Semiorthogonal functions and orthogonal polynomials on the
unit circle
Manuel Alfaroa;∗;1, Mara Jose Canterob;2, Leandro Moralb;2 aDepartamento de Matematicas, Universidad de Zaragoza, 50009 Zaragoza, Spain b
Departamento de Matematica Aplicada, Universidad de Zaragoza, 50009 Zaragoza, Spain
Received 7 November 1997; received in revised form 5 May 1998
Abstract
We establish a bijection between Hermitian functionals on the linear space of Laurent polynomials and functionals on P×Psatisfying some orthogonality conditions (P denotes the linear space of polynomials with real coecients). This allows us to study some topics about sequences (n) of orthogonal polynomials on the unit circle from a new point of view. Whenever the polynomials n have real coecients, we recover a well known result by Szeg˝o. c 1998 Elsevier Science B.V. All rights reserved.
AMS classication:42C05
Keywords:Hermitian functionals; Orthogonal polynomials; Recurrence formulas
1. Introduction
In this paper we are concerned with an algebraic method introduced in [1] in order to study poly-nomial modications of Hermitian functionals L dened on the linear space of Laurent polynomials and, as consequence, properties of the corresponding orthogonal polynomials (OP) on the unit circle
T. The basic idea is the well known Szeg˝o’s result about the connection between OP on T and OP
on the interval [−1;1], (see [6, Section 11.5] and also [2, Section V.1] or [3, 9.1]); in this situation, since the orthogonality measure on T is symmetric, the associated OP have real coecients. We
will analyze a relation such as Szeg˝o’s one, valid for OP on T with complex coecients.
∗Corresponding author. E-mail: [email protected].
1
This research was supported by DGES PB96-0120-C03-02.
Now, we proceed with some notations. In what follows, we denote P=R[x] the linear space of
polynomials with real coecients, Pn is the subspace of polynomials whose degree is less than or equal to n and P#
n the subset of Pn of polynomials whose degree is exactly n. Besides, denotes
the vector space C[z] of complex polynomials, and n its subspace of all polynomials of degree
less than or equal to n. We let L be the class of Laurent polynomials, that is
L=
( n
X
k=−m
kzk;k∈C|m; n∈N∪ {0} )
:
The paper is organized as follows: In Section 2, we give a bijective correspondence between Hermitian functionals L on L and a class of functionals M on P×P; also, we construct a basis of P×P (which we call semi-orthogonal basis) satisfying some orthogonality conditions with respect toM. Section 3 is devoted to prove that the elements of the semi-orthogonal basis satisfy recurrence formulas connected with recurrence formulas for orthogonal matrix polynomials. In Section 4, the relation between OP on T and the semi-orthogonal basis is analyzed. In the last Section, we give an application of the above method to a modication of the functional L.
Next, we show some technical results that we shall use later.
For z∈C\{0}, let us writex= (z+z−1)=2 andy= (z−z−1)=2i; therefore z=x+ iy, z−1=x−iy,
and x2+y2= 1. Notice that if z= ei, (∈R), then x= cos and y= sin. In any case, we can
express z in terms of x with, as usually, √x2−1¿0 when x¿1.
Let us consider a polynomial of degree 2n−1, n¿1, (z) =2n−1z2n−1+· · ·+0, wherek∈C.
We dene f; g:C\{0} →C, by means of
f(x) = 2−nz−n[z(z) +∗(z)];
g(x) =−i2−nz−n[z(z)
−∗(z)]; (1)
where ∗(z) =z2n−1(z−1) is the so-called reversed polynomial.
Lemma 1.1. Let be a monic polynomial of degree 2n−1 (n¿1) and f; g dened in (1). Then;
there exist unique polynomials R1∈Pn#; R2∈Pn−2; S1∈Pn−1 and S2∈Pn#−1; such that
f(x) =R1(x) +
√
1−x2R 2(x);
g(x) =S1(x) +
√ 1−x2S
2(x):
Conversely; given R1∈Pn#; R2∈Pn−2; S1∈Pn−1 and S2∈Pn#−1 with R1 andS2 monic polynomials;
then there is a unique monic polynomial ∈ of degree 2n−1; such that
R1(x) +
√
1−x2R
2(x) = 2−nz−n[z(z) +∗(z)];
S1(x) +
√ 1−x2S
2(x) = −i2
−nz−n[z(z)
Proof. From (1) we have
f(x) = 2−nz−n
2Xn−1
k=0
(kzk+1+kz2n −k−1)
=Tn(x) + n−1
X
j=0
2j−nRe(
n+j−1+n−j−1)Tj(x)−
√ 1−x2
×
n−1
X
j=1
2−jIm(
2n−j−1−j−1)Un−j−1(x);
where Tj and Uj are respectively the jth Tchebychev polynomials of the rst and second kind (see
[6, p.3]). So, keeping in mind that Im2n−1= 0, we can put
f(x) =
n X
k=0
akTk(x) +
√ 1−x2
n−2
X
k=0
bkUk(x):
with an= Re2n−1= 1:
Then, the desired expression for f is obtained if we take R1(x) =
Pn
k=0akTk(x) and R2(x)
=Pn−2
k=0bkUk(x). The uniqueness is obvious.
The proof for g is similar.
To deduce the converse, it suces to expand the polynomials R1 and S1 (respectively, R2 and S2)
in terms of Tchebychev polynomials of the rst (respectively, second) kind and the proof follows straightforward.
Note that if has real coecients, then R2(x)≡0 andS1(x)≡0. Moreover, if is in addition a
polynomial orthogonal on T with respect to a weight function w, then by Szeg˝o’s theory, R1 andS2
are orthogonal on [−1;1] with respect to the weight functions w(x)√1−x2 and w(x)√1−x2,
respectively.
2. Functionals on P×P induced by Hermitian functionals on P
We denote =P×P, which with the product:
(P1; Q1)·(P2; Q2) = (P1P2+ (1−x2)Q1Q2; P1Q2+P2Q1)
is a conmutative algebra with identity element. For each n¿0, we write n=Pn×Pn−1 with the
convention P−1={0}; n is a subspace of such that dim n= 2n+ 1. Notice that, the mapping (P; Q)→P+√1−x2Q establish a bijective correspondence between and P+√1−x2P which preserves the product; so, in the sequel some times we will use the expression P+√1−x2Q to
denote elements of .
Denition 2.1. Let L be a Hermitian functional on L; then ML:→R given by
ML[(P; Q)] =L
"
P z+z −1
2
!
+ z−z
−1
2i !
Q z+z −1
2
!#
is a linear functional on . We call it, the functional induced by L.
Observe that, if L is denite positive and is the corresponding orthogonality measure on T
(see [3, 4]), we can write:
ML[(P; Q)] =
Z 1
−1
P(x) d(1+2)(x) +
Z 1
−1
√
1−x2Q(x) d(
1−2)(x);
where 1 and 2 are the measures on [−1;1] given by
(
d1(x) =−d(); 06¡;
d2(x) = d(); 6¡2;
x= cos:
Using the induction method, it is easy to deduce:
Proposition 2.2. (i) Any set Bn inductively dened by
B0={1}; Bn=Bn−1∪ {h(1)n ; h
(2)
n }; n¿1; (2)
where h(1)
n ∈Pn#×Pn−2; h(2)n ∈Pn−1×Pn#−1 is a basis of n.
(ii) Any set B dened by means of
B={1} ∪ [
n¿1
{h(1)n ; h(2)n } !
;
where h(1)
n ; h(2)n are as in (i); is a basis of .
It is well known that L denes a bilinear Hermitian functional: h·;·i
L:×→C by means of
h; iL=L((z) (z
−1)):
This bilinear functional is called quasi-denite (positive denite) when the restriction of h·;·iL to
n×n is quasi-denite (positive denite), for each n. This denition is equivalent to the fact
that all principal minors of the moment matrix (hzk; zji
L)
∞
k; j= 0 are dierent from zero (¿0). So the
Gram-Schmidt procedure guarantees the existence of a sequence of monic orthogonal polynomials (SMOP) {n}n∈N such that hn; niL6= 0 (¿0).
Our main purpose is to carry over these results to the set of bilinear functionals on , via an orthogonalization procedure.
Denition 2.3. Let M∈∗, where ∗ denotes the algebraic dual space of . We dene the sym-metric bilinear form h·;·iM:×→R by
We say that M is quasi-denite (positive denite) if the restriction of h;iM to n×n is quasi-denite (positive quasi-denite); for each n∈N.
Notice that the denition of the positivity of M is similar to the one given for Hermitian func-tionals. However, the quasi-deniteness of M does not imply thathF; FiM6= 0 for all F∈M, F6= 0.
Theorem 2.4 (Orthogonalization procedure). Let M∈∗ be a quasi-denite (positive denite)
functional. Then; for each n∈N; there exists a unique basis Bn of n; given by
B0= 1; Bn=Bn−1∪ {fn; gn};
where fn= (R(1)n ; R(2)n )∈Pn#×Pn−2 and gn= (Sn(1); Sn(2))∈Pn−1×Pn#−1; with R(1)n and Sn(2) monic
polynomials; such that
M(fn·xk) =M(gn·xk) = 0; 06k6n−1;
M(fn·√1−x2xk) =M(g n·
√
1−x2xk) = 0; 06k6n
−2:
(3)
Besides; the matrix
Cn=
M(f2
n) M(fngn)
M(fngn) M(g2
n) !
is quasi-denite (positive denite.).
Proof. Let M∈∗ be quasi-denite (positive denite). For n= 1, we consider, according to (2), the following basis {1; f1; g1} of 1 given by
f1(x) = (x+c1;0); g1(x) = (ec1;1):
Let M(f1) =M(g1) = 0. Then,
M(f
1) =M(x) +c1M(1) = 0;
M(g1) =ce1M(1) +M√1−x2= 0
but M(1)6= 0, because of the quasi-deniteness of h·;·i
M on 0. Thus we have uniqueness for c1,
e
c1 in the above system.
Moreover, the matrix
M(1) M(f1) M(g1)
M(f1) M(f2
1 ) M(f1g1)
M(g1) M(f1g1) M(g2
1)
=
M
(1) 0 0 C1
is quasi-denite (positive denite) because of the quasi-deniteness (positivity) of M. Thus C
1 is
If M is induced by a positive denite functional, formulas (3) in Theorem 2.4 leads to four ‘unusual’ orthogonality conditions:
Theorem 2.5. LetML be a functional on induced by a Hermitian positive denite functional L.
Let be the Borel positive measure on T associated with L and 1 and 2 the measures on
[−1;1] given by d1(x) =−d(); 06¡; and d2(x) = d(); 6¡2; with x= cos. Then;
the polynomials R(1)
n and R(2)n ; introduced in Theorem 2:4; satisfy
Z 1
−1
R(1)
n (x)xkd (1+2)(x) +
Z 1
−1
R(2)
n (x)xk
√
1−x2d(
1−2)(x) = 0; 06k6n−1;
Z 1
−1
R(2)
n (x)x k(1
−x2) d(
1+2)(x) +
Z 1
−1
R(1)
n (x)x k√1
−x2d(
1−2)(x) = 0; 06k6n−2:
Two analogous formulas are true for the polynomials S(1)
n and Sn(2).
Corollary 2.6. Let M∈∗ be quasi-denite (positive). Then there is a unique basis B of such
that
B={1} ∪ [
n¿1
{fn; gn} !
;
where fn and gn are as in (3). Besides; the matrix associated with h·;·iM (with respect to B) is
the diagonal-block matrix:
M(1) 0 0 · · ·
0 C1 0 · · ·
0 0 C2 · · ·
· · · ·
:
Denition 2.7. Given M∈∗ quasi-denite, we say that the basis B of from Corollary 2.6 is a semi-orthogonal basis with respect to M.
3. Recurrence formulas
Let us consider a quasi-denite linear functional M on and let B={1} ∪(S
n¿1{fn; gn}) be the
corresponding semi-orthogonal basis. We denote
F0= (1;0)T; Fn= (fn; gn)T; (n¿1):
Also, for p1, p2, q1, q2∈, we write
We dene hP;Qi:2×2→R(2;2) by
hP;Qi= M(p1q1) M(p1q2) M(p2q1) M(p2q2)
!
: (4)
Notice that hQ;Pi=hP;QiT and hfP;Qi=hP; fQi ∀f∈.
Now, we consider xFn, n¿1. It is obvious that xfn, xgn∈n+1 and according to (3), we have
M(xfn(x)·xk) =M(xgn(x)·xk) = 0; 06k6n−2;
M(xfn(x)·xk√1−x2) =M(xgn(x)·xk√1−x2) = 0; 06k6n−3:
That is, xfn and xgn belong to the orthogonal complement of n−2 with respect to n+1. So, there
exist matrices n, n, Mn∈R(2;2) such that
xFn= nFn+1+nFn+MnFn−1; n¿1
holds. By identication of the coecients of xn+1 and xn√1−x2, we have that
n is the 2×2
identity matrix and so we get
xFn=Fn+1+nFn+MnFn−1; n¿1: (5)
In a similar way, √1−x2f
n,
√ 1−x2g
n∈n+1 and
M(√1−x2f
n(x)·xk) =M(
√ 1−x2g
n(x)·xk) = 0; 06k6n−2;
M(xfn(x)·xk√1−x2) =M(xgn(x)·xk√1−x2) = 0; 06k6n−3:
Proceeding as above, we get that there exist two matrices en, fMn∈R(2;2) such that
√
1−x2F
n=JFn+1+enFn+fMnFn−1; n¿1; (6)
where
J= 0 1
−1 0
! :
Taking into account that hFn;Fmi=Cnn; m (n; m¿1), we can obtain, after simple calculations:
hxFn;Fn+1i=Cn+1; h
√
1−x2F
n;Fni=enCn ;
h√1−x2F
n;Fn+1i=JCn+1; hxFn+1;Fni=Mn+1Cn ;
hxFn;Fni=nCn; h
√
1−x2F
n+1;Fni=fMn+1Cn :
So, the coecients for the recurrence relations (5) and (6) are determined by (Cn)n¿1, in the
following way:
n=hxFn;FniC −1
n ; Mn+1=Cn+1C
−1
n ;
e n=h
√
1−x2F
n;FniC −1
n ; fMn+1= −Cn+1JC
−1
n ;
Formulas (5) and (6) does not determine completely M1 andMf1. Relations (7) remain true if the
initial conditions M1={M(1)}−1C1 and Mf1={M(1)}−1C1JT are given.
Summing up we have shown
Theorem 3.1. Let M∈∗ be quasi-denite and let B be the associated semi-orthogonal basis.
Then; there are matrices n; Mn; en; fMn∈R(2;2) such that (5)–(7) hold. Besides Mn and fMn are
regular matrices. When the functional M is positive; then the eigenvalues of Mn are positive.
Proposition 3.2. Let M; Me∈∗ be quasi-denite. If M and Me have the same semi-orthogonal
associated basis B; then there is ∈R\{0} such that
M(F) =Me(F) ∀F∈:
Proof. Let B={1} ∪(Sn¿1{fn; gn}). For each F∈, there exist ak, bk, c∈R such that
F=c+X
k
akfk+ X
j bjgj
holds. But, B is a semi-orthogonal basis with respect to M and Me, and this fact implies M(F) =
cM(1), Me(F) =cMe(1). So, it suces to take =M(1)(Me(1))−1.
The above proposition determines a unique quasi-denite functional M∈∗ (except for a factor
∈R\{0}).
4. Orthogonal polynomials and semi-orthogonal basis
LetL be a quasi-denite Hermitian functional onL. Assume that Lis normalized, i.e., L(1) = 1. We denote by {n}∞n=0 the SMOP associated with L.
For each n¿1, we can dene, as in (1) the functions:
fn(x) = 2−nz−n[z2n−1(z) +2∗n−1(z)];
gn(x) = −i2
−nz−n[z
2n−1(z)−∗2n−1(z)];
(8)
where x= (z+z−1)=2.
(Whenever L is symmetric and, hence, the coecients of 2n−1 are real, formulas (8) should be compared with those given by Szeg˝o [6; (11.5.2)].
Obviously, {1} ∪ {fn; gn}n¿1 constitutes a basis of, according to Lemma 1.1 and Proposition 2.2.
Theorem 4.1. Consider B={1} ∪(Sn¿1{fn; gn}); where fn; gn are given by (8). Then B is a
semi-orthogonal basis with respect to the functional ML∈∗ dened in Denition 2:1: Besides; ML is
Proof. We will write the matrix of the bilinear form h·;·iML with respect to the basis B. Thus we have
ML(fn) =L[2
−nz−n(z
2n−1(z) +2∗n−1(z))] = 0; ∀n¿1;
ML(gn) =L
h
2−nz−ni−1(z
2n−1(z)−2∗n−1(z))
i
= 0; ∀n¿1
because of the orthogonality of {n}∞n=0. Also ML(1) =L(1) = 1.
Now for n¿m¿1, it follows that
ML(f
nfm) = 2
−n−mL(z−n[z
2n−1(z) +∗2n−1(z)]z
−m[z
2m−1(z) +2∗m−1(z)])
= 2−n−mL(
2n−1(z)2m−1(z)·z−n−m+2+2∗n−1(z)2m−1(z)·z−n−m+1
+2n−1(z)∗2m−1(z)·z
−n−m+1+∗
2n−1(z)2∗m−1(z)·z
−n−m):
When m¡n, since
L(2n−1(z)·z−k) =L(∗
2n−1(z)·z
−(k+1)) = 0; k= 0; : : : ;2n
−2;
then ML(f
nfm) = 0:
In a similar way, we can obtain that, for m¡n
ML(fngm) =ML(gmfm) =ML(gngm) = 0:
When n=m, it results that
ML(f2
n ) = 2
−2nL(
2n−1(z)2n−1(z)·z
−2n+2)
+ 2−2n+1L(
2n−1(z)2∗n−1(z)·z
−2n+1) + 2−2nL(∗
2n−1(z)2∗n−1(z)z
−2n):
Then, we have, after straightforward calculations:
L(2n−1(z)2n−1(z)·z−2n+2) =
−e2n−12n(0);
L(2n−1(z)∗
2n−1(z)·z
−2n+1) =e 2n−1;
L(∗
2n−1(z)2∗n−1(z)·z
−2n) =
−e2n−12n(0);
where en=L(n(z)·z−n)6= 0, and thus
ML(fn2) = 2
−(2n−1)e
2n−1{1−Re2n(0)};
ML(fngn) = −2
−(2n−1)e
2n−1Im2n(0);
ML(g2
n) = 2
−(2n−1)e
2n−1{1 + Re2n(0)}:
So, B constitutes a semi-orthogonal basis, according to Denition 2.7. Besides, the matrix of h·;·iML with respect to B is
ML(1) 0 0 · · ·
0 C1 0 · · ·
0 0 C2 · · ·
· · · ·
;
where
Cn= 2 −2n+1e
2n−1
1−Re2n(0) −Im2n(0)
−Im2n(0) 1 + Re2n(0) !
;
i.e., each Cn is quasi-denite (positive denite) if and only if L is quasi-denite (positive).
Proposition 4.2. For each n¿1 the functions fn and gn can be expressed as
fn(x) =
(1−2n(0))2n(z) + (1−2n(0))2∗n(z)
2nzn(1− |
2n(0)|2)
;
gn(x) =
(1 +2n(0))2n(z)−(1 +2n(0))2∗n(z)
2nzni(1− |
2n(0)|2)
:
(10)
Proof. It is an easy consequence of formula (8) and Szeg˝o’s recurrence formulas.
Proposition 4.3. The SMOP related to L; can be expressed as
2n−1(z) = 2n
−1zn−1
{fn(x) + ign(x)};
2n(z) = 2n −1zn
{(1 +2n(0)fn(x) + (1−2n(0))ign(x)}:
Proof. It suces to eliminate 2∗n−1(z) in (8) and 2∗n(z) in (10).
In these conditions, Theorem 4.1 has the following converse:
Theorem 4.4. Let M∈∗ be quasi-denite; such that trace Cn6= 0 for each n¿1. Then; there
exists just one quasi-denite Hermitian functional L∈L∗; such that M is induced by L; that is;
M=ML. Moreover; L is positive denite if and only if M is positive denite.
Proof. Consider a semi-orthogonal basis B={1} ∪(Sn¿1{fn; gn}) related to M. Without loss of
generality, we can assume that M(1) = 1. Then the matrix of h·;·iM with respect to B is
1 0 0 · · · 0 C1 0 · · ·
0 0 C2 · · ·
· · · ·
with
Cn=
M(f2
n) M(fngn)
M(gnfn) M(g2
n) !
:
We will obtain a Hermitian functional L such that the associated SMOP satisfy formulas (9). This system provides that
e2n−1= 22n
−2
{M(f2
n ) +M(g
2
n)}
with e2n−16= 0 because of trace Cn6= 0 (¿0). Now, the quasi-deniteness of M implies that
06= DetCn=M(fn2)M(g
2
n)−M(fngn)2=e −1 2n−12
2−4n(1
− |2n(0)|2)
then |2n(0)| 6= 1 and, when M is positive, |2n(0)|¡1.
So, once e2n−1 is known, formulas (9) give us 2n(0) and, because of Proposition 4.3, we have
{n}n∈N. Thus, this sequence {n} is uniquely determined by the matrix sequence (Cn). Now,
Favard’s theorem guarantees the existence of a unique Hermitian, quasi-denite (positive) functional (except for a non-zero factor)L∈∗ such that{n}n∈Nis the related SMOP. Besides, Mis induced
by L in the sense of Denition 2.1.
5. Application
Let L∈L∗ be quasi-denite and M:=ML its induced functional. If we dene Le∈L∗ by:
e
L= (1 2(z+z
−1)−a)L with a∈R, let Me:=Me
e
L ∈∗ be given by
e
M[(P; Q)] =Le[P+√1−x2Q] =L[(x−a)(P+√1−x2Q)] =M[(x−a)(P; Q)]
∀(P; Q)∈. So, Me= (x−a)M is the associated functional of Le, with the quasi-deniteness con-ditions given in Theorem 4.1.
Assume that Me is quasi-denite, and let (Fn) the sequence of semi-orthogonal functions related to Me, Fn= (Fn; Gn)T, n¿1. Then, we have
e
M(xkFn(x)) =M(xk(x−a)Fn(x)) = 0 (06k6n−1);
e
M(xk√1−x2F
n(x)) =M(xk
√
1−x2(x−a)F
n(x)) = 0 (06k6n−2)
from here (x−a)Fn, (x−a)Gn∈n+1 are orthogonal to n with respect to h·;·iM. Thus, there exist
unique matrices An, Bn∈R(2;2) such that
(x−a)Fn=An fn+1+Bn fn:
By comparing the coecients of xn+1 and √1−x2xn we obtain that A
n=I. So, if we assume that
M is quasi-denite, there exists a unique matrix Bn∈R(2;2) such that
When |a| 6= 1, notice that fn(a) has two determinations for=a+
n (a), respectively. Then formula (11) becomes
0 = [ fn++1(a); f−
n+1(a)] +Bn[ fn+(a); f − n (a)]:
Thus, the quasi-deniteness of [ f+
n+1(a); f
−
n+1(a)], (n¿1) implies the existence of Fn (n¿1)
veriying (11) and, moreover, the quasi-deniteness of Me. On the other hand,
n)6= 0, then the same relation for the even terms is obtained. That
is, according to Theorem 4.4, Le is quasi-denite when
Dn() =n∗(
By using formulas (8) and (11), we can obtain the relationship:
(z2−2az+ 1)en(z) =n+2(z) +rnzn(z) +snn∗(z);
where (en) is the SMOP associated with Le. So, we derive the determinantal formula:
(z2
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[5] M.C. Suarez, Polinomios ortogonales relativos a modicaciones de funcionales regulares y hermitianos, Doctoral Dissertation, Universidad de Vigo, 1983.
