is an absolutely continuous measure, i.e., dµ ( x ) = ω ( x ) dx ,we have dσ ( θ ) = 12 ω (cos θ ) | sin θ | dθ. This is the so-called Szeg˝o transformation of probability measures supportedon [ − ,
1] to probability measures supported on T . We write the relation be-tween dµ and dσ through the Szeg˝o transformation as σ = Sz( µ ). Of course,under the previous considerations, we get α n ∈ ( − , , n > . There is a relation between the OPRL associated with a measure dµ supportedon [ − ,
1] and the OPUC associated with the measure σ = Sz( µ ) supportedon T , p n ( x ) = κ n p − α n − ) (cid:0) z − n Φ n ( z ) + z n Φ n (1 /z ) (cid:1) . (1.6)From (1.6) one can obtain a relation between the coefficients of the corre-sponding recurrence relations, see [18], d n +1 = 14 (1 − α n − ) (cid:0) − α n (cid:1) (1 + α n +1 ) , n ≥ , (1.7) b n +1 = 12 [ α n (1 − α n − ) − α n − (1 + α n − )] , n ≥ , (1.8) n perturbed orthogonal polynomials on the real line and the unit circle 5 with the convention α − = −
1. Notice that b n ≡ , n ≥
1, if and only if α n = 0 , n ≥ S -function and the C -function associ-ated with dµ and dσ , respectively, as follows F ( z ) = 1 − z z S ( x ) , or, equivalently, S ( x ) = F ( z ) √ x − , with 2 x = z + z − and z = x − √ x − k onthe real line and the corresponding sequences of monic orthogonal polynomialsobtained on the unit circle via the Szeg˝o transformation. In Section 3, we studythe relation between the associated and anti-associated polynomials of order k on the unit circle and the corresponding sequences of monic orthogonal poly-nomials obtained on the real line via the inverse of the Szeg˝o transformation.In Section 4, we explore the relation between the co-polynomials on the realline (on the unit circle), and the corresponding sequences of monic orthogonalpolynomials obtained on the unit circle (on the real line) via Szeg˝o trans-formation. In Section 5 we investigate the relations between the symmetricpolynomials on [-1,1] and sieved polynomials, and the corresponding sequenceof monic orthogonal polynomials on the unit circle (on the real line) throughthe Szeg˝o transformation. From the sequence of monic orthogonal polynomials { P n } n > we can definethe sequence of associated monic polynomials of order k [8], { P ( k ) n } n > , k > P ( k ) n +1 ( x ) = ( x − b n + k +1 ) P ( k ) n ( x ) − d n + k P ( k ) n − ( x ) , n > , with P ( k ) − ( x ) = 0 and P ( k )0 ( x ) = 1.Here, we study the relation between the associated polynomials of order k on the real line and the corresponding sequence of monic orthogonal polyno-mials obtained on the unit circle via the Szeg˝o transformation. We focus ourattention on the resulting C -function and the Verblumsky coefficients. Theorem 1
Let { ˆ α n } n ≥ be the Verblunsky coefficients for the correspond-ing OPUC related to the associated polynomials of order k on the real line K. Castillo et al. { P ( k ) n } n > , through the Szeg˝o transformation. Then, for a fixed non-negativeinteger k , ˆ α = b k +1 , ˆ α = − d k +1 − ˆ α , ˆ α m = 2 b m + k +1 + (1 + ˆ α m − )ˆ α m − − ˆ α m − , m ≥ , ˆ α m +1 = − d m + k +1 (1 − ˆ α m − )(1 − ˆ α m ) , m ≥ , with ˆ α − = − .Proof Let { ˆ b n } n ≥ and { ˆ d n } n ≥ be the recurrence coefficients for the associ-ated polynomials of order k on the real line { P ( k ) n } n > . From (1.7) and (1.8),for n ≥
0, we have ˆ α n = 2ˆ b n +1 + (1 + ˆ α n − )ˆ α n − (1 − ˆ α n − ) , ˆ α n +1 = − d n +1 (1 − ˆ α n − )(1 − ˆ α n ) . Since ˆ b n = b n + k and ˆ d n = d n + k for all n ≥
1, and assume that ˆ α − = −
1, theresult follows as a consequence of straightforward computations.The S -function S ( k ) ( x ) corresponding to the associated polynomials of order k , can be written as S ( k ) ( x ) ˙= B ( k ) ( x ) S ( x ) , (2.9)where B ( k ) ( x ) = P k ( x ) − P (1) k − ( x ) d k P k − ( x ) − d k P (1) k − ( x ) . Then, applying the Szeg˝o transformation to (2.9) we have the following result.
Theorem 2
Let ˆ F ( k ) ( z ) be the C -function for the corresponding associatedpolynomials of order k through the Szeg˝o transformation. Then z − z ˆ F ( k ) ( z ) ˙= B ( k ) (cid:18) z + z − (cid:19) (cid:18) z − z F ( z ) (cid:19) , with x = z + z − . As a direct consequence of the above theorem, for k = 1 we have a resultproved in [7]. n perturbed orthogonal polynomials on the real line and the unit circle 7 Corollary 1 If F Ω ( z ) denotes the C -function corresponding to the associatedpolynomials of the second kind Ω n ( z ) on T , then ˆ F (1) ( z ) = − (1 − z ) F Ω ( z ) + (1 − z )( z − b z + 1)4 d z . Notice that F Ω ( z ) = F ( z ) . Let us consider a new family of orthogonal polynomials, { P ( − k ) n } n > , whichis obtained by introducing new coefficients b − i ( i = k − , k − , . . . ,
0) on thediagonal, and d − i ( i = k − , k − , . . . ,
0) in the lower subdiagonal of the Jacobimatrix. These polynomials are called anti-associated polynomials of order k ,and were analyzed in [16].The relation between the recurrence coefficients of the anti-associated poly-nomials of order k on the real line and the Verblunsky coefficients for thecorresponding sequence of monic orthogonal polynomials obtained on the unitcircle via the Szeg˝o transformation can be stated as follows. Theorem 3
Let { ˜ α n } n ≥ be the Verblunsky coefficients for the correspon-dingOPUC related to the anti-associated polynomials of order k on the real line { P ( − k ) n } n > , through the Szeg˝o transformation. Then, for a fixed non-negativeinteger k , ˜ α = b − k , ˜ α = − d − k − ˜ α , ˜ α m = 2 b m − k +1 + (1 + ˜ α m − )˜ α m − − ˜ α m − , m ≥ , ˜ α m +1 = − d m − k +1 (1 − ˜ α m − )(1 − ˜ α m ) , m ≥ , with ˆ α − = − .Proof Let { ˜ b n } n ≥ and { ˜ d n } n ≥ be the recurrence coefficients for the anti-associated polynomials of order k on the real line { P ( − k ) n } n > . From (1.7) and(1.8), for n ≥ α n = 2˜ b n +1 + (1 + ˜ α n − )˜ α n − (1 − ˜ α n − ) , ˜ α n +1 = − d n +1 (1 − ˜ α n − )(1 − ˜ α n ) . Since ˜ b n = b n − k and ˜ d n = d n − k for all n ≥
1, and assume that ˜ α − = − S -function S ( − k ) ( x ), corresponding to the anti-associated polynomialsof order k [20], can be written as S ( − k ) ( x ) ˙= B ( − k ) ( x ) S ( x ) , (2.10) K. Castillo et al. where B ( − k ) ( x ) = e d k P ( − k ) k − ( x ) − P ( − k +1) k − ( x ) e d k P ( − k ) k − ( x ) − P ( − k +1) k ( x ) . From (2.10) and the Szeg˝o transformation, we can state the analogue of The-orem 6.
Theorem 4
Let e F ( − k ) ( z ) be the C -function for the corresponding anti-associatedpolynomials of order k through the Szeg˝o transformation. Then z − z e F ( − k ) ( z ) ˙= B ( − k ) (cid:18) z + z − (cid:19) (cid:18) z − z F ( z ) (cid:19) , with x = z + z − . For k = 1, as an analog to corollaryllary 1, we have the result proved in [7]. Corollary 2
Let e F Ω ( z ) be the C -function corresponding to the anti-associatedpolynomials of the second kind on T through the Szeg˝o transformation. Then e F Ω ( z ) = 1 e F ( − ( z ) = e A ( z ) F ( z ) + e B ( z ) e D ( z ) where e A ( z ) = 4 e d z , e B ( z ) = − (1 − z )( z − b z + 1) and e D ( z ) = − (1 − z ) . Let { Φ n } n > be the monic orthogonal polynomial sequence with respect to anontrivial probability measure dσ supported on T . We denote by { Φ ( k ) n } n > be the k -th associated sequence of polynomials of order k > { Φ n } n > , see [14]. In this case they are generated by therecurrence relation Φ ( k ) n +1 ( z ) = zΦ ( k ) n ( z ) − α n + k (cid:16) Φ ( k ) n ( z ) (cid:17) ∗ , n ≥ . Now, we study the relation between the associated polynomials of order k onthe unit circle and the corresponding sequence of monic orthogonal polyno-mials obtained on the real line via the inverse of the Szeg˝o transformation.We focus our attention on the resulting S -function and the parameters of thethree term recurrence relation. Theorem 5
Let { ˆ b n } n ≥ and { ˆ d n } n ≥ be the recurrence coefficients for thecorresponding OPRL related to the associated polynomials of order k on theunit circle { Φ ( k ) n } n > , through the Szeg˝o transformation. Then, for k = 2 m − d = 1 + α m − v m +1 d m +1 , ˆ d n +1 = v n + m ) − v n + m )+1 d n + m +1 , n ≥ , ˆ b = α m − , ˆ b n +1 = b n + m +1 + v n + m ) − − v n + m ) , n ≥ , n perturbed orthogonal polynomials on the real line and the unit circle 9 and for k = 2 m , ˆ d = λd m +1 , ˆ d n +1 = d n + m +1 , n ≥ , ˆ b = α m , ˆ b n +1 = b n + m +1 , n ≥ , with λ = − α m − and v n = (1 + α n )(1 − α n − ) .Proof Let { ˆ α n } n > be the Verblunsky coefficients for the associated polyno-mials of order k with respect to { Φ n } n > . From (1.7) for k = 2 m − n = 0we get ˆ d = 14 (1 − ˆ α − )(1 − ˆ α )(1 + ˆ α ) , = 12 (1 − α m − )(1 + α m ) , = 1 + α m − v m +1 d m +1 . For n ≥
1, ˆ d n +1 = 14 (1 − ˆ α n − )(1 − ˆ α n )(1 + ˆ α n +1 ) , = 14 (1 − α n + m ) − )(1 − α n + m ) − )(1 + α n + m ) ) , = v n + m ) − v n + m )+1 d n + m +1 . On the other hand, from (1.8),ˆ b = 12 [ˆ α (1 − ˆ α − ) − ˆ α − (1 + ˆ α − )] = α m − . For n ≥ b n +1 = 12 [ˆ α n (1 − ˆ α n − ) − ˆ α n − (1 + ˆ α n − )] , = 12 (cid:2) α n + m ) − (1 − α n + m ) − ) − α n + m ) − (1 + α n + m ) − ) (cid:3) , = − v n + m ) − + v n + m ) − , = b n + m +1 + v n + m ) − − v n + m ) . Finally, for k = 2 m , the results follow after similar computations.Consider the C -function F ( k ) ( z ), corresponding to the associated polyno-mials of order k , given by F ( k ) ( x ) ˙= Υ ( k ) ( x ) F ( x ) (3.11)where Υ ( k ) ( x ) = " Φ k ( z ) + Φ ∗ k ( z ) Ω k ( z ) − Ω ∗ k ( z ) Φ k ( z ) − Φ ∗ k ( z ) Ω k ( z ) + Ω ∗ k ( z ) . Then, applying the Szeg˝o transformation to (3.11) we have the following result.
Theorem 6
Let ˆ S ( k ) ( x ) be the S -function for the corresponding associatedpolynomials of order k through the Szeg˝o transformation. Then p x − S ( k ) ( x ) ˙= Υ ( k ) (cid:16) x − p x − (cid:17) (cid:16)p x − S ( x ) (cid:17) , with z = x − √ x − . For k = 2 we have the following result. Corollary 3 ˆ S (2) ( x ) ˙= " P ( x ) − λ − − x ) ( λ − x + b ) S ( x ) , with λ = − α . For more details about this case and the case k = 1 see [6].Let { Φ n } n > be the monic orthogonal polynomial sequence with respectto a nontrivial probability measure dσ supported on T . Let ξ , ξ , . . . , ξ k − be complex numbers with | ξ i | <
1, 0 i k −
1. We denote the anti-associated polynomials of order k of { Φ n } n > , { Φ ( − k ) n } n > , as the sequenceof monic orthogonal polynomials generated by the Verblunsky coefficients { ξ i } k − i =0 S { α n − k } n ≥ k .We now study the relation between the anti-associated polynomials of order k on the unit circle and analyze the corresponding transformation obtained onthe real line using the Szeg˝o transformation. Theorem 7
Let { ˜ b n } n ≥ and { ˜ d n } n ≥ be the recurrence coefficients for thecorresponding OPRL related to the anti-associated polynomials of order k onthe unit circle { Φ ( − k ) n } n > , through the Szeg˝o transformation. Let { ˜ α n } n ≥ = { ξ n } k − n =0 S { α n − k } n ≥ k be the Verblunsky coefficients for { Φ ( − k ) n } n > with ˜ α − = − . Then, for k = 2 m − , ˜ d n +1 = (1 − ξ )(1 + ξ ) , n = 0 , (1 − ξ n − )(1 − ξ n )(1 + ξ n +1 ) , ≤ n ≤ m − , (1 − ξ n − )(1 − ξ n )(1 + α n − m )+2 ) , n = m − , (1 − α n − m ) )(1 − α n − m )+1 )(1 + α n − m )+2 ) , n ≥ m, ˜ b n +1 = ξ , n = 0 , [(1 − ξ n − ) ξ n − (1 + ξ n − ) ξ n − ] , ≤ n ≤ m − , [(1 − α n − m ) ) α n − m )+1 − (1 + α n − m ) ) ξ n − ] , n = m, [(1 − α n − m ) ) α n − m )+1 − (1 + α n − m ) ) α n − m ) − ] , n > m. n perturbed orthogonal polynomials on the real line and the unit circle 11 For k = 2 m , ˜ d n +1 = (1 − ξ )(1 + ξ ) , n = 0 , (1 − ξ n − )(1 − ξ n )(1 + ξ n +1 ) , ≤ n ≤ m − , (1 − ξ n − )(1 − α n − m ) )(1 + α n − m )+1 ) , n = m,d n − m +1 , n > m, ˜ b n +1 = ξ , n = 0 , [(1 − ξ n − ) ξ n − (1 + ξ n − ) ξ n − ] , ≤ n ≤ m − , [(1 − ξ n − ) α n − m ) − (1 + ξ n − ) ξ n − ] , n = m,b n − m +1 , n > m. Proof
From (1.7), for n = 0, we get˜ d = 14 (1 − ˜ α − )(1 − ˜ α )(1 + ˜ α ) = 12 (1 − ξ )(1 + ξ ) . For 1 ≤ n ≤ m − d n +1 = 14 (1 − ˜ α n − )(1 − ˜ α n )(1 + ˜ α n +1 ) , = 14 (1 − ξ n − )(1 − ξ n )(1 + ξ n +1 ) . For n = m −
1, ˜ d n +1 = 14 (1 − ˜ α n − )(1 − ˜ α n )(1 + ˜ α n +1 ) , = 14 (1 − ξ n − )(1 − ξ n )(1 + α n − m )+2 ) . Finally, for n ≥ m ,˜ d n +1 = 14 (1 − ˜ α n − )(1 − ˜ α n )(1 + ˜ α n +1 ) , = 14 (1 − α n − m ) )(1 − α n − m )+1 )(1 + α n − m )+2 ) . On the other hand, from (1.8), for n = 0,˜ b = 12 [(1 − ˜ α − )˜ α − (1 + ˜ α − )˜ α − ] = ξ . For 1 ≤ n ≤ m − b n +1 = 12 [(1 − ˜ α n − )˜ α n − (1 + ˜ α n − )˜ α n − ] , = 12 [(1 − ξ n − ) ξ n − (1 + ξ n − ) ξ n − ] . For n = m ˜ b n +1 = 12 [(1 − ˜ α n − )˜ α n − (1 + ˜ α n − )˜ α n − ] , = 12 [(1 − α n − m ) ) α n − m )+1 − (1 + α n − m ) ) ξ n − ] . Finally, for n > m ,˜ b n +1 = 12 [(1 − ˜ α n − )˜ α n − (1 + ˜ α n − )˜ α n − ] , = 12 [(1 − α n − m ) ) α n − m )+1 − (1 + α n − m ) ) α n − m ) − ] . When, for k = 2 m , the results follow in a similar way.Consider the C -function F ( − k ) ( z ) corresponding to the anti-associated poly-nomials of order k , given by F ( − k ) ( x ) ˙= Υ ( − k ) ( x ) F ( x ) , (3.12)where Υ ( − k ) ( x ) = " e Ω k ( z ) + e Ω ∗ k ( z ) e Ω ∗ k ( z ) − e Ω k ( z ) e Φ ∗ k ( z ) − e Φ k ( z ) e Φ k ( z ) + e Φ ∗ k ( z ) . Then, applying the Szeg˝o transformation to (3.12) we have the following result.
Theorem 8
Let ˜ S ( − k ) ( x ) be the S -function for the corresponding anti-associatedpolynomials of order k through the Szeg˝o transformation. Then p x − S ( − k ) ( x ) ˙= Υ ( − k ) (cid:16) x − p x − (cid:17) (cid:16)p x − S ( x ) (cid:17) , with z = x − √ x − . For k = 2, we get Corollary 4 ˜ S ( − ( x ) ˙= " ˜ K ( x − ˜ b ) 1˜ K ( x − x + ˜ b S ( x ) , with ˜ K = 1 − ξ ξ . An equivalent result to the previous one when k = 1 can be found in [6]. n perturbed orthogonal polynomials on the real line and the unit circle 13 Let { g n } n > be a sequence of orthogonal polynomials satisfying the three termrecurrence relation for OPRL with new recurrence coefficients, { b n } n ≥ and { d n } n ≥ , i.e., g n +1 ( x ) = ( x − b n +1 ) g n ( x ) − d n g n − ( x ) , with initial conditions g − ( x ) = 0 and g ( x ) = 1, perturbed in a (generalized)co-dilated and/or co-recursive way, namely co-polynomials on the real line (COPRL). In other words, we will consider an arbitrary single modification ofthe recurrence coefficients as follows d n = λ δ n,k k d n , λ k > , (co-dilated case) (4.13) b n = b n + τ k +1 δ n,k +1 , τ k +1 ∈ R . (co-recursive case) (4.14)where k is a fixed non-negative integer number, and δ nk is the Kronecker delta.The modification of the Verblunsky coefficients for the corresponding OPUCassociated with the perturbed recurrence coefficients through the Szeg˝o trans-formation is shown in the following result. Theorem 9
Let { b α n } n ≥ be the Verblunsky coefficients for the correspon-dingOPUC, associated with (4.13) and (4.14) through the Szeg˝o transformation.Then, for a fixed non-negative integer k , b α n = α n , ≤ n < k − , b α k − = α k − + M, b α k = (1 − α k − ) α k + 2 τ k +1 + M α k − − α k − − M , b α m +1 = − d m +1 (1 − b α m − )(1 − b α m ) , n = 2 m + 1 , m ≥ k, b α m = 2 b m +1 + (1 + b α m − ) b α m − − b α m − , n = 2 m, m ≥ k + 1 , where M = 4( λ k − d k (1 − α k − )(1 − α k − ) .Proof From (1.7) and (1.8), for n ≥ α n = 2 b n +1 + (1 + α n − ) α n − (1 − α n − ) ,α n +1 = − d n +1 (1 − α n − )(1 − α n ) . Thus, according to (4.13) and (4.14), b α n = α n , ≤ n < k − , b α k − = − λ k d k (1 − α k − )(1 − α k − ) , b α k = 2( b k +1 + τ k +1 ) + (1 + b α k − ) α k − − b α k − , b α m +1 = − d m +1 (1 − b α m − )(1 − b α m ) , n = 2 m + 1 , m ≥ k, b α m = 2 b m +1 + (1 + b α m − ) b α m − − b α m − , n = 2 m, m ≥ k + 1 , and the theorem follows as a consequence of straightforward computations.Note that the modifications (4.13) and (4.14) imply through the Szeg˝otransformation the modification of all the Verblunsky coefficients greater than k . Consider the S -function S ( x ; λ k , τ k +1 ), associated with the COPRL [3],given by S ( x ; λ k , τ k +1 ) ˙= cof ( M k ) S ( x ) . By applying the Szeg˝o transformation to this equation, we have the followingresult.
Theorem 10
Let F ( z ; λ k , τ k +1 ) be the C -function associated with the pertur-bations (4.13) and (4.14) through the Szeg˝o transformation. Then, z − z F ( z ; λ k , τ k +1 ) ˙= cof (cid:18) M k (cid:18) z + z − (cid:19)(cid:19) (cid:18) z − z F ( z ) (cid:19) , with x = z + z − . For the finite composition of perturbations (4.13) and (4.14), we can con-sider the S -function S ( x ; λ m , τ m +1 ; . . . ; λ k , τ k +1 ), associated with the COPRL P n ( x ; λ m , τ m +1 ; . . . ; λ k , τ k +1 ) [3], given by S ( x ; λ m , τ m +1 ; . . . ; λ k , τ k +1 ) ˙= cof k Y j = m M j S ( x ) . Then, applying the Szeg˝o transformation, we get the following result.
Theorem 11
Let F ( z ; λ m , τ m +1 ; . . . ; λ k , τ k +1 ) be the C -function associatedwith the finite composition of perturbations (4.13) and (4.14) through the Szeg˝otransformation. Then, z − z F ( z ; λ m , τ m +1 ; . . . ; λ k , τ k +1 ) ˙= cof k Y j = m M j (cid:18) z + z − (cid:19) (cid:18) z − z F ( z ) (cid:19) , with x = z + z − . n perturbed orthogonal polynomials on the real line and the unit circle 15 For a fixed non-negative integer number k , let us consider the perturbedVerblunsky coefficients { β n } n ≥ given by β n = η k δ nk + (1 − δ nk ) α n . ( k -modification) (4.15)where η k is an arbitrary complex number. In order to achieve a new sequenceof Verblunsky coefficients, from now on we assume that | η k | < η k = α k .We define a sequence of monic co-polynomials on the unit circle (COPUC, inshort), { Φ n ( · ; k ) } n ≥ , those polynomials generated using { β n } n ≥ through theSzeg˝o recurrences. Analogously, we denote by { Ω n ( · ; k ) } n ≥ the correspondingsecond kind polynomials.Let us consider the C -function F ( z ; l, . . . , m ) associated with the finite com-position of perturbations (4.15) [2], given by F ( z ; l, . . . , m ) ˙= m Y j = l B j ( z ) F σ ( z ) . Then, applying the Szeg˝o transformation, we have the following Theorem.
Theorem 12
Let S ( x ; l, . . . , m ) be the S -function for the corresponding OPRLassociated with the finite composition of perturbations (4.15) through the Szeg˝otransformation. Then, p x − S ( x ; l, . . . , m ) ˙= m Y j = l B j ( x − p x − (cid:16)p x − S µ ( x ) (cid:17) , with z = x − √ x − . { v k } k > , v k = 12 (1 + α k )(1 − α k − ) , (4.16)then we have d k +1 = v k v k +1 , (4.17) b k +1 + 1 = v k − + v k , (4.18)and, we can find a unique factorization J + I = LU , where J is the Jacobi matrix associated with (1.1), I is the identity matrix, L is a lower bidiagonal matrix, and U is a upper bidiagonal matrix, with L = v v
1. . . . . . , U = v v v
1. . . . . . . Thus, from (4.16), we have α k = − v k − α k − , (4.19)or equivalently, α k = − v k − v k − − · · · − v − v . Therefore, from the a sequence { v k } k > we can determine in a very simpleway the Verblunsky coefficients { α k } k > for the measure dσ supported on T .From (4.17) and (4.18), we can find the sequence { v k } k > in terms of therecurrence coefficients { b k } k > and { d k } k > , as follows v k = b k +1 + 1 − d k b k + 1 − · · · − d b + 1 ,v k +1 = d k +1 b k +1 + 1 − d k b k + 1 − · · · − d b + 1 , with k ≥ J at level k , then we have a new sequence { ˜ v n } n > , given by˜ v n = b n +1 + 1 − d n b n + 1 − · · · − d k +1 b k +1 + τ k +1 + 1 − λ k d k b k + 1 − · · · − d b + 1 , ˜ v n +1 = d n +1 ˜ v n , with k ≥
0. This can be summarized in the following Theorem.
Theorem 13
Let { ˜ v n } n > be the new sequence associated with (4.13) and (4.14) . Then ˜ v n = v n , ≤ n ≤ k − , ˜ v k = v k + (1 − λ k ) v k − + τ k +1 , ˜ v m +1 = d m +1 ˜ v m , ˜ v m +1) = b m +2 + 1 − ˜ v m +1 , m ≥ k. Therefore, as we mentioned previously, we can compute the new Verblunskycoefficients directly from the sequence { ˜ v n } n > as follows. Theorem 14
Let { b α n } n > be the Verblunsky coefficients for the correspon-ding OPUC associated with the perturbations (4.13) and (4.14) through Szeg˝otransformation. Then, b α n = α n , ≤ n ≤ k − , b α k = α k + 2[(1 − λ k ) v k − + τ k +1 ]1 − α k − , b α n = − v n − b α n − , n ≥ k + 1 . n perturbed orthogonal polynomials on the real line and the unit circle 17 Proof
From (4.19) and Theorem 13, for n = 2 k we have b α k = − v k − α k − = − v k + (1 − λ k ) v k − + τ k +1 ]1 − α k − and the theorem follows.This is an alternative way to compute the perturbed Verblunsky coefficientsthrough the Szeg˝o transformation using the LU factorization. − , { S n } n > be a sequence of monic symmetric polynomials orthogonal withrespect to an even weight function supported on a symmetric subset of [ − , S n +1 ( x ) = xS n ( x ) − d n S n − ( x ) , d n = 0 , d = 1 , n > , with initial conditions S − ( x ) = 0 and S ( x ) = 1, see [4].Let { γ n } n > be the Verblunsky coefficients for the corresponding OPUC, { Φ n } n ≥ , related to the symmetric OPRL { S n } n > , through Szeg˝o transfor-mation. Then, γ n = 0 , γ n +1 = − d n +1 − γ n − , n ≥ γ − = − b n = 0 for every n ≥
0, we have γ n = 0 , n ≥ d n +1 = 14 (1 − γ n − )(1 + γ n +1 ) , n ≥ , and (5.20) follows.Let { b γ n } n ≥ be the Verblunsky coefficients for the corresponding OPUC,associated with (4.13) through the Szeg˝o transformation. Then, for a fixednon-negative integer k , b γ n = γ n = 0 , n ≥ , b γ n − = γ n − , ≤ n < k, b γ k − = γ k − + 4( λ k − d k − γ k − , b γ n +1 = − d n +1 (1 − b γ n − ) , n ≥ k. Notice that the modification (4.13) yields, from the Szeg˝o transformation,the modification of all odd Verblunsky coefficients greater than k . dσ be a nontrivial probability measure supported on T and let { Φ n } n ≥ bethe corresponding OPUC. For a positive integer ℓ the sieved OPUC { Φ { ℓ } n } n ≥ are defined as those orthogonal polynomials associated with the Verblunskycoefficients { α { ℓ } n } n ≥ given by α { ℓ } n = ( α m − if n + 1 = mℓ, , (5.21)for n ≥
0. We also denote by σ { ℓ } the nontrivial probability measure supportedon T associated with { α { ℓ } n } n ≥ . Note that { Φ { } n } n ≥ are the polynomials { Φ n } n ≥ . The earliest treatment of { Φ { ℓ } n } n ≥ for ℓ ≥ ℓ = 2, then from (5.21) we have { α { } n } n ≥ = { , α , , α , . . . } .Then we have the following result.Let { b { } n } n ≥ and { d { } n } n ≥ be the recurrence coefficients for the corre-sponding OPRL, { P { } n } n ≥ , related to the sieved OPUC { Φ { } n } n ≥ , throughSzeg˝o transformation. Then, b { } n +1 = 0 , d { } n +1 = 14 (1 − α n − )(1 + α n ) , n ≥ . (5.22)From (1.8), since α { } n = 0 for every n ≥
0, we have b { } n +1 = 0 , n ≥
0. Then,from (1.7), we deduce that d { } n +1 = 14 (1 − α { } n − )(1 + α { } n +1 ) , n ≥ . Since { α { } n } n ≥ = { , α , , α , . . . } (5.22) follows.Let { b b { } n } n ≥ and { b d { } n } n ≥ be the recurrence coefficients for the corre-sponding OPRL associated with ( k -modification) through the Szeg˝o transfor-mation. Then, for a fixed non-negative integer k , b d { } n +1 = (cid:18) η k α k (cid:19) δ n +1 ,k +1 (cid:18) − η k − α k (cid:19) δ n +1 ,k +2 d { } n +1 , b b { } n +1 = 0 . Note that this k -modification yields, by using the Szeg˝o transformation, themodification of two consecutive recurrence coefficients d { } k +1 and d { } k +2 . Acknowledgements
The research of the first author is supported by the Portuguese Gov-ernment through the FCT under the grant SFRH/BPD/101139/2014. The research of thefirst and second author is supported by Direcci´on General de Investigaci´on Cient´ıfica yT´ecnica, Ministerio de Econom´ıa y Competitividad of Spain, grant MTM2012–36732–C03–01.n perturbed orthogonal polynomials on the real line and the unit circle 19
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