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

Abstract

Szego's procedure to connect orthogonal polynomials on the unit circle and orthogonal polynomials on [-1,1] is generalized to nonsymmetric measures. It generates the so-called semi-orthogonal functions on the linear space of Laurent polynomials L, and leads to a new orthogonality structure in the module LxL. This structure can be interpreted in terms of a 2x2 matrix measure on [-1,1], and semi-orthogonal functions provide the corresponding sequence of orthogonal matrix polynomials. This gives a connection between orthogonal polynomials on the unit circle and certain classes of matrix orthogonal polynomials on [-1,1]. As an application, the strong asymptotics of these matrix orthogonal polynomials is derived, obtaining an explicit expression for the corresponding Szego's matrix function.

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arXiv:math/0204294v1 [math.CA] 24 Apr 2002
A connection between orthogonal polynomials on the unit circle
and 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.
the 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 can
be interpreted in terms of a 2 × 2 matrix measure on [−1,1], and semi-orthogonal
functions provide the corresponding sequence of orthogonal matrix polynomials. This
gives a connection between orthogonal polynomials on the unit circle and certain classes
of matrix orthogonal polynomials on [−1,1]. As an application, the strong asymptotics
of these matrix orthogonal polynomials is derived, obtaining an explicit expression for
the corresponding Szeg˝ o’s matrix function.
It generates
Keywords and phrases: Orthogonal Polynomials, Semi-orthogonal Functions, Matrix
Orthogonal 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 between
orthogonal 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 to
OP on [−1,1]. For instance, this idea was largely exploited to get asymptotic
properties 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. Recently
it has been shown that above procedure can be generalized to arbitrary measures
on T, giving a connection between any sequence of OP on T and the so-called
semi-orthogonal functions [1, 4].
As we will see, semi-orthogonal functions are given in terms of a sequence
of two-dimensional matrix polynomials. The orthogonality properties of semi-
orthogonal functions implies that these matrix polynomials are quasi-orthogonal
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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 explicitly
constructed 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 real
line.
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 discrete
Sturm-Liouville operators [2, 3], in the analysis of sequences of polynomials sat-
isfying higher order recurrence relations [8], rational approximation and system
theory [10]. Unfortunately their study is much more complicated and few things
are 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]) where
it 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 product
and matrix OP is presented in [8] (which is a consequence of the fact that OP
with respect to a discrete Sobolev inner product satisfy a higher order recurrence
relation, see [16]). These connections are more general than the one given in this
paper because they deal with matrix OP of arbitrary dimension. However, they
link matrix OP with “unknown words”, in the sense that not too much is known
about the different kind of OP that they connect with matrix OP. So, they are
not too useful to get new results for matrix OP.
On the contrary, the connection presented in this paper, although much more
restricted, let us translate results from the more “known word” of scalar OP on
the unit circle to a great variety of two-dimensional matrix OP. So, it provides
many 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 we
have to point out that for certain applications, like the study of the doubly infinite
matrices that appear in discrete Sturm-Liouville problems on the real line, only
two-dimensional matrix OP are needed [3, 23].
As an example of the utility of the present connection we derive the strong
asymptotics of these matrix OP when the corresponding matrix measure belongs
to the Szeg˝ o’s class. General results about this situation can be found in [2], where
a generalization of the connection between the real line and T for matrix OP is
used 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 expression
for the Szeg˝ o’s matrix function that gives the asymptotic behavior and only some
general properties are known. The connection given here let us obtain explicitly
this Szeg˝ o’s matrix function for a class of two-dimensional matrix measures. Other
results about asymptotics of matrix OP, such as ratio and relative asymptotics,
appear in [7] and [25] respectively.
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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 some
proofs there for the convenience of the reader.
First of all we fix some notations. The real vector space of polynomials with
real coefficients is denoted by P, the subspace of P of polynomials with degree less
than or equal to n is Pnand P#
mials whose degree is exactly n. Also, Λ is the complex vector space of Laurent
polynomials, that is, Λ =?∞
arbitrary complex number α, their real and imaginary parts are denoted ℜα and
ℑα respectively.
nis the subset of Pnconstituted by those polyno-
??n
n=0Λ−n,n where Λm,n =
k=mαkzk??αk∈ C?
for
m ≤ n. The elements of Λm,nsuch that αm,αn?= 0 form the subset Λ#
m,n. For an
Taking into account the usual identification between the unit circle T =
{eiθ| θ ∈ [0,2π)} and the interval [0,2π), we talk about a measure on T when
we deal with a measure with support on [0,2π). With this convention, in what
follows dµ is a measure on T with finite moments. Unless we say explicitly that
it is an arbitrary measure on T, we suppose that dµ is a positive measure with
infinite support. Then, the sesquilinear functional?·,·?
?f,g?
is an inner product and, hence, there exists a unique sequence (φn)n≥0of monic
OP with respect to?·,·?
of φn(φ∗
so-called Schur parameters an= φn(0) through the recurrence
dµon Λ defined by
dµ=
?2π
0
f(eiθ)g(eiθ)dµ(θ), f,g ∈ Λ,
dµ. If, as it is usual, φ∗
ndenotes the reversed polynomial
n(z) = znφn(z−1)), then, it is well known that OP are determined by the
φ0(z) = 1,
φn(z) = zφn−1(z) + anφ∗
n−1(z),n ≥ 1.
(1)
If we denote by bnthe coefficient of zn−1in φn(z), from (1) we have that
bn= bn−1+ anan−1,n ≥ 1.(2)
Notice that b0= 0 and
bn=
n
?
k=1
akak−1,n ≥ 1.(3)
We can use (1) to show that the positive constants εn=?φn,φn
εn
εn−1
?
dµare related to
the Schur parameters by
= 1 − |an|2,n ≥ 1.(4)
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This relation implies that |an| < 1 for n ≥ 1 and that the sequence (εn)n≥0
must be strictly decreasing. Besides, (4) gives the following expression for εn
n
?
εn=
k=1
(1 − |ak|2)ε0,n ≥ 1.(5)
Orthonormal polynomials are defined up to a factor with unit module, but
they can be fixed if we ask for their leading coefficients to be real and positive. In
this case we denote the n-th orthonormal polynomial by ϕn, and the corresponding
leading coefficient by κn. It is clear that κn= ε−1/2
increasing.
n
and, thus, (κn)n≥0is strictly
The symmetric measure of dµ is
d? µ(θ) = −dµ(2π − θ),θ ∈ [0,2π),
and the measure dµ is said to be symmetric iff d? µ = dµ. This is equivalent to
equivalent to state that the Schur parameters are real.
affirm that the monic OP have real coefficients, which, in sight of (1), is in fact
With the intention of connecting T with the interval [−1,1], for z ∈ C\{0} we
write x = (z+z−1)/2 and y = (z−z−1)/2i (therefore z = x+iy,z−1= x−iy and
x2+ y2= 1). Both expressions give a transformation in the complex plane that
maps T on the interval [−1,1]. Moreover, they map bijectively onto C\[−1,1] the
exterior of T as well as its interior excepting the origin. So, when restricted to these
domains we can invert the transformations giving, for example, z = x +√x2− 1
(the choice of the square root must be done according to the location of z: exterior
or interior to T). Also, the transformation x = (z + z−1)/2 maps biyectively the
upper as well as the lower closed half T onto [−1,1] (in this case, writing z = eiθ, it
is x = cosθ). So, by composition with the corresponding inverse transformations,
the measure dµ provides two projected measures dν1, dν2on [−1,1], being
dν1(x) = −dµ(arccosx),
dν2(x) = −d? µ(arccosx).
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 dν1, dν2.
To this end, and following [1, 4], we start by introducing previously the so called
semi-orthogonal functions.
(6)
The condition of symmetry for dµ is equivalent to the equality dν1= dν2.
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) =zφ2n−1(z) + φ∗
2n−1(z)
2nzn
,
f(2)
n(z) =zφ2n−1(z) − φ∗
2n−1(z)
i2nzn
,
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where φn, n ≥ 1, are the monic OP with respect to?·,·?
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 real
coefficients. Let us go to summarize some interesting properties of SOF [1, 4]:
dµ.
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(k1)
n
(x) + y f(k2)
n
(x),f(kj)
n
∈ P.
More precisely, f(11)
f(21)
n
∈ Pn−1,f(12)
n
∈ P#
n, f(22)
n
∈ P#
n−1, both monic polynomials, and
n
∈ Pn−2.
ii) The family of functions Bn= {1}∪
all n ≥ 1. The matrix of?·,·?
??n
m=1{f(1)
m ,f(2)
m }
?
is a basis of Λ−n,nfor
dµwith respect to the basis B = ∪n≥1Bnof Λ
is a diagonal-block one




?1 − ℜa2n
ε0
0
0
...
00
0
...
...
...
...
C1
0
...
C2
...



,(7)
where
Cn=ε2n−1
22n−1
−ℑa2n
1 + ℜa2n
−ℑa2n
?
,n ≥ 1, (8)
being anthe Schur parameters related to dµ and εngiven in (5).
Proof. From the definition of φ∗
nwe have that
f(1)
f(2)
n(z) = 2−n(z−n+1φ2n−1(z) + zn−1φ2n−1(z−1)),
n(z) = −i2−n(z−n+1φ2n−1(z) − zn−1φ2n−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 two
such decompositions
f(k)
n(z) = f(k1)
n
(x) + y f(k2)
n
(x) = g(k1)
n
(x) + y g(k2)
n
(x),f(kj)
n
,g(kj)
n
∈ P,
then
f(k)
n(z−1) = f(k1)
n
(x) − y f(k2)
n
(x) = g(k1)
n
(x) − y g(k2)
n
(x),
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