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

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A connection between orthogonal polynomials on the unit circle and matrix orthogonal polynomials on the real line
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    a  r   X   i  v  :  m  a   t   h   /   0   2   0   4   2   9   4  v   1   [  m  a   t   h .   C   A   ]   2   4   A  p  r   2   0   0   2 A connection between orthogonal polynomials on the unit circleand matrix orthogonal polynomials on the real line M.J. Cantero ,  M.P. Ferrer  ,  L. Moral  ,  L. Vel´ azquez  Departamento de Matem´atica Aplicada. Universidad de Zaragoza. Spain. Abstract Szeg˝o’s procedure to connect orthogonal polynomials on the unit circle and ortho-gonal polynomials on  [ − 1 , 1]  is generalized to nonsymmetric measures. It generatesthe so-called semi-orthogonal functions on the linear space of Laurent polynomials  Λ ,and leads to a new orthogonality structure in the module  Λ × Λ . This structure canbe interpreted in terms of a  2  ×  2  matrix measure on  [ − 1 , 1] , and semi-orthogonalfunctions provide the corresponding sequence of orthogonal matrix polynomials. Thisgives a connection between orthogonal polynomials on the unit circle and certain classesof matrix orthogonal polynomials on  [ − 1 , 1] . As an application, the strong asymptoticsof these matrix orthogonal polynomials is derived, obtaining an explicit expression forthe corresponding Szeg˝o’s matrix function. Keywords and phrases:  Orthogonal Polynomials, Semi-orthogonal Functions, MatrixOrthogonal Polynomials, Asymptotic Properties (1991) AMS Mathematics Subject Classification   : 42C05 1. Introduction. Semi-orthogonal functions From long time ago, it is well known that there exists a simple relation betweenorthogonal polynomials (OP) on the unit circle ( T ) and OP on [ − 1 , 1] (see [9, 24]).This close relationship provides a method to translate results from OP on  T  toOP on [ − 1 , 1]. For instance, this idea was largely exploited to get asymptoticproperties of OP on [ − 1 , 1] starting from the asymptotics of OP on  T  [18, 19, 20,24]. However, this relation is valid only for symmetric measures on  T . Recentlyit has been shown that above procedure can be generalized to arbitrary measureson  T , giving a connection between any sequence of OP on  T  and the so-calledsemi-orthogonal functions [1, 4].As we will see, semi-orthogonal functions are given in terms of a sequenceof two-dimensional matrix polynomials. The orthogonality properties of semi-orthogonal functions implies that these matrix polynomials are quasi-orthogonal1  with respect to some two-dimensional matrix measure related to the measure on T . A sequence of matrix OP with respect to this matrix measure can be explicitlyconstructed from above quasi-orthogonal matrix polynomials. This gives a con-nection between OP on  T  and a class of two-dimensional matrix OP on the realline.Matrix OP on the real line appear in the Lanczos method for block matrices[12, 13], in the spectral theory of doubly infinite Jacobi matrices [23] and discreteSturm-Liouville operators [2, 3], in the analysis of sequences of polynomials sat-isfying higher order recurrence relations [8], rational approximation and systemtheory [10]. Unfortunately their study is much more complicated and few thingsare known if compared with the scalar case (some nice surveys are [17, 21, 23]).Previous connections between scalar and matrix OP appear in [14] ([15])whereit is derived a relation between scalar OP on an algebraic harmonic curve (lem-niscata) and matrix OP on the real line (unit circle). Using a similar technique,a connection between scalar OP with respect to a discrete Sobolev inner productand matrix OP is presented in [8] (which is a consequence of the fact that OPwith respect to a discrete Sobolev inner product satisfy a higher order recurrencerelation, see [16]). These connections are more general than the one given in thispaper because they deal with matrix OP of arbitrary dimension. However, theylink matrix OP with “unknown words”, in the sense that not too much is knownabout the different kind of OP that they connect with matrix OP. So, they arenot too useful to get new results for matrix OP.On the contrary, the connection presented in this paper, although much morerestricted, let us translate results from the more “known word” of scalar OP onthe unit circle to a great variety of two-dimensional matrix OP. So, it providesmany models of matrix OP where many things can be known and that, therefore,can be used to get or check some ideas about new results for matrix OP. Here wehave to point out that for certain applications, like the study of the doubly infinitematrices that appear in discrete Sturm-Liouville problems on the real line, onlytwo-dimensional matrix OP are needed [3, 23].As an example of the utility of the present connection we derive the strongasymptotics of these matrix OP when the corresponding matrix measure belongsto the Szeg˝o’s class. General results about this situation can be found in [2], wherea generalization of the connection between the real line and  T  for matrix OP isused again to obtain the asymptotics in the real line from the asymptotics in  T .However, the problem is far from being closed since there is no explicit expressionfor the Szeg˝o’s matrix function that gives the asymptotic behavior and only somegeneral properties are known. The connection given here let us obtain explicitlythis Szeg˝o’s matrix function for a class of two-dimensional matrix measures. Otherresults about asymptotics of matrix OP, such as ratio and relative asymptotics,appear in [7] and [25] respectively.2  Now, we proceed to introduce the starting point of our discussion, the semi-orthogonal functions, summarizing some results in [1, 4] with a sketch of someproofs there for the convenience of the reader.First of all we fix some notations. The real vector space of polynomials withreal coefficients is denoted by P  , the subspace of  P   of polynomials with degree lessthan or equal to  n  is  P  n  and  P  # n  is the subset of   P  n  constituted by those polyno-mials whose degree is exactly  n . Also, Λ is the complex vector space of Laurentpolynomials, that is, Λ =  ∞ n =0  Λ − n,n  where Λ m,n  =  nk = m α k z k  α k  ∈ C   for m ≤ n . The elements of Λ m,n  such that  α m ,α n   = 0 form the subset Λ # m,n . For anarbitrary complex number  α , their real and imaginary parts are denoted  ℜ α  and ℑ α  respectively.Taking into account the usual identification between the unit circle  T  = { e iθ |  θ  ∈  [0 , 2 π ) }  and the interval [0 , 2 π ), we talk about a measure on  T  whenwe deal with a measure with support on [0 , 2 π ). With this convention, in whatfollows  dµ  is a measure on  T  with finite moments. Unless we say explicitly thatit is an arbitrary measure on  T , we suppose that  dµ  is a positive measure withinfinite support. Then, the sesquilinear functional  · , ·  dµ  on Λ defined by  f,g  dµ  =    2 π 0 f  ( e iθ ) g ( e iθ ) dµ ( θ ) , f,g  ∈ Λ , is an inner product and, hence, there exists a unique sequence ( φ n ) n ≥ 0  of monicOP with respect to  · , ·  dµ . If, as it is usual,  φ ∗ n  denotes the reversed polynomialof   φ n  ( φ ∗ n ( z ) =  z n φ n ( z − 1 )), then, it is well known that OP are determined by theso-called Schur parameters  a n  =  φ n (0) through the recurrence φ 0 ( z ) = 1 ,φ n ( z ) =  zφ n − 1 ( z ) +  a n φ ∗ n − 1 ( z ) , n ≥ 1 . (1)If we denote by  b n  the coefficient of   z n − 1 in  φ n ( z ), from (1) we have that b n  =  b n − 1  +  a n a n − 1 , n ≥ 1 .  (2)Notice that  b 0  = 0 and b n  = n  k =1 a k a k − 1 , n ≥ 1 .  (3)We can use (1) to show that the positive constants  ε n  =  φ n ,φ n  dµ  are related tothe Schur parameters by ε n ε n − 1 = 1 −| a n | 2 , n ≥ 1 .  (4)3  This relation implies that  | a n |  <  1 for  n  ≥  1 and that the sequence ( ε n ) n ≥ 0 must be strictly decreasing. Besides, (4) gives the following expression for  ε n ε n  = n  k =1 (1 −| a k | 2 ) ε 0 , n ≥ 1 .  (5)Orthonormal polynomials are defined up to a factor with unit module, butthey can be fixed if we ask for their leading coefficients to be real and positive. Inthis case we denote the  n -th orthonormal polynomial by  ϕ n , and the correspondingleading coefficient by  κ n . It is clear that  κ n  =  ε − 1 / 2 n  and, thus, ( κ n ) n ≥ 0  is strictlyincreasing.The symmetric measure of   dµ  is d  µ ( θ ) = − dµ (2 π − θ ) , θ  ∈ [0 , 2 π ) , and the measure  dµ  is said to be symmetric iff   d  µ  =  dµ . This is equivalent toaffirm that the monic OP have real coefficients, which, in sight of (1), is in factequivalent to state that the Schur parameters are real.With the intention of connecting T with the interval [ − 1 , 1], for  z  ∈ C \{ 0 } wewrite  x  = ( z + z − 1 ) / 2 and  y  = ( z − z − 1 ) / 2 i  (therefore  z  =  x + iy,z − 1 =  x − iy  and x 2 +  y 2 = 1). Both expressions give a transformation in the complex plane thatmaps  T on the interval [ − 1 , 1]. Moreover, they map bijectively onto C \ [ − 1 , 1] theexterior of  T as well as its interior excepting the srcin. So, when restricted to thesedomains we can invert the transformations giving, for example,  z  =  x  + √  x 2 − 1(the choice of the square root must be done according to the location of   z : exterioror interior to  T ). Also, the transformation  x  = ( z  +  z − 1 ) / 2 maps biyectively theupper as well as the lower closed half  T onto [ − 1 , 1] (in this case, writing  z  =  e iθ , itis  x  = cos θ ). So, by composition with the corresponding inverse transformations,the measure  dµ  provides two projected measures  dν  1 ,  dν  2  on [ − 1 , 1], being dν  1 ( x ) = − dµ (arccos x ) ,dν  2 ( x ) = − d  µ (arccos x ) . (6)The condition of symmetry for  dµ  is equivalent to the equality  dν  1  =  dν  2 .Now, we wish to arrive at a family of polynomials with real coefficients, or-thogonal with respect to an inner product defined through the measures  dν  1 ,  dν  2 .To this end, and following [1, 4], we start by introducing previously the so calledsemi-orthogonal functions. Definition 1.  The semi-orthogonal functions (SOF) associated to the measure dµ  are the functions  f  ( k ) n  : C \{ 0 }→ C , n ≥ 1 , k  = 1 , 2, defined by f  (1) n  ( z ) =  zφ 2 n − 1 ( z ) +  φ ∗ 2 n − 1 ( z )2 n z n  ,f  (2) n  ( z ) =  zφ 2 n − 1 ( z ) − φ ∗ 2 n − 1 ( z ) i 2 n z n  , 4  where  φ n , n ≥ 1, are the monic OP with respect to  · , ·  dµ .The expressions in above definition are the same used in Szeg˝o’s method,with the difference that we consider here monic OP with complex instead of realcoefficients. Let us go to summarize some interesting properties of SOF [1, 4]: Proposition 1.  The SOF associated to   dµ  satisfy:i)  f  ( k ) n  ( z − 1 ) =  f  ( k ) n  ( z )  and there is a unique decomposition f  ( k ) n  ( z ) =  f  ( k 1) n  ( x ) + yf  ( k 2) n  ( x ) , f  ( kj ) n  ∈P  . More precisely,  f  (11) n  ∈ P  # n  ,  f  (22) n  ∈ P  # n − 1 , both monic polynomials, and  f  (21) n  ∈P  n − 1 ,f  (12) n  ∈P  n − 2 .ii) The family of functions   B n  = { 1 }∪  nm =1 { f  (1) m  ,f  (2) m  }  is a basis of   Λ − n,n  for all   n  ≥  1 . The matrix of   · , ·  dµ  with respect to the basis   B  =  ∪ n ≥ 1 B n  of   Λ is a diagonal-block one   ε 0  0 0  ... 0  C  1  0  ... 0 0  C  2  ... ......... ...  ,  (7) where  C  n  =  ε 2 n − 1 2 2 n − 1  1 −ℜ a 2 n  −ℑ a 2 n −ℑ a 2 n  1 + ℜ a 2 n  , n ≥ 1 ,  (8) being   a n  the Schur parameters related to   dµ  and   ε n  given in (5). Proof.  From the definition of   φ ∗ n  we have that f  (1) n  ( z ) = 2 − n ( z − n +1 φ 2 n − 1 ( z ) +  z n − 1 φ 2 n − 1 ( z − 1 )) ,f  (2) n  ( z ) = − i 2 − n ( z − n +1 φ 2 n − 1 ( z ) − z n − 1 φ 2 n − 1 ( z − 1 )) , and, thus,  f  ( k ) n  ( z − 1 ) =  f  ( k ) n  ( z ).If the decomposition given in i) exits, it must be unique. If we suppose twosuch decompositions f  ( k ) n  ( z ) =  f  ( k 1) n  ( x ) + yf  ( k 2) n  ( x ) =  g ( k 1) n  ( x ) +  yg ( k 2) n  ( x ) , f  ( kj ) n  ,g ( kj ) n  ∈P  , then f  ( k ) n  ( z − 1 ) =  f  ( k 1) n  ( x ) − yf  ( k 2) n  ( x ) =  g ( k 1) n  ( x ) − yg ( k 2) n  ( x ) , 5
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