Rev.Acad.Canar.Cienc., XI (Núms. 1-2), 127-152 (1999)
canary .tex; 14/09/1999; 13:61
ELEMENTS OF A THEORY OF ORTHOGONAL
RATIONAL FUNCTIONS
Adhemar Bultheel1
Department of Computer Science, K.U.Leuven, Belgium.
Pablo González-Vera2
Department Análisis Math., Univ. La Laguna, Tenerife, Spain.
Erik Hendriksen
Department of Mathematics, University of Amsterdam, The Netherlands.
Olav Njastad
Department of Math. Se., Norwegian Univ. of Science and Technology, Trondheim, Norway
Abstract
We give a brief survey to sorne basic elements of the theory of orthogonal rational functions.
Two main cases are treated separately: I. Ali the poles are outside the closed unit disk, and the
orthogonality measures have support in the unit circle. II. Ali the poles are on the extended real
line, and the orthogonality measures have support in the real line. These situations generalize
the theory of orthogonal polynomials on the unit circle (Szegéí polynomials) and the theory of
orthognal polynomials on the real line.
En este trabajo, exponemos brevemente, los elementos básicos de la teoría de funciones
racionales ortogonales. Nos centraremos en dos casos fundamentales: I. Todos los polos se
encuentran fuera del disco unidad cerrado, estando las medidas de ortogonalidad soportadas
sobre la circunferencia unidad. II. Todos los polos se hallan sobre la recta real extendida, y
las medidas de ortogonalidad con soporte en el eje real. Tales situaciones generalizan la teoría
de polinomios ortogonales sobre la circunferencia unidad (Polinomios de Szegéí) y la teoría de
polinomios ortogonales sobre la recta real.
Keywords: orthogonal polynomials, orthogonal rational functions, varying measure, complex
approximation, numerical quadrature, moment problems.
1The work ofthis author is partially supported by the Fund for Scientific Research (FWO), project "Orthogonal
systems and their applications", grant #G.0278.97 and the Belgian Programme on Interuniversity Poles
of Attraction, initiated by the Belgian State, Prime Minister's Office for Science, Technology and Culture. The
scientific responsibility rests with the author.
2The work of the second author was partially supported by the scientific research project of the Spanish
D.G.E.S. under contract PB96-1029.
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1 INTRODUCTION.
A sequence { \Pn};:"=0 of polynomials is said to be an orthogonal polynomial sequence if \Pn is a
polynomial of degree n and it is orthogonal to ali polynomials of lower degree. The orthogonality
may be with respect to a linear functional Mor a measure μ (with support in C). In particular,
the most widely studied cases of such general orthogonal polynomials arise when the support
μ is contained in the real line or the complex unit circle. For general treatises on orthogonal
polynomials we refer to [1], [33], [43], [44, 45], [47], [51], [59], [74], [75], [76].
Polynomials may be viewed as rational functions whose poles are ali fixed at infinity. By
fixing a sequence of poles {In} in the extended complex plane (i.e., on the Riemann sphere), we
obtain a theory of orthogonal rational functions. The poles can in principie be taken anywhere
in the extended plane. Sorne of the /k can be repeated, possibly an infinite number of times.
The sequence is fixed once and for ali, and the order in which the /k occur (possible repetitions
included) is also given.
We consider generalizations of the two special cases indicated above: orthogonality on the
unit circle and on the real line. Polynomials orthogonal on the unit circle are generalized to
orthogonal rational functions with poles outside the closed unit disk. Polynomials orthogonal
on the real line are generalized to orthogonal rational functions with poles on the extended real
line. There is a substantial difference between the two cases, since in the former case the poles
lie outside the "support curve" of the orthogonality measures, while in the latter case the poles
belong to the "support curve". (We do not by this mean that the support of the orthogonality
measure need consist of the whole unit circle on the whole real line.) The orthogonal rational
functions will behave differently. The main reason for this is that in the former case reflection
of the poles with respect to the unit circle produce different points, while in the latter case
reflection of the poles, with respect to the real line produce the same points, doubling the poles
in a sense.
The cause of the difference between the two cases is then not any differenc~ between the unit
circle and the (extended) real line, but between the ways the poles are placed in relation to the
"support curve", whether circle on line. By the Cayley transform z -+ w = :+: the extended
real line is mapped to the unit circle and the extended upper half plane to the unit disk. This
transformation maps ali rational functions to ali rational functions. Thus we may consider
rational functions orthogonal on the real line with poles in the lower half plane as analogs
to rational functions orthogonal on the unit circle with poles outside the unit disk. Similarly
we may consider rational functions orthogonal on the unit circle with poles on the unit circle
as analogs to rational functions orthogonal on the real line with poles on the extended real
line. Orthogonal polynomials on the unit circle correspond to orthogonal rational functions on
the real line with their only pole (infinitely repeated) at the point -i. Similarly orthogonal
polynomials on the real line correspond to orthogonal rational functions on the unit circle with
their only pole (infinitely repeated) at the point -l.
The case of the real line and the unit circle which are linked by the Cayley transform are
essentially the same, and can be treated within a common framework. A unified and rather
extensive treatment is given in the monograph "Orthogonal Rational Functions" [34] by the
present authors. In this paper we give a very brief introduction to sorne basic elements of this
theory. We here treat the situation with poles outside the "support curve" specified to the unit
circle case, and the situation with poles in the "support curve" specified to the real line case.
The generalizations of the classical polynomial situations are thus clearly seen.
From a purely mathematical point of view the theory of orthogonal rational functions was as
far as we know initiated by Djrbashian about 1960. See [37, 38, 39, 40, 41, 42]. lndependently,
partly from an applied point of view, the same constructions were used by Bultheel, Bultheel
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and Dewilde, Dewilde and Dym about 1980. See [2, 3, 4], [36]. A general theory has been
worked out in a series of papers by the present authors. See [6]-[35], and the monograph [34].
A number of contributions have also been given by Li, Pan and by Li and Pan. See [52]-[53],
[68]-[73]. The periodic case, where the 'Yk consist of a finite number of points cyclically repeated,
was studied by González-Vera, Hendriksen and Njastad. See [46], [48]-[49], [60]-[65]. When the
poles of the orthogonal rational functions consist of the origin and the point at infinity, infinitely
repeated, the rational functions are Laurent polynomials. For a presentation of the basic theory
of orthogonal Laurent polynomials and related topics we refer to the survey article [50] by Jones
and Njastad, and the references given there.
The whole theory of orthogonal rational functions is related to the theory of polynomials
orthogonal with respect to varying measures, first extensively studied by Lopez. We refer to
[54]-[58].
We shall make use of the following notation. C denotes the complex plane, D the open unit
disk, U the open upper half plane, 11' the unit circle, R the real line. Furthermore C denotes the
extended complex plane (the point at infinity added), lR the closure of R in C, Ü the closure
of u in C. We also write JE = e \ {D u 11'} and V = e\ {U u R}.
, The substar transform f. of a function J is defined as follows.
In the unit circle situation:
f.(z) = f(l/z).
In the real line situation:
f.(z) = f(z).
The Riesz-Herglotz-Nevanlinna transform O(·,μ) of a finite measure μis defined as follows.
In the unit circle situation:
In the real line situation:
1t +z
O(z, μ) = -dμ(t).
T t - z O( z ,μ) = -i· 1 1- -+d tμz (t).
IR t - Z
(1.1)
(1.2)
The function (1.1) maps D into the right half plane, the function (1.2) maps U into the right
half plane.
2 FUNCTION SPACES 1
Let { an}:;"=1 be an arbitrary sequence of points in D. We introduce the Blaschke factors (k
defined by
k = 1,2, ... .
We set by convention ~ = -1 when ak =O, so that (k(z) = z when ak = O. The Blaschke
products Bn are defined by
Bo = 1, Bn(z) = IT (k( z ), n = 1, 2, .. . .
k= I
We note that Bn(z ) = zn for ali n if ak =O for ali k.
We shall use the notation 7r n for the dertominator polynomial in the rational function Bn,
i.e.,
n
7ro=l, 7rn(z) = IT(l-akz), n=l,2, . . ..
k=I
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We shall also use the notation
wo=l, wn(z)= IT(z-ok), n=l,2, . . ..
k=I
We observe that we may write
Wn(z) = 7r!(z),
where 7r! denotes the superstar transform of the polynomial 7rn (see e.g. [45],[47]). Hence
wn(z) 7r!(z)
Bn(z) = 1/n_(_) = 1/n_(_),
1ínZ 1ínZ
n
1/n = rr Zk.
k=I
We shall study spaces Ln and [, of rational functions. The space Ln is given by
Ln = Span{Bo, B1, ... , Bn}, n =O, 1, 2, ...
and we set
00
n=O
(In [34) is used í,00 for this union, and [, for the closure of this space in an Lrspace. We
shall here not have occasion to consider this closure, and use for convenience [, for the space
of rational functions itself.) A function f belongs to Ln if and only if it is of the form
J(z) = p(z) '
7r,.(z)
where p is a polynomial of degree at most n.
We shall write
Ln• = {!. : f E Ln}, c. = {/. : f E C}.
We then have
00
Ln• = Span{Bo., Bi., ... , Bn.}, C. = U CM
n=l
and we observe that
Bk.(z) = [Bk( z)t1.
We shall in this paper work with the standard basis { B0 , B¡, .. . , Bn, . .. } for C. Severa! other
basis {Ca, C¡, .. . , Cn, .. . } for L with the property Ln = Span{ Ca, C1, ... , Cn} for every n have
been studied and may be useful. One such basis is
1 1 1
{l, -1 _ ,1 -_ Bi(z), ... ,1 -_ Bn-1( z), ... }.
- Q¡Z - 02Z - OnZ
When ali the points °'k are distinct,
1 1
{1,1 - , . . . ,~, ... }
-a¡,Z 1-0nZ
is such a basis, and when °'k =J O for ali k,
1 1
{1, - (-)'ººº'-(-)' º º'} 7r¡ Z 7r n Z
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is such a basis.
The superstar transform f" of a function in Ln \ Ln.:1 is defined as
f*(z) = Bn(z)f.(z).
We note that f" is a function in Ln . We find that
(when an f- O). In particular B~(z) =l.
We call an the /eading coefficient of the function f ( z) = ¿~=O akBk( z) ( with respect to the
basis {Bo, B1, . . . , Bn, . . . } ). We observe that an = f*(o:n)· If the leading coefficient is 1, the
function f is said to be monic.
General reference: Sections 2.1-2.2 of [34].
3 ORTHOGONAL FUNCTIONS 1
Let M be a linear functional defined on the linear space [, + C.. Sin ce ali the points O:k are
contained in D, ali the factors (z - o:k) are different from ali the factors (z - ( o:k)-1 ). Thus we
find by partial fraction decomposition that C + C. is the same as the product space [, · C..
We shall assume that
M[f.J = M[f] for f E C + C. (3.1)
and
M[f ·f.] > O for f E C, f t U. (3.2)
(We recall that here f.(z) = f(l/z).) For convenience we normalize M such that M[l) =l.
Typical examples are functionals represented by positive measures as follows: Let μ be a
finite positive measure on 'll', and define
M[f] = i f(t)dμ(t) for f E C +C •.
We easily verify that M satisfies (3.1)-(3.2).
The functional M gives rise to an inner product (-, ·) on the space [, through the formula
(f,g) = M[f · g.), f,g E C.
We denote by {cpn}~=O the orthonormal basis for [, associated with the sequence {Cn}, with
leading coefficient cp~ ( o:n) real and positive. Thus we ha ve
Ln = Span {<;?o, <;?1, ... , <;?n}
and
(cpj, <;?k) = Ój,k·
When o:k = O for ali k, the functions <;?n are simply the (normalized) Szego polynomials determined
by M .
We may write <;?n in the form
Pn(z)
<;?n(z) = -(-),
11'n z
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(3.3)
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where Pn is a polynomial of exact degree n. We then have
(3.4)
Sin ce 'Pn is orthogonal to ali the functions zm [7r n-1 ( z )t1 for m = O, 1, . . . , n - 1, we may write
Mn[pn(z) · (zm).] =O, m =O, 1, ... , n - 1,
where
Mn[f] = M [ J(z) 1 ·
(1 - anz) YÚ1 - akz)(l - akz).
k=I
Thus the sequence {Pn}::"=o of polynomials is orthogonal with respect to the sequence of varying
complex inner products (-,·)non 'f given by
In particular, if M[f] is given by M[f] = frf(t)dμ(t), then the sequence {Pn}::"=o is orthogonal
with respect to the sequence of varying complex measures μn given by
dμ(t)
dμn(t) = n-I
(1 - ant) rr 11 - aktl2
k=I
The functions of the second kind 1/;n associated with { <t'n} are defined as follows:
1/;o = 1
n = 1,2, ...
(Here M operates on its argument as a function of t.) We may write
.t. ( ) = qn(z)
'f/n Z ( ) ,
7rn Z
(3.5)
where qn is a polynomial of degree at most n, and thus ¡/;n E .Cn.
We may also write
1/;n(z) = M [.tt +_ zz {fJ((tz)) 'Pn(t) - 'Pn(z) }] , n = 1,2, ... ,
where J is any function in .C(n-i) .. We find that the superstar transform ¡/;~ is given by
where gis any function in .Cn• satisfying g(l/an) =O.
General reference: Sections 2.2 and 4.2 of [34] .
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4 RECURSION 1
The orthonormal function 'Pn satisfy a recurrence relation which has the same structure as the
Szegéí recursion for polynomials, and which reduces to this Szegéí recursion in the polynomial
case.
Theorem 4.1 The functions t.pn, <p~ satis/y a recursion of the following form:
[ t.p~ ( Z) ] = en 1 - ~ Z [ Un O ] [ 1 >.n ] [ (n-1 ( Z) O ] [ 'P~-1 ( Z) ] , n = 1, 2, ... ,
'Pn(z) 1 - CTnZ O Vn >.n 1 O 1 '-Pn-1(z)
( 4.1)
with initial conditions <po = 'Po = l. Here en is a positive constant, the constant un, where
lunl = 1, is chosen such that t.p~(an) >O, and Vn = UnZn-1Zn. Final/y >.n is given by
( 4.2)
The coefficient >.n satisfies 1>.n1 < l.
The expression ( 4.2) for the coefficient >.n is not very useful, since it uses function values of
'-Pn and <p~ to compute these functions. More practica! expressions for >.n are given by
__ \ '-Pk(z), zl-=- ~: 'Pn-1(z))
>.n = -Zn-1 \ l _ ) , - On-lZ *
'Pk(z), l _ Q,;"z 'Pn-i(z)
k=0,1, .. . ,n-1."
The consta.nt en can be obtained as the square root of
2 1 - lanl2 1
en= 1 - lan-112. 1 - l>.n12· (4.3)
In the polynomial situation, i.e. when an =O for ali n, the formula (4.1) takes the form
or
[ <p~(z) ] = [ enUnZ enun>.n ] [ 'P~-1(z) ] ,
'Pn(z) enun>.nz enUn 'Pn-l(z)
which has the form of the Szegéí recursion for Szegéí polynomials.
The functions of the second kind satisfy a recurrence relation very similar to that satisfied
by the orthogonal functions.
Theorem 4.2 The functions 1/Jn, 'lj;~ satis/y the recurrence relation
n = 1, 2, . . . , where the recurrence coefficients are the same as those in Theorem 4 .1. The initial
conditions are 1/Jo = 1/;0 = 1.
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We have seen that orthonormal rational functions satisfy a recurrence relation as given
in Theorem 4.1. A converse of this is also true. The following Favard type theorem follows
immediately from results in Section 8.1 in [34] formulated in terms of measures, but can also
be proved directly without recourse to representation theory for functionals.
Theorem 4.3 Let {>.n}:;"=1 be a sequence of complex numbers such that i>.nl < 1, and /et
positive numbers en be determined through the formula (4.3). Define the functions 'Pn recursively
by <po = 1,
n = 1, 2, ... , where Un is chosen such that lunl = 1, <p~(an) >O. Then the functions <p~ satisfy
the recursion <p~ = 1,
n = 1, 2, ... , with Vn = UnZn-1Zn. Furthermore there exists a linear functional M on [, + [..
such that { 'Pn} are the correspo11ding orthonormal functions.
General reference: Section 4.1-4.2 and 8.1 of [34].
5 QUADRATURE 1
It can be shown that ali the zeros of 'Pn are contained in lDl. They may be multiple zeros of
any order. Thus these zeros are not suitable as nodes in a quadrature on T. To obtain such
quadrature formulas, we introduce para-orthogonal functions of order n. These are functions of
the form
They satisfy
Qn(z, r) = 'Pn(z) + T<p~(z), TE C, T =/= 0.
(Qn(z, r), Bk(z)) =O, k = 1, 2, ... , n - 1,
(Qn(z, r), 1) =/=O, (Qn(z, r), Bn(z)) =/=O.
(5.1)
(5.2)
It can be shown that a function that satisfies (5.1)-(5.2) is para-orthogonal as defined above.
With the para-orthogonal functions Qn(z,r) we associate function of the second kind
Pn(z, r) given by
Pn(z,r) = 1/Jn(z)- r¡/;~(z).
Thes(:! functions may also be produced by the formula
Pn(z,r)=M [tt+-zz {fJ((tz)) Qn(t,r)-Qn(z,r) }] ,
where f is any function in L(n-l)• satisfying J(l/ an) = O.
In the following we assume that lrl = l.
We may write
n = 2,3, . . . , (5.3)
Since ali the zeros of Pn are contained in lDl and Ir 1 = IT/n 1 = 1, we find that Pn ( z) + TT/nZnPn• ( z)
is a polynomial of exact degree n for ali r .
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Theorem 5.1 The para-orthogonal function Qn(z,r) has exactly n zeros, ali of which are
simple and lie on 'll'.
Note that since an ri '][' for ali n, the zeros of Qn(z, r) are the same as the zeros of the
polynomial Pn(z) + TTJnZnPn•(z).
The zeros of a para-orthogonal rational functions are nodes in a rational Szego quadrature
formula.
( Pn((nk,r)
Theorem 5.2 Let (nk(r), k = 1, ... , n, be the zeros ofQn(z, r), and set Ank r) = 21" Q' (t" )
i..,nk n i..,nk, T
Then the rational Szeg/J quadrature formula
n
M(~J = L Ank(r)R((nk(r)) (5.4)
k=I
is exact for every R E Ln-1 +Len-!)• = .Ln-1 ·Len-!)•·
An alternative expression for the weight Ank(r) is
We point out that the degree of exactness of this formula is one less than maximal in the sense
that the formula has 2n parameters (ni ( T) , ... , (nn ( T), An1 ( T), ... , Ann ( T), while the dimension
of the space Ln-I + Lcn-i)* is only 2n - l. The formula is thus not analogous to a Gaussian
quadrature formula, but it is a direct generalization of Szego quadrature formulas in the
polynomial case. In the polynomial situation the space Ln-1 + L(n-1)• reduces to the space
A-(n-I),n-I of Laurent polynomials of the form L(z) = I;~;;;~en- I) akzk. Note that there is one
Szego quadrature formula for every r, while Gaussian quadrature formulas are unique.
The following result follows from Theorem 5.2 and the fact that the argument in formula
(5.3) belon~ to Ln-1 + L(n-1) ..
Theorem 5.3 Let (nk(r), Ank(r), k = 1, .. . , n, be as in Theorem 5.2. Then we have
(5.5)
J or n = 1, 2 ....
General reference: Sections 5.1-5.4 of [34].
6 INTERPOLATION AND CONVERGENCE 1
We shall in this section for convenience assume that the functional M is derived from a positive
measure on 'll' as described in Section 2. We recall the definition (1.1) of the Riesz-HerglotzNevanlinna
transform !1(z,μ). We shall discuss how the rational functions -¡/;n/'Pn, ¡/;~ /c.p~ and
-Pn(z,r)/Qn(z,r) (for lrl = 1) interpolate and converge to !1(z,μ). .
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Theorem 6.1 For lrl = 1 the quotients -Pn(z,r)/Qn(z,r) interpolate n(z,μ) at the table
{O,oo,a¡, l/a1, ... ,an-1il/an-d in thefollowing sense:
n(z,μ) + ~n~Z,T~ = g(z)zBn-1(z)
n z,r
(6.1)
!1 ( z, μ ) + QPn ((z , r)) = h ( z ) z -l[ Bn-1 ( z )]-1
n Z,T
(6.2)
for n = 1, 2, ... , where g is holomorphic in D and h is holomorphic in K
Since ak E][)) and 1/ak tf. ][))U'][' for ali k we may also write (6.1)-(6.2) in the form
Pn(z,r) ( ) ( ) n(z,μ) + Q ( ) = zG z Wn-1 z
n z,r
n ( z,μ ) + QPn (( Z , T)) = z -1 H(z ) ?rn-1 ( z),
n Z,T
with G holomorphic in D, H holomorphic in E.
For the orthonormal functions and the functions of the second kind an extra interpolation
condition is satisfied. On the other hand, at a part of the interpolation table only linearized
interpolation is obtained.
Theorem 6.2 The quotients _'fu and 4 interpolate !1(z,μ) at the tables {O,oo,ai,l/a1, .. . ,
'Pn 'Pn
ªn-1,l/an-1il/an} and {O,oo,a1,l/a¡, ... ,an-1,l/an-l,an}, respective/y, in the fol/owing
sense:
!1(z,μ)<pn(z) + 1/Jn(z) = zG(z)Bn-1(z)
n(z,μ) + 1/Jn((z)) = z-1h(z)[Bn(z)¡-1
4'n Z
1/J~(z)
!1(z,μ) - -(-) = zg(z)Bn(z)
<p~ z
n(z, μ)<pn.(z) - 1/Jn.(z) = z-1 H(z)[Bn-1 (z)t 1,
where g and G are holomorphic in D, h and H are holomorphic in K
(6.3)
(6.4)
(6.5)
(6.6)
Recall the formulas (3.3)-(3.4) and (3.5). Ali the zeros of 4'n líe in D, ali the zeros of 4'n•
and <p~ líe in E. We may therefore write (6.3)-(6.6) in the following form:
!1(z,μ)pn(z) + qn(z) = zr(z)wn-1(z)
n(z,μ) + qn((z)) = z-1o(z)7rn(z)
Pn z
!1(z,μ) - qn.((z)) = z1(z)wn(z)
Pn• z
n(z,μ)pno(z) - qn.(z) = z- 1 ~(z)7rn-1(z),
where 'Y and r are holomorphic in D, o and ~ are holomorphic in E.
In the polynomial situation ( an = O for ali n) the expressions zBn_1(z), zBn(z), [zBn_1(z)¡-1,
[zBn(z)J-1 reduce to zn,zn+1, z -n,z-(n+1l. Cf. formulas in [51].
Since 1/Jn/4'n and 1/J~/<p~ are rational functions of type [n/n], the content of Theorem 6.2
may be expressed as follows: -1/Jn/4'n is the [n/n] multipoint Padé approximant to !1(z,μ)
at the table {O, oo, a1, l/a1, . .. , ªn-li 1/ an-1, l/an}, and 1/J~/<p~ is the [n/n] multipoint Padé
approximant to !1(z,μ) at the table {O,oo,a1,l/a1, ... ,an-1,l/an-1,an}·
We close this section with a theorem concerning convergence of the interpolating functions.
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Theorem 6.3 Assume that the condition L.::::°:1 (1 - lanl) = oo is satisfied. Then the following
hold:
A. {1/>~(z)/cp~(z)} converges to D(z, μ) locally uniformly in lD>.
B. {-1/>n(z)/'Pn(z)} converges to D(z, μ) local/y uniformly in JE.
C. {-Pn(z,r)/Qn(z,r)} converges to D(z,μ) locally uniformly in lD>UlE.
General reference: Section 6.1-6.2 and 9.2 in [34].
7 LINEAR FRACTIONAL TRANSFORMATIONS 1
We shall in this section discuss a system of nested disks associated with the functional M. The
results can be proved by applying a Liouville-Ostrogradskii type formula
*( )·'· ( ) + ( )·'·*( ) = 2 (1 - lanl2 'Pn Z 'l'n Z 'Pn Z 'l'n Z ( )( )zB_n (z)) Z - an 1 - anZ
and Christoffel-Darboux type formulas
cp~(z)~ - 'Pn(z)~ ~ ( )-( -)
= L_¿ 'Pk z 'Pk w '
1 - (n(z)(n(w) k=O
cp~(z)~ +cpn(z)~ + ~ = - ~'Pk(z)i/>k(w),
1 - (n(z)(n(w) 1 - zw k=O
1/;~(z)~ -1/;n(z)~ = ~ 1/>k(z)i/>k(w).
1 - (n(z)(n(w) k=O
We set
lDb = { z E lD> : z # ak for k = 1, 2, ... }
~={zElE:z#l/ak for k=l,2, ... }.
For a fixed point z E lDb U~ the values of s = -Pn(z, r)/Qn(z, r) describe a circle Kn(z ) when
r take ali values in 'll'. The closed disk b.n(z) bounded by Kn( z ) is described by
n-1 ( )
s E b.n(z) """""' 2 2s+.S {o} L_¿ l,,Pk(z) - S<pk(z)I :S I ¡2' k=O 1 - z
and its radius rn(z) is given by
2lzl [ n-l 2]-l
rn(z) = l - lzl2 IBn-1(z)l {; l'Pk(z)I
The system of disks {t.n(z)} is nested, i.e. b.n+i(z) e b.n( z ). The intersection
00
b.oo(z) = n b.n(z)
n=l
(7.1)
is therefore either a proper closed disk or a single point. It follows from (7.1) that t.00(z)
is a single point if and only if the sequence {[IBn(z)I ¿;~;;;¿ fcpk(z)i2J-1} tends to zero. When
L.::::"=1 ( 1 - lan 1) < oo this condition is equivalent to divergen ce of the series L:~o l'Pk( z )12.
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Theorem 7.1 Let zo E lilb U lEo and assume that l:.00 (z) is a proper disk. Then l:.00 (z) is a
proper disk for every z E lilb U E,¡_, and the series L~o l<t'k(z)l2 and L~o l1f>k(z)l2 converge
loca/ly uniformly in lilb U lEo U 1l'.
(In [34) only locally uniform convergence in lilb UlEo is stated, but the proof gives the stronger
result.)
As a consequence of this theorem we get a dichotomy:
Either l:.00 (z) is a proper disk for every z E lilb U lEo, the limit circle case, or l:.00 (z) is a single
point for every z E lilb U lEo, the limit point case.
Finally we mention that if ¿:;~= 1 (1 - lanl) = oo, then l:.00 (z) reduces to a point for every
z E lilb U lEo and we have the limit point case.
General reference: Section 10.2 of [34).
8 MOMENT PROBLEMS 1
As before we suppose that we are given a linear functional M on .[,+C., satisfying (3.1)-(3.2).
By the moment problem for M we mean: Find measures μ on 1l' with infinite support such that
M[f] = ¡ f(t)dμ(t) (8.1)
for al! f E L. A measure with this property is called a solution of the moment problem. The
problem is determinate if there is exactly one solution, indeterminate if there is more than one
solution. Note that because of (3.1) we also have M[g] = fyg(t)dμ(t) foral! g E.[,. when μ is
a solution.
For μ to be a solution, it is of course sufficient that (8.1) is satisfied for every element of
sorne basis for .C,. For example, (8.1) is equivalent to
M[Bn] = ¡ Bn(t)dμ(t), n =O, 1, 2, ....
The constants M[Bn], or M[Cn] for any basis { Cn}, may be considered as moments of M, and
this moti vates the expression moment problem for (8.1 ).
We recall the quadrature formulas (5.4) of Section 5. We define the measures μn(·,T) as
the discrete measure with support { (n1 ( T), ... , (nn( T)} and mass Ank( T) at (nk( T), k = 1, . . . , n.
From the quadrature formulas (5.4) we find that
when m < n. Similarly we may write (5.5) as
Pn(z,T) 1 t + z Q ( ) = - -dμn(t,T) = -S1(z,μn(·,T).
n z, T T t - Z
(8.2)
It follows from (8.2) that S1(z,μn(,T)) E Kn(z) when z E lIJb UJEo.
From Helly's selection and convergence theorems it can be deduced that for every z on the
boundary K00 (z) of l:.00 (z) there is a subsequence of a sequence {μn(·,Tn)} which converges
to a solution v and such that the corresponding subsequence of {S1( z, μn(·,Tn))} converges to
S1(z, v). Thus for every boundary point s of l:.00(z) there is a solution v of the moment problem
such that S1(z, v) =s. The set of solutions of the moment problem is easily seen to be convex,
from which it follows that for every s E l:.00(z) there is a solution μ such that S1( z, μ) =s. On
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the other hand, by the aid of Bessel's inequaÜty it can be shown that !1(z,μ) C iloo(z) for any
solution μ of the moment problem.
Summing up this discussion, we get:
Theorem 8.1 Let z E lDb U lli-0. Then il00(z) consists of exactly ali values !1(z,μ), where μis
a solution of the moment problem.
Since a measure is uniquely determined by its transform, this implies:
Corollary 8.2 The moment problem is indeterminate in the limit circle case, determinate in
the limit point case. In particular the problem is always determinate when I::'.°=1 (1- lanl) = oo.
We note that if the sequence { an} consists of a finite number of points repeated in sorne
way, then I::'.°=1 ( 1 - lan 1) = oo and hence the moment problem is determínate. In particular
this is the case in the polynomial situation, when an = O for ali n.
General reference: Section 10.1, 10.3 of [34].
9 FUNCTION SPACES 11
Let { an}~=l be a sequence of points in the extended real line IR. For technical reasons we
assume there is a point in i which is different from ali the ªn· There is no restriction in
assuming this point to be at the origin, i.e., that an i= O for ali n.
We shall make use of the factors Zk defined by
z
Zk(z) = _1 , k= 1,2, ...
1 - ªk z
and the products bn defined by
bo = 1, bn(z) = IJ Zk(z), n = 1, 2, ....
k=I
We shall here use the notation Wn as follows:
Thus we may write
n
wo = 1, wn(z) = IJ(l - a¡;- 1z), n = 1,2,. .. .
k=I
zn
bn(z)=-(-)' n=0,1,2, ....
w,.. z
In particular, bn(z) = zn for ali n when ak = oo for ali k. We note that Zko(z)
bn.(z) = bn(z ), Wn.(z) = wn(z). (Recall that here f.(z) = f( z).)
We shall again study spaces Ln and L of rational functions. These are given by
Ln = Span{bo, b1, ... , bn}, n =O, 1, 2, ...
and
00
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A function f belongs to Ln if and only if it can be written in the form
J(z) = p(z) ,
wn(z)
where p is a polynomial of degree at most n. In particular Ln equals the space of polynomials
of degree at most n if ak = oo for al! k.
As in Section 2 we could work with other simple bases { Co, c1, ... , en, .. . } for [, such that
Ln = Span { eo, e¡, .. . , en}· For example, if al! the points Ctk are finite and distinct,
1 1
{l, -1 , ..• , -1 , ... } 1 - et1 Z 1 - Ctn Z
is such a base, and when al! ak are finite,
1 1
{l, -(-)' ... ' -(-)' ... }
W¡ Z Wn Z
is such a base.
A function f in Ln has the representation J ( z) = L::~=O akbk ( z). We cal! an the leading
coefficient off ( with respect to the basis {bn} ). When an = 1, the function is said to be monic.
General reference: Section 11.1 of [34].
10 ORTHOGONAL FUNCTIONS 11
Let M be a linear functional defined on the linear space [, · L, satisfying
M[f.] = M[f], f E [, · [, (10.l)
and
M[f · J.] > O, f E L, f °t O. (10.2)
Without loss of generality we assume that M[l] = l.
Note that in the present situation we have L. = L . In the previous situation we had
[,·L. = [,+L., and it was sufficient for M to be defined on [,+L •. In the present situation
we have in general [, · [, # [, + L. = L, and we need to require M to be defined on [, · L . The
equality [, · [, = [, holds when the sequence {ak} consists of points which are al! repeated an
infinite number of times in sorne order. In particular this is the case when ak = oo for al! k
(the polynomial case).
Examples of functionals M satisfying (10.1 )-(10.2) can be obtained as follows:
Let μ be a positive measure on R with the property that al! functions in[,·[, are integrable.
Define M by
M[F] = l F(t)dμ(t), FE[,· L. (10.3)
Then clearly M satisfies (10.1)-(10.2).
The functional M gives rise to an inner product (-, ·) on the space [, through the formula
(J,g)=M[f·g.], f,gEL . (10.4)
Let {y:in}::"=o be an orthonormal basis associated with the sequence {Ln}· Le.,
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and
(cpj, 'Pk) = Ójk·
The leading coefficient of 'Pn shall be chosen to be real and p<Jsit.ivc.
The function 'Pn may be represented in thc form
where Pn is a polynomial of exact degree n. Since 'Pn is orthogonal to ali functions (JÍ the form
zm /wn-I (z), m =O, 1, ... , n - 1, we may write
Mn(pn(z)·zm]=O, m=O,l, ... ,n -1,
where
M [!] - M [ J( z ) ]
n - (1 - a;;-1z)wn-1(z)2 ·
This means that the sequence {Pn} of polynomials is orthogonal with respect to the sequence
of varying (not necessarily positive) inner products (., -)n given by
Thus if M[f] is given by (10.3), then the sequence {Pn} is orthogonal with res'pect to the
sequence of varying measures μn given by
The functions of the second kind are defined as follows:
1f;0(z)=iz
1/Jn(z) = M [ -i.-1- +{ ctzp n(t) - 'Pn(z)} ] ,
t-z
These functions have the form
qn(z)
1/Jn(z) = -(-)'n = 1,2, . .. ,
Wn Z
n = 1,2, ....
where qn is a polynomial of degree at most n. Thus 1/Jn E .Cn for n = 1, 2, ....
General reference: Sections 11.1-11.2 of [34].
11 RECURSION 11
The function 'Pn = Pn is called regular if Pn ( °'n-1) # O, singular otherwise. \Vhen the sequen ce
Wn
{'Pn} is regular, i.e., when all cpn are regular, it satisfies a three-term recurrence relation which
generalizes the recursion for orthonormal pol;ynomials.
Theorem 11.1 Assume that the orthonormal sequence { 'Pn} is regular. Thrn tht Junctions
{ 'Pn, 1/Jn} satisfy a recurrence relation of t~e form
. -1 e 1 - ªn-2z + n 1 - a;;- 1 z
[ 0n-2(z) ] .
'fn-2(z)
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with n = 2,3, .. . and ao = oo . . The constants An,En,Cn satisfy
1 - a;;-~2ªn-l
An +En ~ Ü
CXn
(11.1)
(11.2)
In the polynomial case (i.e., an = oo for all n) ali the polynomials 'Pn are regular. Hence we
obtain [ i:~;j] = (Anz +En) [ i~~(~)) ] + Cn [ i:=~~;j ] , n = 2, 3, ...
which has the form of the classical recursion for orthonormal functions.
As in the classical situation, a converse of Theorem 11.1 is true. The argument is, however,
rather more complicated. This is so partly because from a given recursion for functions in C
we need to define a functional M on C · C with respect to which the functions are orthogonal.
Theorem 11.2 Let {cpn} be a sequence of functions such that cp0 = 1, cp1(z) = 1ul:_>. with -a1 z
K ~ O, 'Pn E .Cn \ .Cn-l for n = 1, 2, .... Assume that there exist constants An, En, Cn, n =
2,3, . . . satisfying {11.1)-(11.2} such that
( z 1 - a;;-~ 2 z) 1 - a;;-~2 z
'Pn(z) = An 1 _ a-lz +En l _ a-lz 'Pn-1(z) + Cn l _ a-lz 'Pn-2(z)
n n n
for n = 2, 3,.. .. Then there exists a linear functional M on C · C such that { 'Pn} forms a
regular orthonormal sequence with respect to the inner product {10.4).
General reference: Sections 11.1, 11.9 of [34].
12 QUADRATURE 11
A quasi-orthogonal rational function of order n is a function of the form
1 - a;;-~ 1 z
Qn(z, r) = 'Pn(z) + T _ 1 'Pn-1(z), TE C.
1- Qn Z
(Qn(z,oo) means cp,¡_ 1(z).) These functions satisfy
(Qn(z, r), bk(z)) =O, k =O, 1, ... , n - 2,
and ali functions satisfying this conditions is of form (12.1).
(12.1)
With the quasi-orthogonal functions we associate functions of the second kind given by
. 1-a;;-~ 1 z
Pn(z, r) = tPn(z) + T 1 tPn-1(z).
1 - a~ z
These functions may also be described by the formula
Pn(z,r) = M [-i1 +tz { l -a~11 t Qn(t,z)-Qn(z,r)}], n = 2,3, ....
t - Z 1 - Qn_1t
(12.2)
We inay write
Q ( ) _ Pn(z,r}
n z, T - ( ) ,
Wn Z
P ( ) = qn ( Z, T)
nZ,T ()'
Wn Z
where Pn and qn are polynomials of degree at most n .
In the following we shall assume that T E R.
A value of T for which none of the points {O, a¡, ... , an} are zeros of Pn(z, r ) is called a
regular value for 'Pn · There can be at most n non-regular values.
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Theorem 12 .1 A ssume that 'Pn is regular and that r is a regular value for 'fin. Titen Q r. ( z, r)
has n simple zeros, all lying in r E R \{O, a 1 , .•. , an}.
Note that the zeros are the same as the zeros of the numerator polynomial Pn ( z, r ).
Theorem 12.2 Assume that 'Pn is regular and that T is a regular value for 'fin. Let enk( T ), k =
l , ... ,n, be the zeros ofQn(z,r), and set
A (r) _ i Pn(enk(r),r)
nk - 1 + '' nk(r) 2 Q'n ( >'n k ( T ) ' T ) .
Then the quadrature formula
n
M[R] = L Ank(r)R(enk(r))
k=I
is exact for every RE Ln-1 · Ln-1 ·
An alternative expression for the weight Ank(r) is
The degree of exactness of this formula is one less than maximal in the sense that the formula has
2n parameters An1 ( T ), . . . , Ann( r), ~ni ( r), ... , ~nn( T ), while the dimension of the space Ln-1 ·Ln-1
is only 2n - l. When r = O, the degree of exactness is increased by one, and it is therefore
natural to call this formula a rational Gaussian quadrature formula.
Theorem 12.3 Assume that 'Pn is regular and that r = O is a regular va/u.e for 'Pn . Then the
quadrature formula
n
M[R] = L Ank(O)R(~nk (O))
k=O
is exact for every R E Ln · Ln-1 .
The following result follows from Theorem.12.2 and the fact that the a~gument in formula
(12.2) belongs to Ln- 1 · Ln- 1 ·
Theorem 12.4 Let ~nk(r), Ank(r), k = 1, ... , n, be as in Theorem 12.2. Then we have
General reference: Sections 11.5-11.6, 11.10 of [34).
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13 INTERPOLATION AND CONVERGENCE 11
As in Section 6 we shall here only consider the situation when the functional M is given as
an integral. This means that there exists a positive measure μ on R such that all functions
in .C · .C are integrable, with M given by formula (10.3). We recall the definition (1.2) of the
Riesz-Herglotz-Nevanlinna transform n(z,μ) of μ.
Theorem 13.1 Let 'Pn be regular and /et T be a regular value for 'Pn. Then the quotient - QPn((z,T))
n z,r
interpolates n(z, μ) at the table { i, -i, a1, a¡, .. . ) ªn-1) ªn-d in the following sense:
_Pn(i,r) =n(· )
Q ( . ) i, μ )
n z, T
Pn(-i,r) _ ,...( . )
- - " -i μ
Qn(-i,r) '
lim { Pn(z,r) }(k) + n<kl(z ,μ) =O
z-+crm Qn(z, r)
(subscript means differentiation} for k = O, 1, . . . , a! - 1, m = 1, 2, ... , n - 1, where a!
denotes the multiplicity of am in the sequence {a¡, a1, ... , am, am, ... , O'n-li ªn-d. The limit
is to be understood as angular limtt in arbitrary regions E < larg(z - am)I < 71' - E, E > O,
k=l, ... ,n-1.
For the püre quotients t/Jn/'Pn we have a somewhat stronger result.
Theorem 13.2 Assume that 'Pn is regular and that T = O is a regular value for 'Pn· Then
the quotient -t/Jn/'Pn interpolates n(z, μ) at the table { i, -i, a¡, a1) ... ) ªn-!, ªn-1) an} in the
f ollowing sense:
tPn((z)) + n(z,μ) = (z - i)(z + i)fn(z)wn(z.)wn-1(z),
'Pn Z
where r n is holomorphic in e\ R and bounded in any region € < iarg(z - ak)I < 71' - E, € > o,
k=l, ... ,n.
Since Y!n. is a rational function of type [n/n] the meaning of Theorem 13.2 is: The function
'Pn
-t/Jn/'Pn is the (n/n] multipoint Padé approximant to n(z, μ) at the table { i, -i, a¡, a 1, ... , ªn-i,
On-1,on}·
Let μn( ·, r) denote the discrete measure with support ffn1 ( T ), . .. ) enk( T)} and mass Ank( T)
at enk(r), k = 1, ... ) n.
Theorem 13.3 Assume that the sequence { 'Pn} is regular, and for each n /et Tn be a regular
value of 'Pn. lf the sequence {μn. ( ·, Tn•)} converges to μ, then
local/y uniformly in U.
For the sequence { tPn/'Pn} the following result holds.
Theorem 13.4 Assume that {'Pn} is a regular sequence. Let {'Pn.} be a subsequence of {'Pn}
such that T = O is a regular value for each 'Pn• and such that {μn.(-, O)} converges toμ. Then
l. [ tPn.(z)] ,...( ) lill --- - H Zμ
k-+oo 'Pnk ( Z) - '
local/y uniformly in U.
General reference: Section 11.10-11.11 of (34].
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14 LINEAR FRACTIONAL TRANSFORMATIONS 11
Again we shall make use of Liouville-Ostrogradskii type formulas and Christoffel-Darboux type
formulas. These have the form
and
n-1
= En(w - z) L<f'k(z)rpk(w)
k=O
1/!n( w)1/Jn-l (z)(l - a;;-1w )(1 - a;;-~ 1 z) -1/!n(z)1/!n-l ( w )(1 - a;;-1 z )(1 - a;;-~ 1 w)
=En(w-z) [~1/!k(z)¡/Jk(w)-1].
Here En is a constant, En -=J O if <f'n is regular, En = O if <f'n is singular.
We shall for simplicity assume that the sequence { <,c>n} is regular. The concluding results of
this and the next section are, however, true without this assumption.
The results of this section rely on the formulas above.
For a fixed z E e\ R the mapping T-+ _QPn((z,r)) transforms R to a circle Kn(z). We denote
n Z 1'T
by ~n(z) the closed disk bounded by Kn(z). This disk is described by
n-1 lz - i¡2
s E ~n(z) ~ ll - sl2 + """'11/Jk(z) + S<f'k(z)l2 ~ (s + s)--_-,
L., z-z
k=O
and its radius rn(z) is given by
(14.1)
We have 1/Jn(i) = -<,c>n(i), thus ~n(i) reduces to a point.
The sequence {~n(z)} is nested i.e., ~n+ 1 (z) C ~n(z), andas in Section 7 the intersection
00
~oo(z) = n ~n(z)
n=l
is either a proper closed disk ora single point. It follows from (14.1) that for z -=J i,~00 (z) is a
single point if and only if the series 2:::~1 l<f'k(z)l2 diverges. ·
We set ICo = C \ {R U { i} U { -i}}. The following invariance result holds.
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Theorem 14.1 Let z0 E Ci and assume that ó 00 (zo) is a proper disk. Then ó 00(z) is-a proper
disk for every z E Ci and the series l:~o l'Pk( z) 12 and l:~o l'Pk( z )12 converge local/y uniformly
in Ci.
Thus again we get a dichotomy:
Either ó 00(z) is a proper disk for every z E Ci, the limit circle case, or ó 00(z) is single
point for every z E Ci, the limit point case.
General reference: Sections 11.3-11.4, 11.7 of [34].
15 MOMENT PROBLEMS 11
We assume that we are given a linear functional M on .C · .C satisfying (10.1)-(10.2) . A positive
measure μ on R with infinite support is said to solve the moment problem on .C if
M[/] = L J(t)dμ(t),f E .e, (15.1)
and to solve the moment problem on .C · .C if
M[F] = L F(t)dμ(t), FE .e . .e, (15.2)
·A measure which solves the moment problem on .C · .C also solves the problem orí .C, since
.e e .e . .c.
Clearly it is sufficient for μ to solve the moment problem on .C or on .C · .C that (15.1) or
(15.2) is satisfied for the functions in sorne generating system for .C or .C · .C. For example,
(15.1) is equivalent to
M[bm] = L bm(t)dμ(t), m =O, 1, 2, ... ,
and (15.2) is equivalent to
M[bm. bn] = L bm(t)bn(t)dμ(t), m, n =o, 1, 2, ....
The constants M[bm] or M[bmbn] may be termed moments, from which the expression moment
problems arise.
We sha/I a/so in this section assume that the sequence { 'Pn} is regular.
We recall the measures μn(-, r) introduced in Section 13. Let z E Ci. As in Section 8 we
find that for every s on the boundary of ó 00(z) there is a subsequence of a sequence {μn(-, Tn)}
which converges to a solution v of the moment problem on .C and such that !1(z, v) = s. It
should be noted that in order to carry out the proof, we need to know that M is defined on
.C ·.C. From the convexity of the set of solutions, it follows that for every s E ó 00(z) there is a
solution μ of the moment problem on .C such that n(z, μ) =s. By the aid of Bessel's inequality
it can be shown that n(z,μ) E ó 00 (z ) for every solution of the moment problem on .C..C. Thus
we have:
Theorem 15.1 Let z E Ci. Then
{!1( z, μ): μ solution on .C · .e} C ó 00(z) e {n(z,μ): μ solution on .C}.
Again since a measure is determined by its transform, we conclude:
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Corollary 15.2 The moment problem on .C · .C {if salvable) is determinate in the limit poinl
case. The moment problem on C is indeterminate in the limit circle case.
We note that if the sequence { an} consists of points which are all repeated an infinite
number of times, i.e., if .C = .C · .C, then the moment problem is determinate exactly in the limit
point case.
General reference: Section 11.8 of (34] .
16 CONCLUSION
We have presented sorne of the basic features of the theory oforthogonal rational functions. We
refer to the reference list for detailed treatments both of the topics we have discussed here and
of problem areas that we have not considered. There is much room left for studies of the case
when the poles can be partly on the "support curve", partly outside it. Dewilde and Dym (36]
considered a situation of this kind, anda few remarks can be found in (34]. Sorne applications
of orthogonal rational functions in the area of signa! processing and system theory are discussed
in Chapter 12 of (34], in (28] and in (66]-(67]. .
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