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A class of matrix orthogonal polynomials on the unit circle
A Class of Matrix Orthogonal Francisco
Marcellin
Polynomials
on the Unit Circle
Espacol
De-partamento de Matembticas E.T.S.Z. Zndustriales Uniuersidad Madrid,
Politknica
de Madrid
Spain
and Isabel
Rodriguez
Gonzalez
Dqartamento de Matemhticas E.T.S.Z. Zndustriales de Gijh Universidad de Oviedo Oviedo, Spain
Submitted by Vicente Hernbdez
ABSTRACT The study of matrix orthogonal polynomials on the unit circle is connected with problems of applied mathematics such as filtering theory, prediction theory, and scattering theory. In this paper, some families of matrix orthogonal polynomials on the unit circle are given explicitly. The decomposition II = II( A(z)) @zIl( A( z)) @ ... @z’l-‘II(A(z)), w h ere II is the linear space of complex algebraic polynomials, is the main idea for this approach.
1.
INTRODUCTION
Let l? be the lemniscate defined by IA(.z)l = 1, where A(z) = Xf=oaizi (ai E C, ah f 0) has simple roots. r is the union of a finite number of Jordan curves: We which (1)
r = U{= Iri. can suppose ai
(2) vj,
defined,
over every
ri, a mass distribution
q(z)
for
is nondecreasing with an infinite number of points of increase. k E iv: I/r, ZP dq(z)l < + co.
LZNEAR ALGEBRA
AND ITS APPLZCATZONS 121:23%241
(1989)
0 Elsevier Science Publishing Co., Inc., 1989 655 Avenue of the Americas, New York, NY 10010
00243795/89/$3.50
233
234
F. MARCELtiN
We can define a mass distribution
Hence,
and
an inner product
a sequence
u(z)
AND I. RODRiGUEZ
on I by
on II is given by
of orthogonal
polynomials,
associated
to it, is obtained:
{p,(z)]% (see 131). We consider the basis of II: (1,~ ,..., zhP1, A(z), zA(z) ,..., .zhPIA(z), . . . . A(z)“, zA(z)” ,..., zh-‘A(z)” , . . . }. Because of the isometric character of the operator A : II + II [A(P(z)) = A(z)I the moment matrix (d,,)T,Z=,, for the above inner product, has a block Toeplitz If T, is the principal minor of order h( n + l), we have
‘Do T,=
D,
...
Dj =
’
D,
D; Do ... Dn_1 ................... ,w D;pl ... Do I
’
djh+l,O ” ’ . . . djh+(h-l),l djh.1 dj/z+l,l a,,,;; . .;E;,;;;,;; . * : : : . ’ ;Egp$;
I
djh,O
structure.
djh+(h-l),O
\
. /
It is a well-known result (see [ 11) that the matrix (d,,)~,=, induces a matrix measure Q(O) on the unit circle whose moments are Dj, that is, Dj =
(1/2~)~~“eijedfJ(6). In this paper we obtain the left (right) orthogonal polynomial sequences associated with Q(z) from the polynomial sequences associated with u(z). The main idea is the decomposition
where
z ‘II
[ A( z)] is the linear space spanned by {
2, ZQqZ), zqz)2
,*..I
2%4(z)” } )...
.
235
MATRIX ORTI-IOGONAL POLYNOMIALS
For every polynomial P( z ) E ll, the decomposition P(z) = R,(z) + zR,( z ) + . . . + z~-~R~_~(z), where R,(z) E II[A(z)], exists. Then, R,(z) will be called the i th component of P(z).
2.
OBTAINING A SEQUENCE OF LEFT ORTHOGONAL POLYNOMIALS For all n EN, we have P:(z)
= z”[A,(l/z)]*,
where M* = g’, /
A,(z)
= [I, zZ ,...,
z”Z]T,-’
and
\ d) :
\ (4,
is a left orthogonal polynomial sequence with respect to the matrix measure fi( 8). If A i i are the adjoint matrices of the elements of the matrix T,,, and An its determinant, then from the above matrix expression we obtain
l,kh+lZ
i
k
k=O
A,(z)=;
Ib
n
k=O g
A2,kh+lZk
. . .
k=O
l,kh+2Z .;.
i
k
* . . * ;.
A2,kh+2zk
k=O * . E
2
Ah,kh+gk
k=O
.
..*.+...
“. . . ; : :
i k=O . . .i
Ah,kh+2Zk A,;;,,,;;
l,kh+hZ k=O
k=O
k=O
If (AH,h_l)‘“h+h-l denotes the orthogonal complement of AH,,_ 1 with respect to H,,h+h_i, then there exists a basis in (AHnh_l)‘nh+h-l represented by { *Ah +h_ r( z)}p:J, whose elements are characterized by the condition (~~h+&1(z),zj)=8~~
‘I
vi,j=o,l,...,h-1
(see [3]). It is very easy to obtain the following formula (see [5]):
236
F. MARCELLiiN
.
e
c
s;
‘tt
--
I
I ’ -8 I
h
N
.$
Y
1
-a I --l I
I --------____I
I
I
I
-c s
I I
2 ’ c
I
-2 .. .. .., . . .
I I
N
. . ._ . .
I
I
3
0.
-a” :
-:
1
2. q
I
------------I .
31
I .
2
.
I
4
II
AND I. RODRiGUEZ
MATRIX ORTHOGONAL
237
POLYNOMIALS
PROPOSITION2.1. Vi = 0, 1,. . . , h - 1 the coefficients for the components of *A,,+ h_ Xz) are the same as the polynomials placed in the i + 1 th column of A,(z). Proof. Expanding the determinantal expression for qAh+h_ i(z) through the elements of the i + lth row, we obtain
+~h-lk~oA~+l,iil+hA
+ .-.
and we can see that the coefficients in the jth component of q,$ + h_ i( z) are n the same as the element ( j + 1, i + 1) in the polynomial matrix A ,,( z ). REMAK We can obtain a sequence of left matrix orthogonal polynomials on the unit circle from the basis { *Ah + h_ r( z)}f~~, because /
n
”
CA
kh+l,lZ
“-k
=$ ”
n
-&h+B,lZn-k
c
\
‘%h+,,,lZn-k
k=O
n-k n-k n-k . . . kh+ l,Zz kh+h,ZZ 54 kh+2.ZZ b 54 k=O k=O k=O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b \ k-0
kh+l,hZ
n-k
k
Akh+2,hZn-k
...
k=O
k
,
Akh+h,hZn-k
k=O
and applying Proposition 2.1, the coefficients the components of the family { *Ah+h_ i(z)}. 3.
“’
k-0
k=O
P,‘(z)
c
Aij are obtained in terms of
OBTAINING A SEQUENCE OF RIGHT ORTHOGONAL POLYNOMIALS
In [l] and [2] the following expression is obtained for a right matrix orthogonal polynomial sequence:
P,“(z) = [ZJZ
,...)
z”Z]T,-1
(“’ ,
,;, By a very simple calculation, we obtain
n=0,1,2
,....
Moreover,
nh+l.kh+ZZ
CA
i
k
nh+2.
kh+2Z
k
.
“.
i
nh+h,kh+2’
-%h+,,.kh+lZk
CA
”
k=O k
nh+l,kh+hl k-0
d O,nh+k-1
,,....,,....
d O.nh
__________-____-__---T---------__------__-__-_~_--~----_---_-------__---_.
”
“’
.,,,,,,,,,_,_,,,.__.,..,,...,...._,,,.,,,....,,......._.................~..............
*:
Do
“.
i k=O
Anh+h,kh+hZk
dh,nh+h-1
“’
.
. . . .
...
.”
.
.
.
d ZhLl,nh+h&l
. .
Ztl-lA(Z)
. .
&-1,nh . .
I
I
I
/
, . . .
/ ,
dn/,,nh . .
” . . .
,
j
nh.nh+hml
..,,,...,.,...,............
A(-)”
‘..
__.
“0
A(Z)
.
d h.nh .
0” D“-1
*,
Qnjh~+,, _ I( z), 1~ j < h, by
&,i2,kh+h~k
we can define the polynomials
k-0
iA
k=O k=O k=O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CA
Anh+2,kh+lZk
k
n
t
n
k k=O
nh+l.kh+lZ
k=O
d nh+h-l,nh+hm
zh-'A(z)"
dnh+h~l.nh . .._.._.
MATRIX ORTHOGONAL POLYNOMIALS
239
These polynomials satisfy the conditions
(QL-lb), w) =0
VP(z)E h-17
(Q~,,+I,_l(~),~kA(~)“)= ”
(la)
kzj-17 o’k’h-l’ k=j-1.
1,
(za)
PROPOSITION 3.1. Vj = 1,2,. . . , h the coefficients for the components of fi’,, + ,, _ 1(z) are the same as the polynomials corresponding to the j th column 0 of Y( 2 )*
Proof. obtain
Expanding Q,$,+h_l(~) by the elements of the nh + jth row, we
Anh+j,kh+lA(Z)k
Q
+z i
k=O
Anh+j,kh+2A(Z)k+
”
. +zh-’
k=O
nh+j,kh+hA(Z)k 1
.
By comparison with the matrix expression of P,“
PROPOSITION 3.2. Let { P,,(z)}~=,, be an orthogonal polynomial sequence defined on the lemniscate IT. Then Vj = 1,2,. , . , h there exist a{, al,. . . , a;EQ: such that
Q ;r’,+,-,(z)=aiP,,+,_,(z)+
...
+a{PJz).
F. MARCELLhN
240
AND I. RODRiGUEZ
We consider the decomposition
Proof.
rI
nh+h-1
=K[P,,h+h-,(z)]
@ ‘..
@KIPnhb)l
@nInh&l,
where
K[ P(z)] is the linear space spanned by P(z). For j = 1,2,. . . , h, is orthogonal to Il ,,h_ 1; hence, there exist unique scalars Q!,+,-,(4 a;,..., ui E C such that
j = 1,2
Q,$,+&1(Z)=@,,h+h_l(z)+ ‘.. +@,&),
,...,
h.
In order to compute the coefficients above, we can use the inner product of zhp‘A( z)“. The resulting regular triangubyA(z)", d(z)",..., Q ;t',+,-@> lar system gives the solution:
o=
. I=
0 =~f(P,,,+~_~(t),
Now Vj=1,2,...,
aif-,+~(Pnh+j-l(2)~
&‘A(z)“)+
,jm‘A(z)“)+
..
h the coefficients ai, ai_,,
Qrih+h-,(z)=u~P,h+h-,(~)+~~pnh+h_~(~)+
...
+~~(P,~(z),zj~‘A(t)‘*),
. +ai(P,,Jz),
zh-‘A(z)“).
. . . , ui_ j+z vanish, so that
..’
+a~_j+,P,,h+j_l(Z).
n
A matrix interpretation for the moment problem on a particular class of lemniscates is presented in [4]. Some open problems related to the algorithmic implementation of the Schur formulas for matrix orthogonal polynomials on the unit circle has been considered in [5].
MATRIX
ORTHOGONAL
POLYNOMIALS
241
REFERENCES P. Delsarte, Y. Genin, and Y. Kamp, Orthogonal polynomial matrices on the unit circle, ZEEE Trans. Circuits and Systems CAS-25 (3):149-160 (1978). J. Geronimo, Matrix orthogonal polynomials on the unit circle, J. Math. Phys. 22(7): 1359- 1365 (1981). F. Marcellan, Polinomios Ortogonales sobre Cassinianas, Tesis Doctoral, Zaragoza, 1976. F. Marcellan and L. Moral, Polinomios ortogonales sobre la lemniscata de Bernoulli: Una interpretacibn matricial de1 problema de 10s momentos, in X ~ornudas HisparwLusas de Matemhticas, Murcia, 1985. I. Rodriguez, Polinomios Matriciales Ortogonales sobre Curvas, Tesis Doctoral, Oviedo, 1986. Received
4 January 1988; final manuscript
accepted 14 October 1988
Marcellin
Polynomials
on the Unit Circle
Espacol
De-partamento de Matembticas E.T.S.Z. Zndustriales Uniuersidad Madrid,
Politknica
de Madrid
Spain
and Isabel
Rodriguez
Gonzalez
Dqartamento de Matemhticas E.T.S.Z. Zndustriales de Gijh Universidad de Oviedo Oviedo, Spain
Submitted by Vicente Hernbdez
ABSTRACT The study of matrix orthogonal polynomials on the unit circle is connected with problems of applied mathematics such as filtering theory, prediction theory, and scattering theory. In this paper, some families of matrix orthogonal polynomials on the unit circle are given explicitly. The decomposition II = II( A(z)) @zIl( A( z)) @ ... @z’l-‘II(A(z)), w h ere II is the linear space of complex algebraic polynomials, is the main idea for this approach.
1.
INTRODUCTION
Let l? be the lemniscate defined by IA(.z)l = 1, where A(z) = Xf=oaizi (ai E C, ah f 0) has simple roots. r is the union of a finite number of Jordan curves: We which (1)
r = U{= Iri. can suppose ai
(2) vj,
defined,
over every
ri, a mass distribution
q(z)
for
is nondecreasing with an infinite number of points of increase. k E iv: I/r, ZP dq(z)l < + co.
LZNEAR ALGEBRA
AND ITS APPLZCATZONS 121:23%241
(1989)
0 Elsevier Science Publishing Co., Inc., 1989 655 Avenue of the Americas, New York, NY 10010
00243795/89/$3.50
233
234
F. MARCELtiN
We can define a mass distribution
Hence,
and
an inner product
a sequence
u(z)
AND I. RODRiGUEZ
on I by
on II is given by
of orthogonal
polynomials,
associated
to it, is obtained:
{p,(z)]% (see 131). We consider the basis of II: (1,~ ,..., zhP1, A(z), zA(z) ,..., .zhPIA(z), . . . . A(z)“, zA(z)” ,..., zh-‘A(z)” , . . . }. Because of the isometric character of the operator A : II + II [A(P(z)) = A(z)I the moment matrix (d,,)T,Z=,, for the above inner product, has a block Toeplitz If T, is the principal minor of order h( n + l), we have
‘Do T,=
D,
...
Dj =
’
D,
D; Do ... Dn_1 ................... ,w D;pl ... Do I
’
djh+l,O ” ’ . . . djh+(h-l),l djh.1 dj/z+l,l a,,,;; . .;E;,;;;,;; . * : : : . ’ ;Egp$;
I
djh,O
structure.
djh+(h-l),O
\
. /
It is a well-known result (see [ 11) that the matrix (d,,)~,=, induces a matrix measure Q(O) on the unit circle whose moments are Dj, that is, Dj =
(1/2~)~~“eijedfJ(6). In this paper we obtain the left (right) orthogonal polynomial sequences associated with Q(z) from the polynomial sequences associated with u(z). The main idea is the decomposition
where
z ‘II
[ A( z)] is the linear space spanned by {
2, ZQqZ), zqz)2
,*..I
2%4(z)” } )...
.
235
MATRIX ORTI-IOGONAL POLYNOMIALS
For every polynomial P( z ) E ll, the decomposition P(z) = R,(z) + zR,( z ) + . . . + z~-~R~_~(z), where R,(z) E II[A(z)], exists. Then, R,(z) will be called the i th component of P(z).
2.
OBTAINING A SEQUENCE OF LEFT ORTHOGONAL POLYNOMIALS For all n EN, we have P:(z)
= z”[A,(l/z)]*,
where M* = g’, /
A,(z)
= [I, zZ ,...,
z”Z]T,-’
and
\ d) :
\ (4,
is a left orthogonal polynomial sequence with respect to the matrix measure fi( 8). If A i i are the adjoint matrices of the elements of the matrix T,,, and An its determinant, then from the above matrix expression we obtain
l,kh+lZ
i
k
k=O
A,(z)=;
Ib
n
k=O g
A2,kh+lZk
. . .
k=O
l,kh+2Z .;.
i
k
* . . * ;.
A2,kh+2zk
k=O * . E
2
Ah,kh+gk
k=O
.
..*.+...
“. . . ; : :
i k=O . . .i
Ah,kh+2Zk A,;;,,,;;
l,kh+hZ k=O
k=O
k=O
If (AH,h_l)‘“h+h-l denotes the orthogonal complement of AH,,_ 1 with respect to H,,h+h_i, then there exists a basis in (AHnh_l)‘nh+h-l represented by { *Ah +h_ r( z)}p:J, whose elements are characterized by the condition (~~h+&1(z),zj)=8~~
‘I
vi,j=o,l,...,h-1
(see [3]). It is very easy to obtain the following formula (see [5]):
236
F. MARCELLiiN
.
e
c
s;
‘tt
--
I
I ’ -8 I
h
N
.$
Y
1
-a I --l I
I --------____I
I
I
I
-c s
I I
2 ’ c
I
-2 .. .. .., . . .
I I
N
. . ._ . .
I
I
3
0.
-a” :
-:
1
2. q
I
------------I .
31
I .
2
.
I
4
II
AND I. RODRiGUEZ
MATRIX ORTHOGONAL
237
POLYNOMIALS
PROPOSITION2.1. Vi = 0, 1,. . . , h - 1 the coefficients for the components of *A,,+ h_ Xz) are the same as the polynomials placed in the i + 1 th column of A,(z). Proof. Expanding the determinantal expression for qAh+h_ i(z) through the elements of the i + lth row, we obtain
+~h-lk~oA~+l,iil+hA
+ .-.
and we can see that the coefficients in the jth component of q,$ + h_ i( z) are n the same as the element ( j + 1, i + 1) in the polynomial matrix A ,,( z ). REMAK We can obtain a sequence of left matrix orthogonal polynomials on the unit circle from the basis { *Ah + h_ r( z)}f~~, because /
n
”
CA
kh+l,lZ
“-k
=$ ”
n
-&h+B,lZn-k
c
\
‘%h+,,,lZn-k
k=O
n-k n-k n-k . . . kh+ l,Zz kh+h,ZZ 54 kh+2.ZZ b 54 k=O k=O k=O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b \ k-0
kh+l,hZ
n-k
k
Akh+2,hZn-k
...
k=O
k
,
Akh+h,hZn-k
k=O
and applying Proposition 2.1, the coefficients the components of the family { *Ah+h_ i(z)}. 3.
“’
k-0
k=O
P,‘(z)
c
Aij are obtained in terms of
OBTAINING A SEQUENCE OF RIGHT ORTHOGONAL POLYNOMIALS
In [l] and [2] the following expression is obtained for a right matrix orthogonal polynomial sequence:
P,“(z) = [ZJZ
,...)
z”Z]T,-1
(“’ ,
,;, By a very simple calculation, we obtain
n=0,1,2
,....
Moreover,
nh+l.kh+ZZ
CA
i
k
nh+2.
kh+2Z
k
.
“.
i
nh+h,kh+2’
-%h+,,.kh+lZk
CA
”
k=O k
nh+l,kh+hl k-0
d O,nh+k-1
,,....,,....
d O.nh
__________-____-__---T---------__------__-__-_~_--~----_---_-------__---_.
”
“’
.,,,,,,,,,_,_,,,.__.,..,,...,...._,,,.,,,....,,......._.................~..............
*:
Do
“.
i k=O
Anh+h,kh+hZk
dh,nh+h-1
“’
.
. . . .
...
.”
.
.
.
d ZhLl,nh+h&l
. .
Ztl-lA(Z)
. .
&-1,nh . .
I
I
I
/
, . . .
/ ,
dn/,,nh . .
” . . .
,
j
nh.nh+hml
..,,,...,.,...,............
A(-)”
‘..
__.
“0
A(Z)
.
d h.nh .
0” D“-1
*,
Qnjh~+,, _ I( z), 1~ j < h, by
&,i2,kh+h~k
we can define the polynomials
k-0
iA
k=O k=O k=O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CA
Anh+2,kh+lZk
k
n
t
n
k k=O
nh+l.kh+lZ
k=O
d nh+h-l,nh+hm
zh-'A(z)"
dnh+h~l.nh . .._.._.
MATRIX ORTHOGONAL POLYNOMIALS
239
These polynomials satisfy the conditions
(QL-lb), w) =0
VP(z)E h-17
(Q~,,+I,_l(~),~kA(~)“)= ”
(la)
kzj-17 o’k’h-l’ k=j-1.
1,
(za)
PROPOSITION 3.1. Vj = 1,2,. . . , h the coefficients for the components of fi’,, + ,, _ 1(z) are the same as the polynomials corresponding to the j th column 0 of Y( 2 )*
Proof. obtain
Expanding Q,$,+h_l(~) by the elements of the nh + jth row, we
Anh+j,kh+lA(Z)k
Q
+z i
k=O
Anh+j,kh+2A(Z)k+
”
. +zh-’
k=O
nh+j,kh+hA(Z)k 1
.
By comparison with the matrix expression of P,“
PROPOSITION 3.2. Let { P,,(z)}~=,, be an orthogonal polynomial sequence defined on the lemniscate IT. Then Vj = 1,2,. , . , h there exist a{, al,. . . , a;EQ: such that
Q ;r’,+,-,(z)=aiP,,+,_,(z)+
...
+a{PJz).
F. MARCELLhN
240
AND I. RODRiGUEZ
We consider the decomposition
Proof.
rI
nh+h-1
=K[P,,h+h-,(z)]
@ ‘..
@KIPnhb)l
@nInh&l,
where
K[ P(z)] is the linear space spanned by P(z). For j = 1,2,. . . , h, is orthogonal to Il ,,h_ 1; hence, there exist unique scalars Q!,+,-,(4 a;,..., ui E C such that
j = 1,2
Q,$,+&1(Z)=@,,h+h_l(z)+ ‘.. +@,&),
,...,
h.
In order to compute the coefficients above, we can use the inner product of zhp‘A( z)“. The resulting regular triangubyA(z)", d(z)",..., Q ;t',+,-@> lar system gives the solution:
o=
. I=
0 =~f(P,,,+~_~(t),
Now Vj=1,2,...,
aif-,+~(Pnh+j-l(2)~
&‘A(z)“)+
,jm‘A(z)“)+
..
h the coefficients ai, ai_,,
Qrih+h-,(z)=u~P,h+h-,(~)+~~pnh+h_~(~)+
...
+~~(P,~(z),zj~‘A(t)‘*),
. +ai(P,,Jz),
zh-‘A(z)“).
. . . , ui_ j+z vanish, so that
..’
+a~_j+,P,,h+j_l(Z).
n
A matrix interpretation for the moment problem on a particular class of lemniscates is presented in [4]. Some open problems related to the algorithmic implementation of the Schur formulas for matrix orthogonal polynomials on the unit circle has been considered in [5].
MATRIX
ORTHOGONAL
POLYNOMIALS
241
REFERENCES P. Delsarte, Y. Genin, and Y. Kamp, Orthogonal polynomial matrices on the unit circle, ZEEE Trans. Circuits and Systems CAS-25 (3):149-160 (1978). J. Geronimo, Matrix orthogonal polynomials on the unit circle, J. Math. Phys. 22(7): 1359- 1365 (1981). F. Marcellan, Polinomios Ortogonales sobre Cassinianas, Tesis Doctoral, Zaragoza, 1976. F. Marcellan and L. Moral, Polinomios ortogonales sobre la lemniscata de Bernoulli: Una interpretacibn matricial de1 problema de 10s momentos, in X ~ornudas HisparwLusas de Matemhticas, Murcia, 1985. I. Rodriguez, Polinomios Matriciales Ortogonales sobre Curvas, Tesis Doctoral, Oviedo, 1986. Received
4 January 1988; final manuscript
accepted 14 October 1988