Table Of ContentChristoffel formula for kernel polynomials on the unit
circle
C.F. Braccialia, A. Mart´ınez-Finkelshteinb, A. Sri Rangaa∗and D.O. Veronesec
a
DMAp, IBILCE, UNESP - Universidade Estadual Paulista,
7
15054-000,Sa˜o Jos´e do Rio Preto, SP, Brazil.
1
0 a
Departamento de Matema´ticas, Universidad de Almer´ıa, 04120 Almer´ıa,
2
and Instituto Carlos I de F´ısica Teo´rica and Computacional, Granada University, Spain
n
a cICTE, UFTM - Universidade Federal do Triaˆngulo Mineiro,
J
38064-200Uberaba, MG, Brazil.
8
1
January 19, 2017
]
A
C
.
h
Abstract
t
a
m Given a nontrivial positive measure µ on the unit circle, the associated
n
[ Christoffel-Darboux kernels are Kn(z,w;µ) = k=0ϕk(w;µ)ϕk(z;µ), n ≥ 0,
where ϕk(·;µ) are the orthonormalpolynomials with respect to the measure µ.
1 P
Letthepositivemeasureν ontheunitcirclebegivenbydν(z)=|G2m(z)|dµ(z),
v
5 where G2m is a conjugate reciprocalpolynomialofexact degree2m. We estab-
9 lish a determinantal formula expressing {Kn(z,w;ν)}n≥0 directly in terms of
9 {Kn(z,w;µ)}n≥0.
4 Furthermore, we consider the special case of w =1; it is knownthat appro-
0
priately normalized polynomials Kn(z,1;µ) satisfy a recurrence relation whose
.
1 coefficients are given in terms of two sets of real parameters {cn(µ)}∞n=1 and
0 {gn(µ)}∞n=1,with0<gn <1forn≥1. Thedoublesequence{(cn(µ),gn(µ))}∞n=1
7
characterizesthemeasureµ. Anaturalquestionabouttherelationbetweenthe
1
: parameterscn(µ),gn(µ),associatedwithµ,andthesequencescn(ν),gn(ν),cor-
v responding to ν, is also addressed.
i
X Finally, examples are considered, such as the Geronimus weight (a measure
supported on an arc of T), a class of measures given by basic hypergeometric
r
a functions,andaclassofmeasureswithhypergeometricorthogonalpolynomials.
Keywords: Orthogonal functions, Christoffel formulas, three term recurrence rela-
tion, orthogonal polynomials on the unit circle.
1 Introduction
Given a nontrivial positive measure µ on the unit circle T := {ζ = eiθ: 0 ≤ θ ≤ 2π}
the associated orthonormal polynomials ϕ (z;µ) = κ zn +lower degree terms are
n n
defined by κ = κ (µ)> 0 and
n n
2π
ϕ (ζ;µ)ϕ (ζ;µ)dµ(ζ) = ϕ (eiθ;µ)ϕ (eiθ;µ)dµ(eiθ)= δ ,
m n m n m,n
ZT Z0
∗[email protected] (corresponding author)
for m,n = 0,1,2,..., where δ stands for the Kronecker delta. These are
m,n
orthogonal polynomials on the unit circle, or in short, OPUC. A recent complete
treatise on OPUC is the monograph [24]. Among their fundamental properties is
that all their zeros belong to the open unit disk D := {z ∈ C: |z| < 1}.
The reproducing kernels K (z, w;µ) (also known as Christoffel–Darboux kernels
n
or simply CD kernels) associated with the measure µ are given by
n
K (z,w;µ) = ϕ (w;µ)ϕ (z;µ), n≥ 0. (1.1)
n j j
j=0
X
They have been the subject of study in many recent contributions including the
review[25]ontheiruseinthespectraltheory oforthogonalpolynomialsandrandom
matrices.
In what follows we use the standard notation for the reversed (or conjugate-
reciprocal) polynomials: if q is an algebraic polynomial of degree n, then
q∗(z) := znq(1/z).
Withthisnotation, thewell-known Christoffel-Darbouxformulasays thatforz 6= w,
ϕ (w;µ)ϕ (z;µ)−ϕ∗ (w;µ)ϕ∗ (z;µ)
K (z,w;µ) = n+1 n+1 n+1 n+1
n
wz−1
(1.2)
wzϕ (w;µ)ϕ (z;µ)−ϕ∗(w;µ)ϕ∗(z;µ)
= n n n n , n≥ 0.
wz−1
Notice that K (z,0;µ) = ϕ∗(0;µ)ϕ∗(z;µ). On the other hand, if w ∈ T, then all
n n n
zeros ofK (z,w;µ) (asapolynomialinz)lieonT, anduptoanormalization factor,
n
(wz −1)K (z,w;µ) is a so-called para-orthogonal polynomial of degree n+1. For
n
information concerning para-orthogonal polynomials we refer to [4] and references
therein.
Amultiplicationofthegivenmeasureµbyafactorthatispositiveonitssupport,
supp(µ), yields a new measure and a correspondingset of OPUC and of CD kernels.
It is a natural question to ask whether there is an explicit connection between these
two sets.
In this paper we are interested in the case when the factor is of the form |g|2,
whereg is a polynomial. For the orthogonality on thereal axis, this is thecontent of
the so-called Christoffel formula (see, for example, [29]), which was extended in [20]
to cover OPUC (see also [22], which generalizes [20] and constitutes a nice survey
of related results obtained prior to 1999, as well as some recent related results in
[1, 2]). In these cases there is a determinantal expression for the “new” orthogonal
polynomials in terms of those orthogonal with respect to µ.
One of the goals of this paper is to obtain such a determinantal formula for the
CD kernels on T. Observe that this kind of expressions is not a trivial consequence
of the analogous formulas for OPUC.
RecallthattheclassicalFej´er–Riesztheorem(see[19,§1.12])saysthateverynon-
negative trigonometric polynomial f(θ) can be written as |g(z)|2, z = eiθ, where g
is an algebraic polynomial non-vanishing in D. Equivalently, we can say that f(θ)
is of the form z−mG(z), where G is a self-reciprocal polynomial (i.e., G∗ = G) of
degree 2m.
Motivated by this result, we slightly weaken the assumptions of Fej´er and Riesz
and require the trigonometric multiplication factor of µ to be non-negative only
2
within the support of µ. More precisely, let G be a self-reciprocal polynomial of
2m
exact degree 2m, m ∈ N, and non-negative on supp(µ), and let
G (ζ)
dν(ζ)= 2m dµ(ζ)= |G (ζ)|dµ(ζ), ζ ∈ T, (1.3)
ζm 2m
which is also a positive measure on T.
We denote by
z ,z ,...,z
1 2 2m
the zeros of G . Those of them not on T must appear in pairs symmetric with
2m
respect to T; notice that no z is = 0. However, unlike in the case of Fej´er–Riesz,
j
if supp(µ) 6= T, zeros of G on T also can be simple, as long as the hypothe-
2m
sis of positivity of G (ζ)/ζm on supp(µ) (i.e., the positivity of ν on supp(µ)) is
2m
preserved.1
Inwhatfollows wewillbemainly interested inthecase whenall z ’sarepairwise
j
distinct (or equivalently, when all zeros of G are simple).
2m
Definition 1.1. Given m ∈ N, we call a set P = {p ,p ,...,p } of 2m+1 not
0 1 2m
identically 0 algebraic polynomials p admissible if p (z) ≡ 1, p (z) = zm,
j 0 2m
min{j,m}
p (z) = b zi, j = 1,...,2m−1, (1.4)
j ij
j=max{0,j−m}
X
and either one of the following three conditions is satisfied:
degp < j for j = 1,...,2m, (1.5)
j
degp < m for j = 1,...,2m−1, (1.6)
j
or
p (0) = 0 for j = 1,...,2m. (1.7)
j
Theorem 1.1. For an admissible set P = {p ,p ,...,p } and w ∈ C define
0 1 2m
Q (z,w) := p (z)K (z,w;µ), j = 0,1,...,2m, (1.8)
j j n+2m−j
and the (2m+1)×(2m+1) matrix
Q (z,w) Q (z,w) ... Q (z,w)
0 1 2m
Q (z ,w) Q (z ,w) ... Q (z ,w)
0 1 1 1 2m 1
Q(z,w) = Q(P)(z,w) := .. .. .. .. . (1.9)
. . . .
Q (z ,w) Q (z ,w) ... Q (z ,w)
0 2m 1 2m 2m 2m
(P)
Then, there exists a polynomial C (w) = C (w) of degree ≤ (2m+1)(n+m) such
n n
that
detQ(z,w) = C (w)G (z)K (z,w;ν). (1.10)
n 2m n
1For instance, if supp(µ) = {eiθ : 0 < θ1 ≤ θ ≤ θ2 < 2π}, we can consider the self reciprocal
polynomial G2(z) = e−iα/2e−iβ/2(z−eiα)(z−eiβ), with 0 < α ≤ θ1 and θ2 ≤ β < 2π. Then the
rational function
G2(ζ) G2(eiθ) θ−α β−θ
= =4sin sin
ζ eiθ 2 2
is positive on supp(µ), butnot on theentire T;see Example 4.3 in Section 4 below.
3
Observe that for certain values of w, both sides of (1.10) can vanish, in which
casetheidentity in(1.10)isformallycorrect, butpractically useless. Thus,anatural
question is about sufficient conditions for C 6= 0.
n
Theorem 1.2. Let all the zeros of G be simple. For an admissible set P =
2m
{p ,p ,...,p } and w ∈ C, with the notations of Theorem 1.1, if either
0 1 2m
i) |w| ≥ 1, condition (1.5) holds and polynomials Q (·,w),Q (·,w),...,Q (·,w)
1 2 2m
are linearly independent, or
ii) |w| ≤ 1, condition (1.6)holdsandpolynomials Q (·,w),Q (·,w),...,Q (·,w)
0 1 2m−1
are linearly independent, or
iii) 0< |w| ≤ 1, condition (1.7) holds and polynomials Q (·,1/w), Q (·,1/w), ...,
1 2
Q (·,1/w) are linearly independent,
2m
(P)
then C (w)6= 0.
n
Remark 1.1. If the polynomial G has non-simple zeros, then the results above
2m
stillholdifonereplacesthepolynomialsineachrowofthematrixQbytherespective
derivativesinaccordancewiththeorderofmultiplicity. Forexample,ifz 6= z = z ,
1 2 3
then the fourth row of Q must be replaced by
Q′(z ,w),Q′(z ,w),··· ,Q′ (z ,w),Q′ (z ,w),
0 2 1 2 2m−1 2 2m 2
where Q′ stands for its derivative with respect to z.
j
Given an admissible set P = {p ,p ,...,p }, we define P = {p ,p ,...,p },
0 1 2m 0 1 2m
with
pj(z) := zjpj(1/z) =zj−deg(pj)p∗j(z), j = 0,1,.b..,2mb. b (b1.11)
Observe that (P)= P.
b
Propositiond1b.3. A set of polynomials P = {p0,p1,...,p2m} is admissible if and
only if P = {p ,p ,...,p } is. Moreover, P satisfies (1.5) (resp., (1.7)), then P
0 1 2m
satisfies (1.7) (resp., (1.5)).
Addibtionalbly, bif w 6= 0b, b
b 1
C(P)(w) 6= 0 ⇔ C(P) 6= 0. (1.12)
n n w
(cid:18) (cid:19)
It would be nice to have a simple recipe for constructing an admissible set P for
which (1.10) renders a non-trivial identity for the CD kernel K (z,w;ν). Obviously,
n
there is no “universal” P such that = C(P)(w) in (1.10) is 6= 0 for all w ∈ C.
n
However, there is a simple admissible set that guarantees this, at least for |w| = 1.
It is easy to check that P = {p ,p ,...,p }, with
0 1 2m
zj/2, if j is even,
p (z) = z⌊j/2⌋ = j = 0,1,...,2m, (1.13)
j (z(j−1)/2, if j is odd,
isadmissibleandsatisfiesboth(1.5)and(1.6). ThecorrespondingP = {p ,p ,...,p }
0 1 2m
is
p (z) = z⌊j+1/2⌋ = zj/2, if j is even, j = 0,1,...,b2m, b b (1.14b)
j
(z(j+1)/2, if j is odd,
satisfyinbg, by Proposition 1.3, condition (1.7).
4
Proposition 1.4. Let the admissible set of polynomials P = {p ,p ,...,p } be
0 1 2m
given by (1.13), and let w be such that
K (0,w;µ) 6= 0 and degK (·,w;µ) = n for n ∈ N. (1.15)
n n
Then Q (·,w), Q (·,w), ..., Q (·,w) are linearly independent. In particular, it
0 1 2m
holds for |w| = 1.
Corollary 1.5. For the admissible sets of polynomials P and P, given by (1.13)
b
(P) (P)
and (1.14), respectively, both C (w) 6= 0 and C (w) 6= 0 in (1.10) when |w| = 1.
n n
b
Example 1.1. Let us consider the normalized Lebesgue measure on T,
1
dµ(ζ)= |dζ|, ζ ∈ T. (1.16)
2π
Then
1−wn+1zn+1
K (z,w;µ) = , n ≥ 0,
n
1−wz
so that all w 6= 0 satisfy conditions (1.15) from Proposition 1.4. In particular, for all
(P)
such w, and for the admissible set P given by (1.13), C (w) 6= 0 in (1.10). How-
n
ever, K (z,0;µ) ≡ 1, which implies that polynomials Q (z,0) in (1.8) are linearly
n j
dependent for any choice of the admissible set of polynomials P, and hence, one
cannot find an admissible set for which detQ(·,0) 6≡ 0. Clearly, we still can recover
K (z,0;ν) = ϕ∗(0;ν)ϕ∗(z;ν) by taking limit,
n n n
1
G (z)K (z,0;ν) = lim detQ(z,w).
2m n w→0C(P)(w)
n
The proofs of the assertions above are gathered in Section 2.
In Section 3 we consider an interesting particular case of w = 1, for which
K (·,1;µ) constitute an instance of paraorthogonal polynomials on T. A convenient
n
“symmetrization” of these polynomials was found in [8]; it was shown there that
the appropriately normalized K (·,1;µ), that we denote by R (·;µ) (see the precise
n n
definition in Section 3) satisfy a three term recurrence relation of the form
R (z;µ) = [(1+ic )z+(1−ic )]R (z;µ)
n+1 n+1 n+1 n
(1.17)
− 4(1−g )g zR (z;µ),
n n+1 n−1
for n ≥ 0, with R (z) = 0 and R (z) = 1. Sequences {c }∞ = {c (µ)}∞
−1 0 n n=1 n n=1
and {g }∞ = {g (µ)}∞ are both real, with 0 < g < 1 for n ≥ 1. As shown in
n n=1 n n=1 n
[4,6,8],thedoublesequence{(c (µ),g (µ))}∞ isaparametrization ofthemeasure
n n n=1
µ, alternative to its Verblunsky coefficients. Thus, a natural question is the relation
between the parameters c (µ), g (µ), associated with µ, and the sequences c (ν),
n n n
g (ν), correspondingto ν. These questions will be addressed in Section 3. Since the
n
statement of the corresponding results requires introducing a considerable piece of
notation, we postpone it to the aforementioned section.
Finally, in Section 4 we consider four different applications of our formulas: a
rather straightforward case when µ is the Lebesgue measure on T, the Geronimus
weight (a measure supported on an arc of T), a class of measures given by basic
hypergeometric functions, and a class of measures with hypergeometric OPUC.
5
2 Proof of the Christoffel formula for kernels
First we discuss a characterization of the kernel polynomials K .
n
Let w ∈ C be fixed. With the positive measure µ on T we consider the complex-
valued measure on T given by,
dµ (ζ) =(1−wζ)dµ(ζ).
w
The following simple lemma will be useful in the forthcoming proofs:
Lemma 2.1. Let f be an integrable function on T such that either one of the fol-
lowing condition is satisfied:
i) |w| ≤ 1 and
|f(ζ)|dµ (ζ) = 0, (2.1)
w
Z
or
ii) |w| ≥ 1 and
ζ|f(ζ)|dµ (ζ) = 0. (2.2)
w
Z
Then f = 0 µ-a.e. (in case when |w| =6 1) and µ -a.e., otherwise.
T\{w}
In particular, if f is a polynomial and µ has an infinite number of points of
(cid:12)
increase, then i) or ii) imply that f ≡ 0. (cid:12)
Proof. Consider i) first. For w = 0 the statement is trivial, so let 0 < |w| ≤ 1. By
assumptions of the lemma,
1
|f(ζ)| −ζ dµ(ζ)= 0,
w
Z (cid:18) (cid:19)
and hence,
1
|f(ζ)| −eiθζ dµ(ζ) = 0, θ = −arg(w).
|w|
Z (cid:18) (cid:19)
In particular, taking the real part, we get
1
|f(ζ)|Re −eiθζ dµ(ζ) = 0,
|w|
Z (cid:18) (cid:19)
and it remains to notice that
1
Re −eiθζ > 0,
|w|
(cid:18) (cid:19)
unless |w| = 1 and ζ =w.
In the case ii), we have that
|f(ζ)| w−ζ dµ(ζ) = |f(ζ)|(w−ζ)dµ(ζ) =0,
Z Z
(cid:0) (cid:1)
so that
|f(ζ)| |w|−eiθζ dµ(ζ) = 0,
Z (cid:16) (cid:17)
and again,
Re |w|−eiθζ > 0,
(cid:16) (cid:17)
unless |w| = 1 and ζ =w.
6
Lemma 2.2. For a fixed w ∈ C and n ∈ N, the CD kernel K (z,w;µ) is a poly-
n
nomial in z of degree ≤ n, characterized up to a constant factor by the following
orthogonality relations,
ζsK (ζ,w;µ)dµ (ζ)= 0, 1 ≤ s ≤ n, (2.3)
n w
Z
and the additional condition
a) if |w| ≥ 1, then K (z,w;µ) is of degree exactly n;
n
b) if |w| ≤ 1, then
K (ζ,w;µ)dµ (ζ)6= 0.
n w
Z
Proof. Orthogonalityconditions(2.3)areastraightforwardconsequenceof (1.2)and
of the well-known relations for the reversed polynomials,
ζsϕ∗(ζ;µ)dµ(ζ) = 0, 1 ≤ s ≤ n, and ϕ∗(ζ;µ)dµ(ζ) 6= 0.
n n
Z Z
Since for |w| ≥ 1, ϕ (w;µ) 6= 0, usingthe definition (1.1) of K (z,w;µ) we conclude
n n
that it is a polynomial in z of degree = n. For |w| ≤ 1, using (1.2) and the fact that
ϕ∗ does not vanish inside or on the unit disk, we see that
n+1
K (ζ,w;µ)dµ (ζ)= K (ζ,w;µ)(1−wζ)dµ(ζ)
n w n
Z Z
= ϕ∗ (w;µ) ϕ∗ (ζ;µ)dµ(ζ) 6= 0.
n+1 n+1
Z
Let us prove that these relations characterize the CD kernel.
Assume that |w| ≥ 1. If P is a polynomial of degree exactly n, satisfying
ζsP(ζ)dµ (ζ)= 0, 1 ≤ s≤ n, (2.4)
w
Z
then there exists a constant c6= 0 such that L(z) := cK (z,w;µ)−P(z) is of degree
n
≤ n−1. By hypothesis, L satisfies the same orthogonality conditions, so that
0 = ζL(ζ)L(ζ)dµ (ζ)= ζ|L(ζ)|2µ (ζ),
w w
Z Z
and it remains to apply Lemma 2.1, ii), to conclude that L ≡ 0.
On the other hand, if P of degree ≤ n satisfies (2.4) and is such that
P(ζ)dµ (ζ)6= 0,
w
Z
then there exists a non-zero constant, let us denote it by c again, such that
(P(ζ)−cK (ζ,w;µ))dµ (ζ)= 0.
n w
Z
Combining it with (2.3) and (2.4) we get that for s = 0,1,...,n,
ζs(P(ζ)−cK (ζ,w;µ))dµ (ζ)= ζs(P(ζ)−cK (ζ,w;µ))(1−wζ)dµ(ζ)= 0.
n w n
Z Z
It means that
(P(ζ)−cK (ζ,w;µ))(1−wζ)= constϕ (ζ;µ).
n n+1
Since ϕ cannot vanish at 1/w, we conclude that P(ζ)≡ cK (ζ,w;µ).
n+1 n
7
In order to prove Theorem 1.1 we need some preparatory steps.
With the notation of Section 1, it is immediate to check that detQ(z,w) is an
algebraic polynomialinz (of degree≤ (n+2m)) andinw (ofdegree≤ (m+n)(2m+
1)). Furthermore, for each w ∈ C it vanishes at the zeros z ,...,z of G . Thus,
1 2m 2m
we can write
detQ(z,w) = G (z)A (z,w), (2.5)
2m n
where A is an algebraic polynomial in z (of degree ≤ n) and in w. We need to
n
show that for each w ∈ C, there exists a constant C such that
A (z,w) =CK (z,w;ν). (2.6)
n n
If this is established, the polynomial dependence of C from w (as well as on the
admissible set P chosen, see Definition 1.1) is a straightforward consequence of
(2.5)–(2.6).
We prove (2.6) by appealing to the characterization of K given in Lemma 2.2.
n
By (2.3) and the definition of ν, kernels K (z,w;ν) satisfy
n
ζsG (ζ)K (ζ,w;ν)dµ (ζ) = 0, m+1 ≤ s ≤ m+n.
2m n w
Z
Thus, a necessary condition for (2.6) is that
ζsG (ζ)A (ζ)dµ (ζ) = 0, m+1 ≤ s ≤ m+n.
2m n w
Z
This is always true, and it is an immediate consequence of the following lemma:
Lemma 2.3. For j = 0,1,...,2m,
ζsQ (ζ,w)dµ (ζ) = 0, s = m+1,...,m+n. (2.7)
j w
Z
Thus,
ζsdetQ(ζ,w)dµ (ζ) =0, m+1 ≤ s ≤ m+n. (2.8)
w
Z
Proof. In order to calculate
ζsp (ζ)K (ζ,w;µ)dµ (ζ) for m+1 ≤ s ≤ m+n, 0≤ j ≤ 2m, (2.9)
j n+2m−j w
Z
with account of (1.4), it is sufficient to find the values of
ζs−rK (ζ,w;µ)dµ (ζ) for m+1 ≤ s ≤ m+n, 0≤ j ≤ 2m, (2.10)
n+2m−j w
Z
for max{0,j −m}≤ r ≤ min{j,m}.
Since r ≤ min{j,m} ≤ m, we get
s−r ≥ m+1−r ≥ 1.
Analogously, from r ≥ max{0,j −m}≥ j −m, we conclude that
s−r ≤ m+n−r ≤ n+2m−j.
By (2.3) it follows that all integrals in (2.10) (and consequently, in (2.9)) vanish,
which yields (2.7)–(2.8).
8
So, the necessary condition (orthogonality) always holds. Now we go for a suffi-
cient condition, given by a) and b) of Lemma 2.2.
Recall first the following well known fact, that we state just as a remark.
Remark 2.1. By the maximum principle, |ϕ∗(z;µ)| > |ϕ (z;µ)| for |z| < 1, and
n n
|ϕ∗(z;µ)| < |ϕ (z;µ)| for |z| > 1. By (1.2),
n n
K (z,w;µ) = 0 ⇔ ϕ (w;µ)ϕ (z;µ)−ϕ∗ (w;µ)ϕ∗ (z;µ) = 0,
n n+1 n+1 n+1 n+1
which for |w| > 1 can be rewritten as
ϕ (z;µ) ϕ∗ (w;µ)
n+1 = n+1 .
ϕ∗ (z;µ) ϕ (w;µ)
n+1 (cid:18) n+1 (cid:19)
The right hand side is of absolute value < 1, so that equality is possible only for
|z| < 1. Same analysis is valid for the other case and we conclude that the zeros of
K (z,w;µ) (as a polynomial of z) are of absolute value > 1 (respectively, = 1 or
n
< 1) if |w| < 1 (respectively, |w| = 1 or |w| < 1).
Lemma 2.4. Let A be defined by (2.5) and w ∈ C fixed. The following conditions
n
are necessary and sufficient for A (·,w) ≡ 0:
n
i) |w| ≥ 1 and
deg(detQ(·,w)) < n+2m; (2.11)
ii) |w| ≤ 1 and
ζmdetQ(ζ,w)dµ (ζ) = 0. (2.12)
w
Z
Proof. A (·,w) ≡ 0is clearly asufficientcondition in i) andii) for(2.11)and(2.12),
n
respectively. So, we prove that this is also necessary.
For i), if degA (·,w) < n (which is equivalent to deg(detQ(·,w)) < n+2m),
n
then by (2.8),
0 = ζm+1A (ζ,w)detQ(ζ,w)dµ (ζ)= ζ|A (ζ,w)|2|G (ζ)|dµ (ζ),
n w n 2m w
Z Z
and we use again Lemma 2.1, ii), to conclude that A (·,w) ≡ 0.
n
For ii), if
ζmdetQ(ζ,w)dµ (ζ) = A (ζ,w)|G (ζ)|dµ (ζ)= 0,
w n 2m w
Z Z
then by (2.8) we have in fact that for s = 0,1,...,n,
ζsA (ζ,w)|G (ζ)|dµ (ζ)= ζsA (ζ,w)(1−wζ)|G (ζ)|dµ(ζ) = 0.
n 2m w n 2m
Z Z
Since degA (·,w) ≤ n, we conclude that
n
A (ζ,w)(1−wζ)= cϕ (ζ;ν),
n n+1
But ϕ (ζ;ν) cannot vanish at ζ = 1/w ∈/ D, which implies that c = 0, and
n+1
A (·,w) ≡ 0.
n
9
A combination of Lemmas 2.3 and 2.4 constitutes the proof of Theorem 1.1.
Indeed,let|w| ≥1. Considertheidentity(2.5);byLemma2.4,i),eitherA (·,w) ≡ 0
n
or degA (·,w) = n. In the latter case by the characterization in Lemma 2.2, a),
n
A (z,w) = CK (ζ,w;ν).
n n
On the other hand, if |w| ≤ 1, then again by Lemma 2.4, ii), either A ≡ 0 or
n
A (ζ)|G (ζ)|dµ (ζ)6= 0,
n 2m w
Z
in which case by Lemma 2.2, b), A coincides, up to a constant factor, with
n
K (·,w;ν).
n
Now we turn to Theorem 1.2.
Checking(2.11)or(2.12)isnotstraightforward. Seekingamoreexplicitalgebraic
condition, weintroduceanotation fortheminorsofthematrixQ: theone, obtained
by deleting its first row and column,
Q (z ,w) ... Q (z ,w)
1 1 2m 1
∆0(w) = det ... ... ... , (2.13)
Q (z ,w) ... Q (z ,w)
1 2m 2m 2m
and the minor obtained from Q by deleting its first row and its last column,
Q (z ,w) ... Q (z ,w)
0 1 2m−1 1
∆m(w) = det ... ... ... . (2.14)
Q (z ,w) ... Q (z ,w)
0 2m 2m−1 2m
Lemma 2.5. Let A be defined by (2.5) and w ∈ C fixed, and let either one of the
n
following conditions hold:
• (1.5) with |w| ≥ 1;
• (1.7) with |w| ≤ 1.
Then
A (·,w) ≡ 0 ⇔ ∆ (w) = 0.
n 0
On the other hand, if condition (1.6) holds with |w| ≤1, then
A (·,w) ≡ 0 ⇔ ∆ (w) = 0.
n m
Proof. Under assumptions of the first part, observe that
G (z)A (z,w) = detQ(z,w) = ∆(n)(w)K (z,w)+span {Q ,...,Q }.
2m n m n+2m 1 2m
(2.15)
We prove the (⇒) part first, assuming ∆ (w) 6= 0. If (1.5), then the lead-
0
ing coefficient of detQ(·,w) is ∆(n)(w)× leading coefficient of K (·,w) 6= 0, so
m n+2m
A (·,w) 6≡ 0. And if (1.7) takes place, then (see Remark 2.1)
n
detQ(0,w) = ∆ (w)K (0,w) 6= 0,
0 n+2m
so A (·,w) 6≡ 0 again.
n
10
