Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol99.Issue1-2.1998:
Journal of Computational and Applied Mathematics 99 (1998) 3–14
Semiorthogonal functions and orthogonal polynomials on the
unit circle
Manuel Alfaroa; ∗; 1 , Mara Jose Canterob; 2 , Leandro Moralb; 2
a
b
Departamento de Matematicas, Universidad de Zaragoza, 50009 Zaragoza, Spain
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 × P satisfying 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
c 1998 Elsevier
view. Whenever the polynomials n have real coecients, we recover a well known result by Szeg˝o.
Science B.V. All rights reserved.
AMS classi cation: 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 polynomial modi cations of Hermitian functionals L de ned 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].
This research was supported by DGES PB96-0120-C03-02.
2
This research was supported by UZ97-CIE-10 (Universidad de Zaragoza).
1
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 4 0 - X
4
M. Alfaro et al. / Journal of Computational and Applied Mathematics 99 (1998) 3–14
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 Pn# 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
)
k z k ; 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
to M. 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 modi cation of the functional L.
Next, we show some technical results that we shall use later.
For z ∈ C\{0}, let us write x = (z + z −1 )=2 and y = (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−1 z 2n−1 +· · ·+ 0 , where k ∈ C.
We de ne f; g : C\{0} → C, by means of
f(x) = 2−n z −n [z(z) + ∗ (z)];
g(x) = −i2−n z −n [z(z) − ∗ (z)];
(1)
where ∗ (z) = z 2n−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 de ned 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 − x2 R2 (x);
√
g(x) = S1 (x) + 1 − x2 S2 (x):
#
Conversely; given R1 ∈ Pn# ; R2 ∈ Pn−2 ; S1 ∈ Pn−1 and S2 ∈ Pn−1
with R1 and S2 monic polynomials;
then there is a unique monic polynomial ∈ of degree 2n − 1; such that
R1 (x) +
√
S1 (x) +
√
1 − x2 R2 (x) = 2−n z −n [z(z) + ∗ (z)];
1 − x2 S2 (x) = − i2−n z −n [z(z) − ∗ (z)]:
M. Alfaro et al. / Journal of Computational and Applied Mathematics 99 (1998) 3–14
5
Proof. From (1) we have
f(x) = 2−n z −n
2n−1
X
( k z k+1 + k z 2n−k−1 )
k=0
= Tn (x) +
n−1
X
j=0
×
n−1
X
j=1
2j−n Re( n+j−1 + n−j−1 )Tj (x) −
√
1 − x2
2−j Im( 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 Im 2n−1 = 0, we can put
f(x) =
n
X
k=0
ak Tk (x) +
√
1 − x2
n−2
X
bk Uk (x):
k=0
with an = Re 2n−1 = 1:
P
Then,
the desired expression for f is obtained if we take R1 (x) = nk=0 ak Tk (x) and R2 (x)
Pn−2
= k=0 bk Uk (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 and S1 (x) ≡ 0. Moreover, if is in addition a
polynomial orthogonal on T with respect to a weight function w, then by
1 and S2
√Szeg˝o’s theory, R√
2
are orthogonal on [−1; 1] with respect to the weight functions w(x) 1 − x 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 ) = (P1 P2 + (1 − x2 )Q1 Q2 ; P1 Q2 + P2 Q1 )
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
2
(P; Q) → P + 1 − x2 Q establish a bijective correspondence between and P + 1 −
√x P which
preserves the product; so, in the sequel some times we will use the expression P + 1 − x2 Q to
denote elements of .
For every Hermitian functional on L, we can de ne a linear functional on in the following way:
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M. Alfaro et al. / Journal of Computational and Applied Mathematics 99 (1998) 3–14
De nition 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 de nite positive and is the corresponding orthogonality measure on T
(see [3, 4]), we can write:
ML [(P; Q)] =
Z
1
P(x) d(1 + 2 )(x) +
Z
1
√
−1
−1
1 − x2 Q(x) d(1 − 2 )(x);
where 1 and 2 are the measures on [−1; 1] given by
(
d1 (x) = −d(); 06¡;
d2 (x) = d();
x = cos :
6¡2;
Using the induction method, it is easy to deduce:
Proposition 2.2. (i) Any set Bn inductively de ned by
B0 = {1};
(2)
Bn = Bn−1 ∪ {h(1)
n ; hn };
n¿1;
(2)
#
(2)
#
where h(1)
n ∈ Pn × Pn−2 ; hn ∈ Pn−1 × Pn−1 is a basis of n .
(ii) Any set B de ned by means of
B = {1} ∪
[
!
(2)
{h(1)
n ; hn }
n¿1
;
(2)
where h(1)
n ; hn are as in (i); is a basis of .
It is well known that L de nes a bilinear Hermitian functional: h·; ·iL : × → C by means of
h; iL = L((z) (z −1 )):
This bilinear functional is called quasi-de nite (positive de nite) when the restriction of h·; ·iL to
n × n is quasi-de nite (positive de nite), for each n. This de nition is equivalent to the fact
that all principal minors of the moment matrix (hz k ; z j iL )∞
k; j = 0 are di erent 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 ; n iL 6= 0 (¿0).
Our main purpose is to carry over these results to the set of bilinear functionals on , via an
orthogonalization procedure.
De nition 2.3. Let M ∈ ∗ , where ∗ denotes the algebraic dual space of . We de ne the symmetric bilinear form h·; ·iM : × → R by
hF; GiM = M (F · G) ;
F; G ∈ :
M. Alfaro et al. / Journal of Computational and Applied Mathematics 99 (1998) 3–14
7
We say that M is quasi-de nite (positive de nite) if the restriction of h ; iM to n × n is quaside nite (positive de nite); for each n ∈ N.
Notice that the de nition of the positivity of M is similar to the one given for Hermitian functionals. However, the quasi-de niteness of M does not imply that hF; FiM 6= 0 for all F ∈ M, F 6= 0.
Theorem 2.4 (Orthogonalization procedure). Let M ∈ ∗ be a quasi-de nite (positive de nite)
functional. Then; for each n ∈ N; there exists a unique basis Bn of n ; given by
Bn = Bn−1 ∪ {fn ; gn };
B0 = 1;
(2)
#
(1)
(2)
#
(1)
where fn = (R(1)
and Sn(2) monic
n ; Rn ) ∈ Pn × Pn−2 and gn = (Sn ; Sn ) ∈ Pn−1 × Pn−1 ; with Rn
polynomials; such that
M(fn · x k ) = M(gn · x k ) = 0; 06k6n − 1;
√
√
M(fn · 1 − x2 x k ) = M(gn · 1 − x2 x k ) = 0;
06k6n − 2:
(3)
Besides; the matrix
Cn =
M(fn2 ) M(fn gn )
M(fn gn ) M(gn2 )
!
is quasi-de nite (positive de nite.).
Proof. Let M ∈ ∗ be quasi-de nite (positive de nite). 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) = (ce1 ; 1):
Let M(f1 ) = M(g1 ) = 0. Then,
M(f1 ) = M(x) + c1 M(1) = 0;
√
M(g1 ) = ce1 M(1) + M
1 − x2 = 0
but M(1) 6= 0, because of the quasi-de niteness of h·; ·iM on 0 . Thus we have uniqueness for c1 ,
ce1 in the above system.
Moreover, the matrix
M(1)
M(f1 )
M(f1 )
M(f12 )
M(g1 ) M(f1 g1 )
M(g1 )
M(f1 g1 )
=
M(g12 )
M(1) 0
0 C1
is quasi-de nite (positive de nite) because of the quasi-de niteness (positivity) of M. Thus C1 is
quasi-de nite (positive de nite). By applying induction, the proof follows straightforward.
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M. Alfaro et al. / Journal of Computational and Applied Mathematics 99 (1998) 3–14
If M is induced by a positive de nite functional, formulas (3) in Theorem 2.4 leads to four
‘unusual’ orthogonality conditions:
Theorem 2.5. Let ML be a functional on induced by a Hermitian positive de nite 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;
(2)
the polynomials R(1)
n and Rn ; introduced in Theorem 2:4; satisfy
Z
1
k
R(1)
n (x)x d (1 + 2 )(x) +
√
k
1 − x2 d(1 − 2 )(x) = 0;
R(2)
n (x)x
1
−1
−1
Z
Z
1
−1
k
2
R(2)
n (x)x (1 − x ) d(1 + 2 )(x) +
Z
1
−1
06k6n − 1;
√
k
R(1)
1 − x2 d(1 − 2 )(x) = 0;
n (x)x
06k6n − 2:
Two analogous formulas are true for the polynomials Sn(1) and Sn(2) .
Corollary 2.6. Let M ∈ ∗ be quasi-de nite (positive). Then there is a unique basis B of such
that
[
B = {1} ∪
!
{fn ; gn } ;
n¿1
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 · · ·
··· ···
De nition 2.7. Given M ∈ ∗ quasi-de nite, 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-de nite linear functional M on and let B = {1} ∪(
corresponding semi-orthogonal basis.
We denote
F0 = (1; 0)T ;
Fn = (fn ; gn )T ;
(n¿1):
Also, for p1 , p2 , q1 , q2 ∈ , we write
P = (p1 ; p2 )T ;
Q = (q1 ; q2 )T ;
(n¿1):
S
n¿1 {fn ; gn })
be the
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M. Alfaro et al. / Journal of Computational and Applied Mathematics 99 (1998) 3–14
We de ne hP; Qi : 2 × 2 → R(2; 2) by
hP; Qi =
M(p1 q1 ) M(p1 q2 )
M(p2 q1 ) M(p2 q2 )
!
(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) · x k ) = M(xgn (x) · x k ) = 0; 06k6n − 2;
√
√
M(xfn (x) · x k 1 − x2 ) = M(xgn (x) · x k 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 =
n Fn+1
+ n Fn + Mn Fn−1 ;
n¿1
√
holds. By identi cation of the coecients of x n+1 and x n 1 − x2 , we have that
identity matrix and so we get
n
is the 2 × 2
xFn = Fn+1 + n Fn + Mn Fn−1 ; n¿1:
√
√
In a similar way, 1 − x2 fn , 1 − x2 gn ∈ n+1 and
√
√
M( 1 − x2 fn (x) · x k ) = M( 1 − x2 gn (x) · x k ) = 0; 06k6n − 2;
√
√
M(xfn (x) · x k 1 − x2 ) = M(xgn (x) · x k 1 − x2 ) = 0; 06k6n − 3:
fn ∈ R(2; 2) such that
Proceeding as above, we get that there exist two matrices en , M
√
fn Fn−1 ; n¿1;
1 − x2 Fn = J Fn+1 + en Fn + M
where
J=
(5)
(6)
!
0 1
:
−1 0
Taking into account that hFn ; Fm i = Cn n; m (n; m¿1), we can obtain, after simple calculations:
√
hxFn ; Fn+1 i = Cn+1 ;
h 1 − x2 Fn ; Fn i = en Cn ;
√
h 1 − x2 Fn ; Fn+1 i = JCn+1 ; hxFn+1 ; Fn i = Mn+1 Cn ;
√
fn+1 Cn :
hxFn ; Fn i = n Cn ;
h 1 − x2 Fn+1 ; Fn i = M
So, the coecients for the recurrence relations (5) and (6) are determined by (Cn )n¿1 , in the
following way:
n = hxFn ; Fn iCn−1 ;
√
en = h 1 − x2 Fn ; Fn iCn−1 ;
Mn+1 = Cn+1 Cn−1 ;
fn+1 = − Cn+1 JC −1 ;
M
n
(n¿1):
(7)
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M. Alfaro et al. / Journal of Computational and Applied Mathematics 99 (1998) 3–14
f1 . Relations (7) remain true if the
Formulas (5) and (6) does not determine completely M1 and M
−1
−1
T
f
initial conditions M1 = {M(1)} C1 and M1 = {M(1)} C1 J are given.
Summing up we have shown
Theorem 3.1. Let M ∈ ∗ be quasi-de nite and let B be the associated semi-orthogonal basis.
fn ∈ R(2; 2) such that (5)–(7) hold. Besides Mn and M
fn are
Then; there are matrices n ; Mn ; en ; M
regular matrices. When the functional M is positive; then the eigenvalues of Mn are positive.
Proposition 3.2. Let M; Me ∈ ∗ be quasi-de nite. If M and Me have the same semi-orthogonal
associated basis B; then there is ∈ R\{0} such that
e
M(F) = M(F)
∀F ∈ :
Proof. Let B = {1} ∪(
F =c +
X
k
S
ak fk +
n¿1 {fn ; gn }).
X
For each F ∈ , there exist ak , bk , c ∈ R such that
bj gj
j
e and this fact implies M(F) =
holds. But, B is a semi-orthogonal basis with respect to M and M,
−1
e
e
e
cM(1), M(F) = cM(1). So, it suces to take = M(1)(M(1)) .
The above proposition determines a unique quasi-de nite functional M ∈ ∗ (except for a factor
∈ R\{0}).
4. Orthogonal polynomials and semi-orthogonal basis
Let L be a quasi-de nite Hermitian functional on L. Assume that L is normalized, i.e., L(1) = 1.
We denote by {n }∞
n=0 the SMOP associated with L.
For each n¿1, we can de ne, as in (1) the functions:
∗ (z)];
fn (x) = 2−n z −n [z2n−1 (z) + 2n−1
g (x) = − i2−n z −n [z
(z) − ∗ (z)];
n
2n−1
(8)
2n−1
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.
S
Theorem 4.1. Consider B = {1} ∪( n¿1 {fn ; gn }); where fn ; gn are given by (8). Then B is a semiorthogonal basis with respect to the functional ML ∈ ∗ de ned in De nition 2:1: Besides; ML is
quasi-de nite and; when L is positive de nite; then ML is positive de nite.
M. Alfaro et al. / Journal of Computational and Applied Mathematics 99 (1998) 3–14
11
Proof. We will write the matrix of the bilinear form h·; ·iML with respect to the basis B. Thus we
have
∗ (z))] = 0;
ML (fn ) = L[2−n z −n (z2n−1 (z) + 2n−1
h
i
∀n¿1;
∗ (z)) = 0;
ML (gn ) = L 2−n z −n i−1 (z2n−1 (z) − 2n−1
∀n¿1
because of the orthogonality of {n }∞
n=0 . Also ML (1) = L(1) = 1.
Now for n¿m¿1, it follows that
∗ (z)]z −m [z
∗
ML (fn fm ) = 2−n−m L(z −n [z2n−1 (z) + 2n−1
2m−1 (z) + 2m−1 (z)])
∗ (z)
−n−m+1
= 2−n−m L(2n−1 (z)2m−1 (z) · z −n−m+2 + 2n−1
2m−1 (z) · z
∗ (z) · z −n−m+1 + ∗ (z)∗ (z) · z −n−m ):
+ 2n−1 (z)2m−1
2n−1
2m−1
When m¡n, since
∗ (z) · z −(k+1) ) = 0;
L(2n−1 (z) · z −k ) = L(2n−1
k = 0; : : : ; 2n − 2;
then ML (fn fm ) = 0:
In a similar way, we can obtain that, for m¡n
ML (fn gm ) = ML (gm fm ) = ML (gn gm ) = 0:
When n = m, it results that
ML (fn2 ) = 2−2n L(2n−1 (z)2n−1 (z) · z −2n+2 )
∗ (z) · z −2n+1 ) + 2−2n L(∗ (z)∗ (z)z −2n ):
+ 2−2n+1 L(2n−1 (z)2n−1
2n−1
2n−1
Then, we have, after straightforward calculations:
L(2n−1 (z)2n−1 (z) · z −2n+2 ) = − e2n−1 2n (0);
∗ (z) · z −2n+1 ) = e
L(2n−1 (z)2n−1
2n−1 ;
∗ (z)∗ (z) · z −2n ) = − e
L(2n−1
2n−1 2n (0);
2n−1
where en = L(n (z) · z −n ) 6= 0, and thus
ML (fn2 ) = 2−(2n−1) e2n−1 {1 − Re 2n (0)};
ML (fn gn ) = − 2−(2n−1) e2n−1 Im 2n (0);
ML (gn2 ) = 2−(2n−1) e2n−1 {1 + Re 2n (0)}:
(9)
12
M. Alfaro et al. / Journal of Computational and Applied Mathematics 99 (1998) 3–14
So, B constitutes a semi-orthogonal basis, according to De nition 2.7. Besides, the matrix of
h·; ·iML with respect to B is
ML (1)
0
0
···
0
C1
0
···
where
−2n+1
Cn = 2
e2n−1
0 ···
0 ···
;
C2 · · ·
··· ···
1 − Re 2n (0) −Im 2n (0)
1 + Re 2n (0)
−Im 2n (0)
!
;
i.e., each Cn is quasi-de nite (positive de nite) if and only if L is quasi-de nite (positive).
Proposition 4.2. For each n¿1 the functions fn and gn can be expressed as
∗ (z)
(1 − 2n (0))2n (z) + (1 − 2n (0))2n
fn (x) =
;
2n z n (1 − |2n (0)|2 )
∗ (z)
(1 + 2n (0))2n (z) − (1 + 2n (0))2n
gn (x) =
:
2n z n i(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−1 z n−1 {fn (x) + ign (x)};
2n (z) = 2n−1 z n {(1 + 2n (0)fn (x) + (1 − 2n (0))ign (x)}:
∗ (z) in (8) and ∗ (z) in (10).
Proof. It suces to eliminate 2n−1
2n
In these conditions, Theorem 4.1 has the following converse:
Theorem 4.4. Let M ∈ ∗ be quasi-de nite; such that trace Cn 6= 0 for each n¿1. Then; there
exists just one quasi-de nite Hermitian functional L ∈ L∗ ; such that M is induced by L; that is;
M = ML . Moreover; L is positive de nite if and only if M is positive de nite.
S
Proof. Consider a semi-orthogonal basis B = {1} ∪( n¿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 · · ·
··· ···
M. Alfaro et al. / Journal of Computational and Applied Mathematics 99 (1998) 3–14
13
with
Cn =
M(fn2 )
M(fn gn )
M(gn fn ) M(gn2 )
!
:
We will obtain a Hermitian functional L such that the associated SMOP satisfy formulas (9).
This system provides that
e2n−1 = 22n−2 {M(fn2 ) + M(gn2 )}
with e2n−1 6= 0 because of trace Cn 6= 0 (¿0). Now, the quasi-de niteness of M implies that
−1
0 6= Det Cn = M(fn2 )M(gn2 ) − M(fn gn )2 = e2n−1
22−4n (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-de nite (positive) functional
(except for a non-zero factor) L ∈ ∗ such that {n }n∈N is the related SMOP. Besides, M is induced
by L in the sense of De nition 2.1.
5. Application
Let L ∈ L∗ be quasi-de nite and M := ML its induced functional. If we de ne Le ∈ L∗ by:
Le = ( 21 (z + z −1 ) − a)L with a ∈ R, let Me := MeLe ∈ ∗ be given by
e +
e
M[(P;
Q)] = L[P
√
1 − x2 Q] = L[(x − a)(P +
√
1 − x2 Q)] = M[(x − a)(P; Q)]
e with the quasi-de niteness con∀(P; Q) ∈ . So, Me = (x − a)M is the associated functional of L,
ditions given in Theorem 4.1.
Assume that Me is quasi-de nite, and let (Fn ) the sequence of semi-orthogonal functions related
e Fn = (Fn ; Gn )T , n¿1. Then, we have
to M,
e k Fn (x)) = M(x k (x − a)Fn (x)) = 0 (06k6n − 1);
M(x
√
√
e k 1 − x2 Fn (x)) = M(x k 1 − x2 (x − a)Fn (x)) = 0 (06k6n − 2)
M(x
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 x n+1 and 1 − x2 x n we obtain that An = I . So, if we assume that
M is quasi-de nite, there exists a unique matrix Bn ∈ R(2; 2) such that
(x − a)Fn = fn+1 + Bn fn :
(11)
14
M. Alfaro et al. / Journal of Computational and Applied Mathematics 99 (1998) 3–14
√
√
When |a| =
6 1, notice that fn (a) has two determinations for = a + a2 − 1 and −1 = a − a2 − 1,
that we denote by fn+ (a) and fn− (a), respectively. Then formula (11) becomes
−
+
(a)] + Bn [ fn+ (a); fn− (a)]:
(a); fn+1
0 = [ fn+1
−
+
Thus, the quasi-de niteness of [ fn+1
(a); fn+1
(a)], (n¿1) implies the existence of Fn (n¿1)
e
veri ying (11) and, moreover, the quasi-de niteness of M.
On the other hand,
det[
fn+ (a);
fn− (a)] =
=
1
i22n
1
∗ ( )
2n−1 ( ) + 2n−1
( ) − ∗ ( )
i22n−2
2n−1
∗
2n−1 ( )
( )
2n−1
2n−1
∗ ( −1 )
−1 2n−1 ( −1 ) + 2n−1
−1
( −1 ) − ∗ ( −1 )
2n−1
2n−1
6= 0:
( −1 )
∗ ( −1 )
2n−1
−1 2n−1
e n · F T ) 6= 0, then the same relation for the even terms is obtained. That
If we claim that trace M(F
n
is, according to Theorem 4.4, Le is quasi-de nite when
Dn ( ) = n∗ ( −1 ) n ( ) − n∗ ( ) −1 n ( −1 ) 6= 0
(12)
∀n¿1 and 6= ± 1. This requirement is the same given in [5] for the quasi-de niteness of Le in
the particular case Le = Re(z − )L.
By using formulas (8) and (11), we can obtain the relationship:
en (z) = n+2 (z) + rn zn (z) + sn ∗ (z);
(z 2 − 2az + 1)
n
e So, we derive the determinantal formula:
en ) is the SMOP associated with L.
where (
n+2 (z)
2
−1
e
(z − 2az + 1)n (z) = Dn ( ) n+2 ( )
( −1 )
n+2
zn (z)
n ( )
−1 n ( −1 )
n
∗
−1
( )
n∗ (z)
∗ ( )
n
en ) in terms of (n ), constrained to the condition (12).
which gives (
References
[1] M.J. Cantero, Polinomios ortogonales sobre la circunferencia unidad. Modi caciones de los parametros de Schur,
Doctoral Dissertation, Universidad de Zaragoza, 1997.
[2] G. Freud, Orthogonal Polynomials, Akademiai Kiado, Budapest and Pergamon Press, Oxford, 1971, 1985.
[3] Y.L. Geronimus, Orthogonal Polynomials, Consultants Bureau, New York, 1961.
[4] Y.L. Geronimus, Polynomials orthogonal on a circle and their applications, Transl. Amer. Math. Soc. 3 Serie 1 (1962)
1–78.
[5] M.C. Suarez, Polinomios ortogonales relativos a modi caciones de funcionales regulares y hermitianos, Doctoral
Dissertation, Universidad de Vigo, 1983.
[6] G. Szego, Orthogonal Polynomials, 4th ed., AMS Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1975.
Semiorthogonal functions and orthogonal polynomials on the
unit circle
Manuel Alfaroa; ∗; 1 , Mara Jose Canterob; 2 , Leandro Moralb; 2
a
b
Departamento de Matematicas, Universidad de Zaragoza, 50009 Zaragoza, Spain
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 × P satisfying 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
c 1998 Elsevier
view. Whenever the polynomials n have real coecients, we recover a well known result by Szeg˝o.
Science B.V. All rights reserved.
AMS classi cation: 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 polynomial modi cations of Hermitian functionals L de ned 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].
This research was supported by DGES PB96-0120-C03-02.
2
This research was supported by UZ97-CIE-10 (Universidad de Zaragoza).
1
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 4 0 - X
4
M. Alfaro et al. / Journal of Computational and Applied Mathematics 99 (1998) 3–14
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 Pn# 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
)
k z k ; 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
to M. 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 modi cation of the functional L.
Next, we show some technical results that we shall use later.
For z ∈ C\{0}, let us write x = (z + z −1 )=2 and y = (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−1 z 2n−1 +· · ·+ 0 , where k ∈ C.
We de ne f; g : C\{0} → C, by means of
f(x) = 2−n z −n [z(z) + ∗ (z)];
g(x) = −i2−n z −n [z(z) − ∗ (z)];
(1)
where ∗ (z) = z 2n−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 de ned 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 − x2 R2 (x);
√
g(x) = S1 (x) + 1 − x2 S2 (x):
#
Conversely; given R1 ∈ Pn# ; R2 ∈ Pn−2 ; S1 ∈ Pn−1 and S2 ∈ Pn−1
with R1 and S2 monic polynomials;
then there is a unique monic polynomial ∈ of degree 2n − 1; such that
R1 (x) +
√
S1 (x) +
√
1 − x2 R2 (x) = 2−n z −n [z(z) + ∗ (z)];
1 − x2 S2 (x) = − i2−n z −n [z(z) − ∗ (z)]:
M. Alfaro et al. / Journal of Computational and Applied Mathematics 99 (1998) 3–14
5
Proof. From (1) we have
f(x) = 2−n z −n
2n−1
X
( k z k+1 + k z 2n−k−1 )
k=0
= Tn (x) +
n−1
X
j=0
×
n−1
X
j=1
2j−n Re( n+j−1 + n−j−1 )Tj (x) −
√
1 − x2
2−j Im( 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 Im 2n−1 = 0, we can put
f(x) =
n
X
k=0
ak Tk (x) +
√
1 − x2
n−2
X
bk Uk (x):
k=0
with an = Re 2n−1 = 1:
P
Then,
the desired expression for f is obtained if we take R1 (x) = nk=0 ak Tk (x) and R2 (x)
Pn−2
= k=0 bk Uk (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 and S1 (x) ≡ 0. Moreover, if is in addition a
polynomial orthogonal on T with respect to a weight function w, then by
1 and S2
√Szeg˝o’s theory, R√
2
are orthogonal on [−1; 1] with respect to the weight functions w(x) 1 − x 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 ) = (P1 P2 + (1 − x2 )Q1 Q2 ; P1 Q2 + P2 Q1 )
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
2
(P; Q) → P + 1 − x2 Q establish a bijective correspondence between and P + 1 −
√x P which
preserves the product; so, in the sequel some times we will use the expression P + 1 − x2 Q to
denote elements of .
For every Hermitian functional on L, we can de ne a linear functional on in the following way:
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M. Alfaro et al. / Journal of Computational and Applied Mathematics 99 (1998) 3–14
De nition 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 de nite positive and is the corresponding orthogonality measure on T
(see [3, 4]), we can write:
ML [(P; Q)] =
Z
1
P(x) d(1 + 2 )(x) +
Z
1
√
−1
−1
1 − x2 Q(x) d(1 − 2 )(x);
where 1 and 2 are the measures on [−1; 1] given by
(
d1 (x) = −d(); 06¡;
d2 (x) = d();
x = cos :
6¡2;
Using the induction method, it is easy to deduce:
Proposition 2.2. (i) Any set Bn inductively de ned by
B0 = {1};
(2)
Bn = Bn−1 ∪ {h(1)
n ; hn };
n¿1;
(2)
#
(2)
#
where h(1)
n ∈ Pn × Pn−2 ; hn ∈ Pn−1 × Pn−1 is a basis of n .
(ii) Any set B de ned by means of
B = {1} ∪
[
!
(2)
{h(1)
n ; hn }
n¿1
;
(2)
where h(1)
n ; hn are as in (i); is a basis of .
It is well known that L de nes a bilinear Hermitian functional: h·; ·iL : × → C by means of
h; iL = L((z) (z −1 )):
This bilinear functional is called quasi-de nite (positive de nite) when the restriction of h·; ·iL to
n × n is quasi-de nite (positive de nite), for each n. This de nition is equivalent to the fact
that all principal minors of the moment matrix (hz k ; z j iL )∞
k; j = 0 are di erent 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 ; n iL 6= 0 (¿0).
Our main purpose is to carry over these results to the set of bilinear functionals on , via an
orthogonalization procedure.
De nition 2.3. Let M ∈ ∗ , where ∗ denotes the algebraic dual space of . We de ne the symmetric bilinear form h·; ·iM : × → R by
hF; GiM = M (F · G) ;
F; G ∈ :
M. Alfaro et al. / Journal of Computational and Applied Mathematics 99 (1998) 3–14
7
We say that M is quasi-de nite (positive de nite) if the restriction of h ; iM to n × n is quaside nite (positive de nite); for each n ∈ N.
Notice that the de nition of the positivity of M is similar to the one given for Hermitian functionals. However, the quasi-de niteness of M does not imply that hF; FiM 6= 0 for all F ∈ M, F 6= 0.
Theorem 2.4 (Orthogonalization procedure). Let M ∈ ∗ be a quasi-de nite (positive de nite)
functional. Then; for each n ∈ N; there exists a unique basis Bn of n ; given by
Bn = Bn−1 ∪ {fn ; gn };
B0 = 1;
(2)
#
(1)
(2)
#
(1)
where fn = (R(1)
and Sn(2) monic
n ; Rn ) ∈ Pn × Pn−2 and gn = (Sn ; Sn ) ∈ Pn−1 × Pn−1 ; with Rn
polynomials; such that
M(fn · x k ) = M(gn · x k ) = 0; 06k6n − 1;
√
√
M(fn · 1 − x2 x k ) = M(gn · 1 − x2 x k ) = 0;
06k6n − 2:
(3)
Besides; the matrix
Cn =
M(fn2 ) M(fn gn )
M(fn gn ) M(gn2 )
!
is quasi-de nite (positive de nite.).
Proof. Let M ∈ ∗ be quasi-de nite (positive de nite). 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) = (ce1 ; 1):
Let M(f1 ) = M(g1 ) = 0. Then,
M(f1 ) = M(x) + c1 M(1) = 0;
√
M(g1 ) = ce1 M(1) + M
1 − x2 = 0
but M(1) 6= 0, because of the quasi-de niteness of h·; ·iM on 0 . Thus we have uniqueness for c1 ,
ce1 in the above system.
Moreover, the matrix
M(1)
M(f1 )
M(f1 )
M(f12 )
M(g1 ) M(f1 g1 )
M(g1 )
M(f1 g1 )
=
M(g12 )
M(1) 0
0 C1
is quasi-de nite (positive de nite) because of the quasi-de niteness (positivity) of M. Thus C1 is
quasi-de nite (positive de nite). By applying induction, the proof follows straightforward.
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M. Alfaro et al. / Journal of Computational and Applied Mathematics 99 (1998) 3–14
If M is induced by a positive de nite functional, formulas (3) in Theorem 2.4 leads to four
‘unusual’ orthogonality conditions:
Theorem 2.5. Let ML be a functional on induced by a Hermitian positive de nite 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;
(2)
the polynomials R(1)
n and Rn ; introduced in Theorem 2:4; satisfy
Z
1
k
R(1)
n (x)x d (1 + 2 )(x) +
√
k
1 − x2 d(1 − 2 )(x) = 0;
R(2)
n (x)x
1
−1
−1
Z
Z
1
−1
k
2
R(2)
n (x)x (1 − x ) d(1 + 2 )(x) +
Z
1
−1
06k6n − 1;
√
k
R(1)
1 − x2 d(1 − 2 )(x) = 0;
n (x)x
06k6n − 2:
Two analogous formulas are true for the polynomials Sn(1) and Sn(2) .
Corollary 2.6. Let M ∈ ∗ be quasi-de nite (positive). Then there is a unique basis B of such
that
[
B = {1} ∪
!
{fn ; gn } ;
n¿1
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 · · ·
··· ···
De nition 2.7. Given M ∈ ∗ quasi-de nite, 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-de nite linear functional M on and let B = {1} ∪(
corresponding semi-orthogonal basis.
We denote
F0 = (1; 0)T ;
Fn = (fn ; gn )T ;
(n¿1):
Also, for p1 , p2 , q1 , q2 ∈ , we write
P = (p1 ; p2 )T ;
Q = (q1 ; q2 )T ;
(n¿1):
S
n¿1 {fn ; gn })
be the
9
M. Alfaro et al. / Journal of Computational and Applied Mathematics 99 (1998) 3–14
We de ne hP; Qi : 2 × 2 → R(2; 2) by
hP; Qi =
M(p1 q1 ) M(p1 q2 )
M(p2 q1 ) M(p2 q2 )
!
(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) · x k ) = M(xgn (x) · x k ) = 0; 06k6n − 2;
√
√
M(xfn (x) · x k 1 − x2 ) = M(xgn (x) · x k 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 =
n Fn+1
+ n Fn + Mn Fn−1 ;
n¿1
√
holds. By identi cation of the coecients of x n+1 and x n 1 − x2 , we have that
identity matrix and so we get
n
is the 2 × 2
xFn = Fn+1 + n Fn + Mn Fn−1 ; n¿1:
√
√
In a similar way, 1 − x2 fn , 1 − x2 gn ∈ n+1 and
√
√
M( 1 − x2 fn (x) · x k ) = M( 1 − x2 gn (x) · x k ) = 0; 06k6n − 2;
√
√
M(xfn (x) · x k 1 − x2 ) = M(xgn (x) · x k 1 − x2 ) = 0; 06k6n − 3:
fn ∈ R(2; 2) such that
Proceeding as above, we get that there exist two matrices en , M
√
fn Fn−1 ; n¿1;
1 − x2 Fn = J Fn+1 + en Fn + M
where
J=
(5)
(6)
!
0 1
:
−1 0
Taking into account that hFn ; Fm i = Cn n; m (n; m¿1), we can obtain, after simple calculations:
√
hxFn ; Fn+1 i = Cn+1 ;
h 1 − x2 Fn ; Fn i = en Cn ;
√
h 1 − x2 Fn ; Fn+1 i = JCn+1 ; hxFn+1 ; Fn i = Mn+1 Cn ;
√
fn+1 Cn :
hxFn ; Fn i = n Cn ;
h 1 − x2 Fn+1 ; Fn i = M
So, the coecients for the recurrence relations (5) and (6) are determined by (Cn )n¿1 , in the
following way:
n = hxFn ; Fn iCn−1 ;
√
en = h 1 − x2 Fn ; Fn iCn−1 ;
Mn+1 = Cn+1 Cn−1 ;
fn+1 = − Cn+1 JC −1 ;
M
n
(n¿1):
(7)
10
M. Alfaro et al. / Journal of Computational and Applied Mathematics 99 (1998) 3–14
f1 . Relations (7) remain true if the
Formulas (5) and (6) does not determine completely M1 and M
−1
−1
T
f
initial conditions M1 = {M(1)} C1 and M1 = {M(1)} C1 J are given.
Summing up we have shown
Theorem 3.1. Let M ∈ ∗ be quasi-de nite and let B be the associated semi-orthogonal basis.
fn ∈ R(2; 2) such that (5)–(7) hold. Besides Mn and M
fn are
Then; there are matrices n ; Mn ; en ; M
regular matrices. When the functional M is positive; then the eigenvalues of Mn are positive.
Proposition 3.2. Let M; Me ∈ ∗ be quasi-de nite. If M and Me have the same semi-orthogonal
associated basis B; then there is ∈ R\{0} such that
e
M(F) = M(F)
∀F ∈ :
Proof. Let B = {1} ∪(
F =c +
X
k
S
ak fk +
n¿1 {fn ; gn }).
X
For each F ∈ , there exist ak , bk , c ∈ R such that
bj gj
j
e and this fact implies M(F) =
holds. But, B is a semi-orthogonal basis with respect to M and M,
−1
e
e
e
cM(1), M(F) = cM(1). So, it suces to take = M(1)(M(1)) .
The above proposition determines a unique quasi-de nite functional M ∈ ∗ (except for a factor
∈ R\{0}).
4. Orthogonal polynomials and semi-orthogonal basis
Let L be a quasi-de nite Hermitian functional on L. Assume that L is normalized, i.e., L(1) = 1.
We denote by {n }∞
n=0 the SMOP associated with L.
For each n¿1, we can de ne, as in (1) the functions:
∗ (z)];
fn (x) = 2−n z −n [z2n−1 (z) + 2n−1
g (x) = − i2−n z −n [z
(z) − ∗ (z)];
n
2n−1
(8)
2n−1
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.
S
Theorem 4.1. Consider B = {1} ∪( n¿1 {fn ; gn }); where fn ; gn are given by (8). Then B is a semiorthogonal basis with respect to the functional ML ∈ ∗ de ned in De nition 2:1: Besides; ML is
quasi-de nite and; when L is positive de nite; then ML is positive de nite.
M. Alfaro et al. / Journal of Computational and Applied Mathematics 99 (1998) 3–14
11
Proof. We will write the matrix of the bilinear form h·; ·iML with respect to the basis B. Thus we
have
∗ (z))] = 0;
ML (fn ) = L[2−n z −n (z2n−1 (z) + 2n−1
h
i
∀n¿1;
∗ (z)) = 0;
ML (gn ) = L 2−n z −n i−1 (z2n−1 (z) − 2n−1
∀n¿1
because of the orthogonality of {n }∞
n=0 . Also ML (1) = L(1) = 1.
Now for n¿m¿1, it follows that
∗ (z)]z −m [z
∗
ML (fn fm ) = 2−n−m L(z −n [z2n−1 (z) + 2n−1
2m−1 (z) + 2m−1 (z)])
∗ (z)
−n−m+1
= 2−n−m L(2n−1 (z)2m−1 (z) · z −n−m+2 + 2n−1
2m−1 (z) · z
∗ (z) · z −n−m+1 + ∗ (z)∗ (z) · z −n−m ):
+ 2n−1 (z)2m−1
2n−1
2m−1
When m¡n, since
∗ (z) · z −(k+1) ) = 0;
L(2n−1 (z) · z −k ) = L(2n−1
k = 0; : : : ; 2n − 2;
then ML (fn fm ) = 0:
In a similar way, we can obtain that, for m¡n
ML (fn gm ) = ML (gm fm ) = ML (gn gm ) = 0:
When n = m, it results that
ML (fn2 ) = 2−2n L(2n−1 (z)2n−1 (z) · z −2n+2 )
∗ (z) · z −2n+1 ) + 2−2n L(∗ (z)∗ (z)z −2n ):
+ 2−2n+1 L(2n−1 (z)2n−1
2n−1
2n−1
Then, we have, after straightforward calculations:
L(2n−1 (z)2n−1 (z) · z −2n+2 ) = − e2n−1 2n (0);
∗ (z) · z −2n+1 ) = e
L(2n−1 (z)2n−1
2n−1 ;
∗ (z)∗ (z) · z −2n ) = − e
L(2n−1
2n−1 2n (0);
2n−1
where en = L(n (z) · z −n ) 6= 0, and thus
ML (fn2 ) = 2−(2n−1) e2n−1 {1 − Re 2n (0)};
ML (fn gn ) = − 2−(2n−1) e2n−1 Im 2n (0);
ML (gn2 ) = 2−(2n−1) e2n−1 {1 + Re 2n (0)}:
(9)
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M. Alfaro et al. / Journal of Computational and Applied Mathematics 99 (1998) 3–14
So, B constitutes a semi-orthogonal basis, according to De nition 2.7. Besides, the matrix of
h·; ·iML with respect to B is
ML (1)
0
0
···
0
C1
0
···
where
−2n+1
Cn = 2
e2n−1
0 ···
0 ···
;
C2 · · ·
··· ···
1 − Re 2n (0) −Im 2n (0)
1 + Re 2n (0)
−Im 2n (0)
!
;
i.e., each Cn is quasi-de nite (positive de nite) if and only if L is quasi-de nite (positive).
Proposition 4.2. For each n¿1 the functions fn and gn can be expressed as
∗ (z)
(1 − 2n (0))2n (z) + (1 − 2n (0))2n
fn (x) =
;
2n z n (1 − |2n (0)|2 )
∗ (z)
(1 + 2n (0))2n (z) − (1 + 2n (0))2n
gn (x) =
:
2n z n i(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−1 z n−1 {fn (x) + ign (x)};
2n (z) = 2n−1 z n {(1 + 2n (0)fn (x) + (1 − 2n (0))ign (x)}:
∗ (z) in (8) and ∗ (z) in (10).
Proof. It suces to eliminate 2n−1
2n
In these conditions, Theorem 4.1 has the following converse:
Theorem 4.4. Let M ∈ ∗ be quasi-de nite; such that trace Cn 6= 0 for each n¿1. Then; there
exists just one quasi-de nite Hermitian functional L ∈ L∗ ; such that M is induced by L; that is;
M = ML . Moreover; L is positive de nite if and only if M is positive de nite.
S
Proof. Consider a semi-orthogonal basis B = {1} ∪( n¿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 · · ·
··· ···
M. Alfaro et al. / Journal of Computational and Applied Mathematics 99 (1998) 3–14
13
with
Cn =
M(fn2 )
M(fn gn )
M(gn fn ) M(gn2 )
!
:
We will obtain a Hermitian functional L such that the associated SMOP satisfy formulas (9).
This system provides that
e2n−1 = 22n−2 {M(fn2 ) + M(gn2 )}
with e2n−1 6= 0 because of trace Cn 6= 0 (¿0). Now, the quasi-de niteness of M implies that
−1
0 6= Det Cn = M(fn2 )M(gn2 ) − M(fn gn )2 = e2n−1
22−4n (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-de nite (positive) functional
(except for a non-zero factor) L ∈ ∗ such that {n }n∈N is the related SMOP. Besides, M is induced
by L in the sense of De nition 2.1.
5. Application
Let L ∈ L∗ be quasi-de nite and M := ML its induced functional. If we de ne Le ∈ L∗ by:
Le = ( 21 (z + z −1 ) − a)L with a ∈ R, let Me := MeLe ∈ ∗ be given by
e +
e
M[(P;
Q)] = L[P
√
1 − x2 Q] = L[(x − a)(P +
√
1 − x2 Q)] = M[(x − a)(P; Q)]
e with the quasi-de niteness con∀(P; Q) ∈ . So, Me = (x − a)M is the associated functional of L,
ditions given in Theorem 4.1.
Assume that Me is quasi-de nite, and let (Fn ) the sequence of semi-orthogonal functions related
e Fn = (Fn ; Gn )T , n¿1. Then, we have
to M,
e k Fn (x)) = M(x k (x − a)Fn (x)) = 0 (06k6n − 1);
M(x
√
√
e k 1 − x2 Fn (x)) = M(x k 1 − x2 (x − a)Fn (x)) = 0 (06k6n − 2)
M(x
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 x n+1 and 1 − x2 x n we obtain that An = I . So, if we assume that
M is quasi-de nite, there exists a unique matrix Bn ∈ R(2; 2) such that
(x − a)Fn = fn+1 + Bn fn :
(11)
14
M. Alfaro et al. / Journal of Computational and Applied Mathematics 99 (1998) 3–14
√
√
When |a| =
6 1, notice that fn (a) has two determinations for = a + a2 − 1 and −1 = a − a2 − 1,
that we denote by fn+ (a) and fn− (a), respectively. Then formula (11) becomes
−
+
(a)] + Bn [ fn+ (a); fn− (a)]:
(a); fn+1
0 = [ fn+1
−
+
Thus, the quasi-de niteness of [ fn+1
(a); fn+1
(a)], (n¿1) implies the existence of Fn (n¿1)
e
veri ying (11) and, moreover, the quasi-de niteness of M.
On the other hand,
det[
fn+ (a);
fn− (a)] =
=
1
i22n
1
∗ ( )
2n−1 ( ) + 2n−1
( ) − ∗ ( )
i22n−2
2n−1
∗
2n−1 ( )
( )
2n−1
2n−1
∗ ( −1 )
−1 2n−1 ( −1 ) + 2n−1
−1
( −1 ) − ∗ ( −1 )
2n−1
2n−1
6= 0:
( −1 )
∗ ( −1 )
2n−1
−1 2n−1
e n · F T ) 6= 0, then the same relation for the even terms is obtained. That
If we claim that trace M(F
n
is, according to Theorem 4.4, Le is quasi-de nite when
Dn ( ) = n∗ ( −1 ) n ( ) − n∗ ( ) −1 n ( −1 ) 6= 0
(12)
∀n¿1 and 6= ± 1. This requirement is the same given in [5] for the quasi-de niteness of Le in
the particular case Le = Re(z − )L.
By using formulas (8) and (11), we can obtain the relationship:
en (z) = n+2 (z) + rn zn (z) + sn ∗ (z);
(z 2 − 2az + 1)
n
e So, we derive the determinantal formula:
en ) is the SMOP associated with L.
where (
n+2 (z)
2
−1
e
(z − 2az + 1)n (z) = Dn ( ) n+2 ( )
( −1 )
n+2
zn (z)
n ( )
−1 n ( −1 )
n
∗
−1
( )
n∗ (z)
∗ ( )
n
en ) in terms of (n ), constrained to the condition (12).
which gives (
References
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Doctoral Dissertation, Universidad de Zaragoza, 1997.
[2] G. Freud, Orthogonal Polynomials, Akademiai Kiado, Budapest and Pergamon Press, Oxford, 1971, 1985.
[3] Y.L. Geronimus, Orthogonal Polynomials, Consultants Bureau, New York, 1961.
[4] Y.L. Geronimus, Polynomials orthogonal on a circle and their applications, Transl. Amer. Math. Soc. 3 Serie 1 (1962)
1–78.
[5] M.C. Suarez, Polinomios ortogonales relativos a modi caciones de funcionales regulares y hermitianos, Doctoral
Dissertation, Universidad de Vigo, 1983.
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