Rev. Acad. Canar. Cienc., XVI (Núms. 1-2), 9-14 (2004) (publicado en julio de 2005)
SEPARATION OF ZEROS OF PARA-ORTHOGONAL
RATIONAL FUNCTIONS
A. Bultheel*, P. González-Vera1, E. Hendriksen1 & Olav Njastad§
Abstract
We generalize a result by L. Golinskii [6] on separation of the zeros of paraorthogonal
polynornials on the unit circle to a similar result for para-orthogonal rational
functions.
Resumen
En este trabajo se extiende un resultado de Golinskii [6] sobre separación de ceros
de polinomios para-ortogonales sobre la circunferencia unidad -al caso de funciones
racionales para-ortoganales.
1 Introduction
Every probability measure on the unit circle gives rise toan orthonormal sequence {Pn}~=o of
polynomials, so called Szegéí polynomials. See for example [7, 8]. Invariant para-orthogonal
polynomials are polynomials of the form c.,[pn(z) + Tp~(z)], ITI = 1, e,,# O, where p~( z) =
znpn(l/z). These polynomials have all their zeros on the unit circle, and they are all simple.
The zeros are nades in a quadrature formula with positive weights which is exact on the space
span{l/zn- 1 , ... , 1, ... , z"- 1 }. See e.g. [7]. An equivalent representation of the invariant
para-orthogonal polynomials is as the class of all polynomials of the form dn [p~ ( z) P~. ( w) -
Pn(z)pn(w)], lwl = 1, dn # O. For a given w, the value z = w is a zero of this polynomial.
It was shown by Golinskii [6] that the zcros of two consecutive of these polynomials (for a
given w) separate each othcr when the zero z = w is not included among thc zeros of thc
polynomial of highest degree.
The aim of this note is to prove a similar result for orthogonal rational funi:;tions on the
unit circlc. In sections 2 and 3 wc givc a brief summary of relevant basic properties of such
functions. For a more comprehensive treatment, see [3]. In section 4 we give a proof of the
indicated result , in the main following the reasoning of Golinskii.
By a quite different approach, Cantero, Moral and Velázquez [5] obtained separation
results that contain the result of Golinskii.
*Department of Computer Science, K.U.Leuven, Belgium. The work of this author is partially supported
by the Fund for Scientific R.esearch (FWO), projects "COR.FU: Constructive study of orthogonal functions",
grant #G.0184.02 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.
1Department Análisis Math., Univ. La Laguna, Tenerife, Spain. The work of this author was partially
supported by the scientific reseaJ"ch project PB96-1029 of the Spanish D.G.E.S.
IDepartment of .Mathematics, University of Amsterdam, The Netherlands.
§Department of Math. Se., Norwegian Univ. of Science and Technology, Trondheim, Norway
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2 Orthogonal rational functions
In the following, ]])) denotes the open unit disk in the complex plane <C. 1f denotes the unit
circle and E the exterior of the closed unit disk.
Let a sequence {an} ~= l of not necessarily distinct points in llJl be given. We define
z - a,,
(o= 1, (n(z) = Z1 n_ -, n = l. 2,.... (2.1)
- O'.nZ
where Zn = -lanl/an if O'.n -f O and z,, = 1 if a,,= O. Furthermore we define the Blaschke
products Bn by
n
Bo = 1, Bn(z) = I1 (k(z), n = l. 2 ... . . (2.2)
k=I
The fun ctions { B0 , B 1 , ... , Bn} span the space [ 11 consisting of all functions of the form
f( z) = P(z)/7í( z), where PE Pn (the space of polynomials of degree at most n) and
n
(2.3)
k=1
In general we define for any function f E Ln \ Ln- l the superstar transform f* by J* ( z) =
Bn(z )f.(z), where J.( z) = f (1/z) . Note that J* also belongs to l n·
Let μbe a probability measure on 1f, with associated inner product (-, ·) given by
(f, g) = h f(t)g(t)dμ(t). (2.4)
We shall use the notation <fin for the elements of the orthonormal basis for [ 11 which is
ordered such that <Po = 1 and cPk E Lk \ Lk-l for k = l. 2, .. . . n. We may then write
efin (z) = Pn( z )/7ín(z), efi~ ( z) = q,,(z )/ 7ín( z) where Pn E P11 , Qn E Pn·
We note that if O'.n = O for all n, then Bn ( z) = z", [ 11 = P n and ef¡1,, ef¡~ are orthonormal
polynomials with respect toμ and their reciprocals. For motivations for studying the rational
generalizations of orthogonal polynomials introduced above, we refer to [3. 4].
Let kr.(z, w) denote the reproducing kernel for Ln, i.e.,
n
kn( z, w) = L cPi (z)ef¡j(w). (2.5)
j=O
The orthonormal functions <fin satisfy
efi~+l (z)<P:i+l ( W) - cPn+l (z )efin+l ( W) = [l - (n+l ( z)(n+ l ( w) ]k,,( z, W ). (2.6)
efi~(z)efi~(w) - (r,(z)(n(w)efin(z)efi11(w) = [1 -(n(z)(n(w)]kn(z, w). (2.7)
It follows easily from these formulas that
l<Pn(z)I < l efi~(z)I for z E]]))
l<Pn(z)I = l efi~(z)I for z E 1f
lefi,,(z)I > l efi~(z)I for z E E.
(2.8)
(Note that l(n(z)I < 1 for z E ]])), l(n(z)I = 1 for z E 1f and l(n( z) I > 1 for z E E.)
Furthermore ali the zeros of ef¡,, lie in ]])). Simple examples ( e.g. with μ the normalized
Lebesgue measure anda,, = O for all n, which gives efin(z) = z") show that the zeros may be
multiple.
For more exhaustive treatments, see e.g., [l , 3].
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3 Para-orthogonal rational functions
Quadrature formulas with positive weights and nodes on 'f have important uses. Of special
interest are such formulas which integrate exactly ali functions in spaces of the form L p,q =
{fg: f E l q,9• E Lp} with as large value of p + q as possible. The zeros of <f>n can not be
used as nodes, since thcy líe in ][]) (and may even be multiple). It turns out that so-called
invariant para-orthogonal functions give rise to such quadrat ure formulas, exact on L n - l ,n - l
(while no quadrature formula as specified can be exact on L n-l,n of on Ln,n-1). See [2, 3l.
A function Qn in Ln is called invariant if Q~(z) = knQn(z) for sorne kn =/= O. It is
called para-orthogonal if (Qn , !) = O for ali f E Ln-l n l n(o:n), where l n(o:n) = {f E Ln :
f(o:n) =O}, while (Qn, 1) =/= O and (Qn, Bn) =/= O. (These concepts are direct generalizations
of corresponding concepts in the polynomial case, ie., when O:n = O for all n. These were
introduced and studied in [71.)
It can be shown that the invariant para-orthogonal rational functions are exactly functions
of the form c.,,Qn(z, T) , Cn =/=O, where
(3 .1)
Furthermore, Qn(z, T) has exactly n simple zeros, ali of them lying on 'f. See [3l .
Now considera function dníln(z, w), dn =/=O, where
íln( z, w) = [<f>~(z)<f>~(w) - <f>n(z)<f>n(w)l. (3.2)
We may write
(3.3)
Because of (2.8) we have for w E 'f that -[::¡:lJ E 'f. Thus íln(z,w) is a function of the
form cnQn(z, T) as in (3. 1). On the other hand, for each T E 'f, there aren values of w
in 'f such that -[::¡:lJ = T. (Note that for a given T , -[:::¡:~l = T may be written as an
algebraic equation of degree nin w, and that according to (2.8), -[:~¡:jl E 'f if and only if
w E 'f. See also [3, Thm. 5.2. ll.) Thus the class of functions of the form cnQn(z, T), e,,=/= O,
T E 'f as given in (3. 1) is exactly the same as the class of functions dníln(z, w), dn =/= O,
w E 'fas given in (3.2).
4 Separation of zeros
In [6l Golinskii showed that in the polynomial case, i.e., when ali O:n equal zero, a certain
separation property of the zeros of two consecutive polynomials íln(z, w) (for fixed w) holds.
We shall prove a similar result in the general rational case. The result as well as the proof
are rather straightforward generalizations of Golinskii 's discussion in the polynomial case.
In the following, w denotes a fixed point on 'f. We observe that [l - (n(z)(n(w)l = O if
and only if z = w. It then follows from (2.5)-(2.6) and (3.2) that z = w is a zero of íln(z, w)
for ali n , and that the remaining zeros of íln ( z, w) are exactly the zeros of kn-l ( z, w).
Now assume that z0 is a common zero of íln(z,w) and íln+l(z ,w) , z0 =/= w. Note that
z0 has to be on 'f. It follows from (2.6) and the definition (3.2) that kn(z0 , w) = O and
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kn_1(zo,w) =O, hence also <f>n(zo)</>n(w) =O. This is impossible sincc all the zeros of <f>n lie
in lll Consequently íln( z, w) and íln+l(z , w) have no common zeros except z = w.
Now for each n !et Zn,k = eiBn.k, k =O, 1, ... , n-1, be the zeros of íl,,(z, w), with z,,,0 = w,
ordered such that
Bn,O < Bn,l < · · · < Bn ,n-1 < Bn,O + 27í. ( 4.1)
Theorem 4.1 The zeros ofíln(z, w) included z = w and the zeros ofíln+l (z , w) not included
z = w separate each other in the sense that
Bn,O < Bn+l,l < Bn,l < Bn+l ,2 < · · · < Bn,n-l < Bn+l,n· (4.2)
Proof. Consider the function
( ) ( ) kn ( Z, W)
rnz =fnz,w = íl ( )"
n Z,W
(4.3)
It follows from the foregoing discussion that the zeros of kn(z, w) are exactly the points
Zn+l,l, ... ' Zn+l,n while the zeros of íln ( z' w) are the points Zn,o, ... ' Zn,n-1 · Thus r n ( z) has
simple zeros at the points Zn+i,1, ... , Zn+J,n and simple poles at the points Zn,o, ... . Zn,n-I ·
(Recall that íln ( z, w) and íln+l ( z, w) ha ve no common zeros except z = w. Also note that
the terms 7ín(z) in the numerator and the denominator cancel.) Expressing kn(z, w) by (2.7)
we may write
r n(z) = <t>;Jz)~ - (n(z)~<l>n(z)~
[1 - (n(z)(n(w)][<f>;Jz)</>~(w) - </>n(z)</>n(w)]
(4.4)
We introduce the function bn defined by
b ( ) = </>n(z)
n z </>~( z)" (4.5)
We note that bn is holomorphic in lDJ U 11' and maps lDJ onto lDJ, 11' onto 11', according to (2.8).
In terms of this function, fn( z) may be written as
rn( z) = 1- (n(z)(n(w)bn(z)bn(w)
[1- (n(z)(n(w)][l - bn(z)bn(w)]
and hence by a simple calculation
r n(z) = ~ [1 + b,,(z)/JJ.W) + 1 + (n(z)~] .
2 1 - bn(z)bn(w) 1 - (n(z)(n(w)
(4.6)
(4.7)
The Méibius transformation z ---+ i:~ maps lDJ onto the open right half-plane lHI and 11' onto
the extended imaginary axis ft. Taking into account the mapping properties of the function
bn stated above, we find that each of the two terms in ( 4. 7) maps lDJ onto lHI and 11' onto ft. The
function r n ( z) then has the same property. In other words, r n ( z) is a lossless Carathéodory
function.
A rational lossless Carathéodory function has the property that the zeros and poles
separate each other. For the sake of completeness, we sketch the proof.
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The function fn(z) may be written in the from
n-1
r " ( z ) = i. c+ ~L> .k_ Z +_ Zn_,k ,
k=O Z - Zn,k
(4.8)
where Ak > O and e is a real constant. (See e.g. [7].) We find that
It follows that
z+z k iO+ iOn k ( When B-+ Bn,k, the term >.k z-z~:i = >.<,0_:,0,.:in 4.8) dominates, hence we may conclude
that
1 o
lim -:-fn(e') = +oo,
o_.o;;.k i
1 o
lim -:-fn(e') =-OO.
o-o~,k i
Thus the image of the are { z = e;o : Bn,k < B < Bn,k+ ¡} by the mapping z -+ f n ( z) is the
whole imaginary axis Il. Consequently (at least) one of the zeros of f n(z) must lie on this
are. Taking into account the ordering for general n indicated in ( 4.1) and the fact that r n ( z)
has the same number of zeros a11d poles, we conclude that (4.2) holds.
This completes the proof of the theorem. D
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