For a Nevanlinna{Pick problem with more than one solution, Rolf Nevanlinna proved that all extremal solutions are inner functions. If the interpolation points are contained in dinitely many cones terminating at the unit circle, it is shown that all extremal solutions are Blaschke products

Stray, Arne

Stray, A.

Revistes Catalanes amb Accés Obert

Publ. Mat. 59 (2015), 45–54
DOI: 10.5565/PUBLMAT 59115 03
BLASCHKE PRODUCTS AND NEVANLINNA–PICK
INTERPOLATION
Arne Stray
Abstract: For a Nevanlinna–Pick problem with more than one solution, Rolf Nevan-
linna proved that all extremal solutions are inner functions. If the interpolation points
are contained in finitely many cones terminating at the unit circle, it is shown that
all extremal solutions are Blaschke products.
2010 Mathematics Subject Classification: 30E05; Secondary: 30J10.
Key words: Blaschke product, minimal interpolation, logarithmic capacity.
Let U denote the set of all analytic functions f bounded by one in
modulus in the unit disc D = {z : |z| < 1}. We shall assume there
are at least two different solutions to the Nevanlinna–Pick interpolation
problem
(∗) f(zn) = wn, n = 1, 2, . . . , f ∈ U.
If this happens, the problem is called indeterminate. Let E denote the
set of all solutions to (∗). R. Nevanlinna found functions P , Q, R, and S,
analytic in D, such that E is parametrized by U in the following way:
E =
{
P −Qw
R− Sw , w ∈ U
}
.
It is part of Nevanlinna’s discovery that for w ≡ eiα, 0 ≤ α < 2pi,
the corresponding solution Iα =
P−Qeiα
R−Seiα is an inner function. It is a
consequence of Nevanlinna’s formula that the “Wertevorrat”
4(z) = {f(z) : f ∈ U, f solves (∗)}
is a disc and that the boundary points of 4(z) is in one-one correspon-
dence with
{Iα(z), 0 ≤ α < 2pi}.
For this reason the set {Iα, 0 ≤ α < 2pi} are often called the extremal
solutions of (∗). In [5] it was proved that for almost all α, Iα is a
Blaschke product, where the exceptional set of α-values has zero loga-
rithmic capacity. If the interpolation problem (∗) involves only a finite
46 A. Stray
number of points, it is a classical fact that all Iα are Blaschke products.
A similar conclusion was obtained in [4] in the rather special situation
where S = {zn} is an interpolating sequence for the bounded analytic
functions in D and where lim sup |wn| is sufficiently small. The present
work complements the results in [4] and [5] as follows. First we find a
rather precise condition on the sequence S = {zn} giving that for any
indeterminate problem (∗) all Iα are Blaschke products. Next we show
that for any closed set K of zero logarithmic capacity on the unit cir-
cle T , there is a Nevanlinna–Pick problem (∗) such that Iα is a Blaschke
product if and only if eiα ∈ T\K.
Theorem 1. Given a Nevanlinna–Pick interpolation problem f(zn) =
wn, n = 1, 2, . . . , f ∈ U with more than one solution, if S = {zn} is
contained in a finite number of Stolz angles, then any extremal solution
is a Blaschke product.
As mentioned above, for an arbitrary indeterminate Nevanlinna–Pick
problem, almost all extremal solutions are Blaschke products, the excep-
tional set of α-values having zero logarithmic capacity. It seems that for
sequences S = {zn} clustering on a small subset of the unit circle, the
exceptional set of α-values should be described by some finer measure
than logarithmic capacity. As an example we mention:
Theorem 2. Given an indeterminate Nevanlinna–Pick interpolation
problem f(zn) = wn, n = 1, 2, . . . , f ∈ U , if S = {zn} has M limit
points on the unit circle, then at most M of the extremal solutions fail
to be a Blaschke product.
Finally we complement Theorem 1 and Theorem 2 with the following:
Theorem 3. Let K denote a compact subset of the unit circle hav-
ing zero logarithmic capacity. Then there is a Nevanlinna–Pick problem
where Iα is a singular inner function if and only if e
iα ∈ K and an
interpolating Blaschke product for other values of eiα.
It seems that Theorem 3 should be true for more general sets K having
zero logarithmic capacity.
Proof of Theorem 1: Let Γ denote the points in the unit disc inside the
cone given by
{z : −φ < arg(1− z) < φ},
where φ is some fixed angle, 0 < φ < pi2 . Let φ < ψ <
pi
2 and let Γ˜ denote
the cone corresponding to ψ.
Blaschke Products and Nevanlinna–Pick Interpolation 47
We assume the interpolation points zn are contained in Γ and a finite
number of rotated copies of Γ. We also consider Γ˜ and its corresponding
rotations.
Let us assume that some extremal solution J of (∗) contains a singular
inner function
I(z) = exp
(
−t1 + z
1− z
)
as a factor, where t is a positive constant. Using Nevanlinna’s param-
etrization formula, we shall obtain a contradiction.
We fix a positive integer N and let
ΓN = {z : z ∈ Γ, 0 < 1− |z| < 2−N}
and
Γ˜N = {z : z ∈ Γ˜, 0 < 1− |z| < 2−N+1}.
Let Π(z) denote the Blaschke product
Π(z) =
∏ |zn|
zn
zn − z
1− znz
and denote by ΠN the subproduct consisting of the factors with its zero
in ΓN . Consider a point z ∈ D such that the pseudohyperbolic distance
σ(z, zn) =
|z − zn|
|1− znz|
exceeds some positive constant a for all n. Then it is well known (see [3,
p. 279]) that
4
∑ (1− |z|)(1− |zn|)
|1− znz|2 ≤ log |Π(z)|
−2
≤
(
1 + 2 log
1
a
)
4
∑ (1− |z|)(1− |zn|)
|1− znz|2 .
(1)
We apply the second inequality in (1) with Π replaced by ΠN and
z ∈ D located on ∂Γ˜N . The constant a in (1) depend on φ and ψ, but
nothing else.
Assume that Mk is the number of zeros of ΠN in {z : 2−k−1 ≤ |z| ≤
2−k}. Since ∣∣∣∣ 11− znz
∣∣∣∣ = ∣∣∣∣ z − zn1− znz
∣∣∣∣ ∣∣∣∣ 1z − zn
∣∣∣∣
48 A. Stray
and z ∈ ∂Γ˜N we have ∣∣∣∣ 1z − zn
∣∣∣∣ ≤ C1− |z|
with C depending only on φ and ψ. We conclude that
log |ΠN (z)|−2 ≤ C
1− |z|
∞∑
k=N
2−kMk <

1− |z|
for given  if N is large. This gives us the lower bound
|ΠN (z)| ≥ e−

1−|z|
on ∂Γ˜N . For the inner function I, assumed to be a factor in some
extremal solution J of (∗), we have for a given small number λ that∣∣∣∣∣ I
1
2
ΠN
∣∣∣∣∣ < λ
on Γ˜N provided N is large, forcing  to be as small as we wish.
In the following we consider the reduced interpolation problem
(∗)N f(zν) = wν , ν = 1, 2, . . . ,
where zν denotes the zeros of ΠN . Let the solutions of (∗)N be parame-
trized by the functions PN , QN , RN , and SN .
Using the method of dual extremal problems as presented in the proof
of Carleson lemma in [2], we can conclude that for given δ > 0, we can
choose λ so small that (∗)N has a solution satisfying
(2) ||h||∞ < δ
and
(3) |h(z)| ≤ δ|ΠN (z)|
uniformly on ∂Γ˜N . Indeed we find h = gI
1
2 where g interpolates JI
−1
2
at the zeros of ΠN .
To obtain g, we just have to replace the Carleson contour Γ in [2]
by ∂Γ˜N and observe that arc length restricted to ∂Γ˜N is a Carleson
measure.
Let us recall that for the general problem (∗), the functions P , Q,
R, and S admit analytic continuation across the unit circle where {zn}
doesn’t cluster. Hence the same is true for the extremal solutions Iα.
We may also represent these extremal solutions as
Jw =
PN −QNw
RN − SNw ,
Blaschke Products and Nevanlinna–Pick Interpolation 49
where w has to be a solution to the Nevanlinna–Pick problem
(∗∗) PN (zµ)−QN (zµ)w(zµ)
RN (zµ)− SN (zµ)w(zµ) = wµ,
where {zµ} represents the points from {zn} not being a zero of ΠN .
Since (∗) is indeterminate, so is also (∗∗) and evidently Ju is an ex-
tremal solution to (∗) if and only if u is an extremal solution to (∗∗).
From (2) it follows that RN belongs to the Hardy space H
2 (see [6]).
This will be shown to contradict certain lower bounds on |RN (z)| on
certain subsets of D.
Now fix a point z on ∂Γ˜N . The radius ρN (z) of the Wertevorrat
4N (z) corresponding to (∗)N satisfies
(1− δ)|ΠN (z)| ≤ ρN (z) ≤ |ΠN (z)|
since the Wertevorrat contains all values of the form h(z)+ζΠN (z) with
|ζ| < 1− δ.
For z ∈ ∂Γ˜N it follows from (2) and (3) that the Wertevorrat4N (z) is
nearly concentric around the origin. It follows from Schwarz lemma
applied to the mapping
w → PN −QNw
RN − SNw
that
(4)
∣∣∣∣c(z)− J(z)ρN (z)
∣∣∣∣ = σ(u(z), a(z)) ≤ t < 1,
where c(z) is the center of 4N (z) and a(z) = SN (z)RN (z) . Here our extremal
solution J to (∗) is written as
J =
PN −QNu
RN − SNu ,
where u is some extremal solution to (∗∗). Since u is inner and analytic
near z = 1, we have the estimate
(5) |u(z)| ≥ 1− c(1− |z|)
near 1. From (4) and (5) we deduce
(6)
∣∣∣∣ SNRN (z)
∣∣∣∣ ≥ 1− d(1− |z|),
where d depends only on t in (4) and c in (5). We then use our lower
bound
ρN (z) =
|ΠN (z)|
|RN (z)|2 − |SN (z)|2 ≥ (1− δ)|ΠN (z)|
50 A. Stray
to obtain the lower bound
(7) |RN (z)|2 ≥ δ
d(1− |z|)
uniformly on ∂Γ˜N . This is in contradiction with the Hardy–Littlewood
maximal theorem since it is well known that RN belongs to the Hardy
space H2. The proof of Theorem 1 is complete.
Proof of Theorem 2: Given the fact that PS −RQ = Π, we find
Iα − Iβ = ((eiα − eiβ)Π)((R− Seiα)(R− Seiβ))−1.
Since ((R−Seiα)(R−Seiβ))−1 is known to be an outer function and each
Iα is analytic near the unit circle except for the limit points of S, this
means that two different extremal solutions of (∗) cannot have a common
singular factor corresponding to a point mass on the unit circle.
Proof of Theorem 3: Let M be a compact subset of the open unit disc D
with zero logarithmic capacity, and consider a universal covering map
φ : D → D\M.
It is well known that φ is an inner function. We shall assume that
0 ∈M and from this it follows that φ must be a singular inner function.
Let u denote a point in D\M and define
φu =
φ− u
1− uφ.
We claim that φu is a Blaschke product. To see this, consider a
point z0 such that φ(z0) = u. We can assume z0 is an interior point
of some fundamental domain Ω for the covering φ. Since φ is univalent
on Ω, there is a small pseudohyperbolic disc D0 around z0 such that
(8) D0 ⊃ {z : |φ(z)− u| < } ∩ Ω
provided  is sufficiently small. If φu contained a singular inner fac-
tor, there would be a Stolz angle Γ of opening close to pi, such that
φ(z) → u uniformly inside Γ as |z| → 1 inside Γ. But then φ ≈ u in
large pseudohyperbolic discs inside Γ and since φ is automorphic, this
contradicts (8).
Let S= {zν} = φ−1{u}. Consider the Nevanlinna–Pick problem
(∗ ∗ ∗) f(zν) = u, ν = 1, 2, . . . , f ∈ U.
It is now evident that any solution f to (∗ ∗ ∗) will be of the form
u+ wφu
1 + uwφu
, where w is an arbitrary analytic function bounded by one
in modulus in D. We also see that the extremal solutions Iα to (∗ ∗ ∗)
Blaschke Products and Nevanlinna–Pick Interpolation 51
corresponds to w = eiα, 0 ≤ α < 2pi. In particular φ = I0. For these
extremal solutions it is evident that Iα is a singular inner function if and
only if
φ 6= u(1− e
iα)
1− |u|2eiα
in the unit disc. Let |ζ0|=1. Consider the mapping
Λ: ζ → u(1− ζ0ζ)
1− |u|2ζζ0
taking the unit circle onto the circle centered at u1+|u|2 and containing the
origin. If K is a compact subset of {ζ : |ζ| = 1} containing the point ζ0
and of zero logarithmic capacity, we shall assume in the following that
M = Λ(K). It follows that Iα is a singular inner function whenever
α ∈ K. If α ∈ T\K, it follows that Iα is a Blaschke product. This
follows in the same way as we showed the φu is a Blaschke product
whenever u ∈ D\M . To complete the proof, we need only show that the
zeros of φ−b
1−bφ form an interpolating sequence if b ∈ D\M . The map φ
is automorphic with respect to a group of cover transformations T of D
and the zeros is just the complete orbit of one fixed zero. If B is one of
the Blaschke products under consideration, we have to verify Carleson’s
product condition at the zeros of B. This condition can be written as
|B′(z)|(1− |z|2) ≥ δ > 0,
where δ doesn’t depend on the zero considered. Since φ is automorphic
one finds that
|B′(z)|(1− |z|2) = |B′(Tz)|(1− |Tz|2)
for z ∈ D and any cover transformation T , and so Carleson’s condition
follows.
This completes the proof of Theorem 3.
There is a strong connection between the problem considered here
and the work by J. R. Akeroyd, D. Khavinson, and H. S. Shapiro in [1].
We say that a sequence in D converges onesidedly to a point on the
unit circle if it converges and stays on one side of a line through the
limitpoint dividing the disc into two parts. The result from [1] we have
in mind is the following: Let B be a Blaschke product whose zero set
is a finite union of sequences converging onesidedly. The ∗-invariant
subspace H2 	BH2 of H2 contains no singular inner functions.
To relate this result to our work, we shall call a Nevanlinna–Pick
problem scaled if it can be solved by a function in H∞ of norm strictly
52 A. Stray
less than one. The problem is called semiscaled if after excluding finitely
many interpolation points, the reduced problem is scaled.
We then have
Theorem 4. Let S = {zn} be a finite union of Blaschke sequences Si,
1 ≤ i ≤ m converging onesidedly to m distinct points ζi on the unit
circle. Suppose the reduced Nevanlinna–Pick problem corresponding to
some Si is semiscaled. If the full problem has more than one solution,
none of the extremal solutions can have a singular inner factor with a
singularity at ζi.
Let us first remark that Theorem 4 follows from the above cited result
from [1] in the special case where the interpolation problem is scaled.
The reason is that, for scaled problems, P − Qζ ∈ H2 	 zBH2 for any
constant ζ. Indeed, integrating over the unit circle gives∫
(P −Qζ)zBhdθ =
∫
(Rζ − S)zh dθ = 0
for any h ∈ H2. The formulas
Q = −BR, P = −BS
valid on the unit circle T , dates back to Nevanlinna’s original paper on
the subject. These formulas also give that P , Q, R, and S all belong to
H2 	 zBH2 in the scaled case. See [3] for details.
Let us prove Theorem 4: We assume ζi = 1 and consider the reduced
scaled interpolation problem corresponding to Si minus a finite number
of points from Si. The set of all solutions to the reduced problem is
parametrized by
Pr −Qrw
Rr − Srw .
Let Π denote the Blaschke product corresponding to the reduced prob-
lem. We assume that the zeros {zn} of Π are located above some non-
vertical line trough 1. Consider the half circle C− in the lower half plane
defined by
1− |z|2 = |1− z|2.
Using equation (1) above it follows easily that for z ∈ C−
lim
z→1
|Π(z)| = 1.
Using Theorem 1 in [6], it follows that the radius ρr(z) of the corre-
sponding Wertevorrat also tends to 1 along C−. Hence the center cz of
Blaschke Products and Nevanlinna–Pick Interpolation 53
the Wertevorrat tends to 0 at the same time. Any extremal solution to
the full problem in Theorem 4 will have the form
J =
Pr −QrI
Rr − SrI ,
where I is an inner function analytic near z = 1. If J contains a singular
inner factor exp(−t 1+z1−z ) we conclude that the pseudohyperbolic distance
σ
(
I(z),
(
Sr(z)
Rr(z)
))
=
|cz − J(z)|
ρr(z)
is strictly less than one as z → 1 along C−. But then we arrive at a
contradiction like in the proof of Theorem 1, since the H2-function Rr
must satisfy
|Rr(z)|2 ≥ d(1− |z|)−1
uniformly on C− near z = 1.
Final remarks. It is not clear if Theorem 4 holds without assuming there
is a semiscaled reduced problem. A solution seems to depend on more
detailed information about the subharmonic function (|R|2 − |S|2)−1.
Let us also mention two consequenses of Theorem 1. It follows im-
mediately that the extremal Blaschke products in that theorem are all
indestructible. Another consequence is that the function
I(z) = exp
(
−1 + z
1− z
)
can never be a minimal solution to any Nevanlinna–Pick problem in a
Stoltz angle terminating at z = 1.
References
[1] J. R. Akeroyd, D. Khavinson, and H. S. Shapiro, Weak com-
pactness in certain star-shift invariant subspaces, J. Funct. Anal.
202(1) (2003), 98–122. DOI: 10.1016/S0022-1236(03)00060-0.
[2] P. L. Duren, “Theory of Hp spaces”, Pure and Applied Mathemat-
ics 38, Academic Press, New York-London, 1970.
[3] J. B. Garnett, “Bounded analytic functions”, Revised first edition,
Graduate Texts in Mathematics 236, Springer, New York, 2007.
[4] A. Stray, Minimal interpolation by Blaschke products, J. London
Math. Soc. (2) 32(3) (1985), 488–496. DOI: 10.1112/jlms/s2-32.
3.488.
54 A. Stray
[5] A. Stray, Minimal interpolation by Blaschke products. II, Bull.
London Math. Soc. 20(4) (1988), 329–332. DOI: 10.1112/blms/20.
4.329.
[6] A. Stray, Interpolating sequences and the Nevanlinna Pick prob-
lem, Publ. Mat. 35(2) (1991), 507–516. DOI: 10.5565/PUBLMAT−
35291−14.
Department of Mathematics
University of Bergen
5020 Bergen
Norway
E-mail address: Arne.Stray@math.uib.no
Primera versio´ rebuda el 17 de setembre de 2013,
darrera versio´ rebuda el 10 de desembre de 2013.


Blaschke products and Nevanlinna-Pick interpolation

Diposit Digital de Documents de la UAB

Publ. Mat. 59 (2015), 45–54DOI: 10.5565/PUBLMAT 59115 03BLASCHKE PRODUCTS AND NEVANLINNA–PICKINTERPOLATIONArne StrayAbstract: For a Nevanlinna–Pick problem with more than one solution, Rolf Nevan-linna proved that all extremal solutions are inner functions. If the interpolation pointsare contained in finitely many cones terminating at the unit circle, it is shown thatall extremal solutions are Blaschke products.2010 Mathematics Subject Classification: 30E05; Secondary: 30J10.Key words: Blaschke product, minimal interpolation, logarithmic capacity.Let U denote the set of all analytic functions f bounded by one inmodulus in the unit disc D = {z : |z| < 1}. We shall assume thereare at least two different solutions to the Nevanlinna–Pick interpolationproblem(∗) f(zn) = wn, n = 1, 2, . . . , f ∈ U.If this happens, the problem is called indeterminate. Let E denote theset of all solutions to (∗). R. Nevanlinna found functions P , Q, R, and S,analytic in D, such that E is parametrized by U in the following way:E ={P −QwR− Sw , w ∈ U}.It is part of Nevanlinna’s discovery that for w ≡ eiα, 0 ≤ α < 2pi,the corresponding solution Iα =P−QeiαR−Seiα is an inner function. It is aconsequence of Nevanlinna’s formula that the “Wertevorrat”4(z) = {f(z) : f ∈ U, f solves (∗)}is a disc and that the boundary points of 4(z) is in one-one correspon-dence with{Iα(z), 0 ≤ α < 2pi}.For this reason the set {Iα, 0 ≤ α < 2pi} are often called the extremalsolutions of (∗). In [5] it was proved that for almost all α, Iα is aBlaschke product, where the exceptional set of α-values has zero loga-rithmic capacity. If the interpolation problem (∗) involves only a finite46 A. Straynumber of points, it is a classical fact that all Iα are Blaschke products.A similar conclusion was obtained in [4] in the rather special situationwhere S = {zn} is an interpolating sequence for the bounded analyticfunctions in D and where lim sup |wn| is sufficiently small. The presentwork complements the results in [4] and [5] as follows. First we find arather precise condition on the sequence S = {zn} giving that for anyindeterminate problem (∗) all Iα are Blaschke products. Next we showthat for any closed set K of zero logarithmic capacity on the unit cir-cle T , there is a Nevanlinna–Pick problem (∗) such that Iα is a Blaschkeproduct if and only if eiα ∈ T\K.Theorem 1. Given a Nevanlinna–Pick interpolation problem f(zn) =wn, n = 1, 2, . . . , f ∈ U with more than one solution, if S = {zn} iscontained in a finite number of Stolz angles, then any extremal solutionis a Blaschke product.As mentioned above, for an arbitrary indeterminate Nevanlinna–Pickproblem, almost all extremal solutions are Blaschke products, the excep-tional set of α-values having zero logarithmic capacity. It seems that forsequences S = {zn} clustering on a small subset of the unit circle, theexceptional set of α-values should be described by some finer measurethan logarithmic capacity. As an example we mention:Theorem 2. Given an indeterminate Nevanlinna–Pick interpolationproblem f(zn) = wn, n = 1, 2, . . . , f ∈ U , if S = {zn} has M limitpoints on the unit circle, then at most M of the extremal solutions failto be a Blaschke product.Finally we complement Theorem 1 and Theorem 2 with the following:Theorem 3. Let K denote a compact subset of the unit circle hav-ing zero logarithmic capacity. Then there is a Nevanlinna–Pick problemwhere Iα is a singular inner function if and only if eiα ∈ K and aninterpolating Blaschke product for other values of eiα.It seems that Theorem 3 should be true for more general sets K havingzero logarithmic capacity.Proof of Theorem 1: Let Γ denote the points in the unit disc inside thecone given by{z : −φ < arg(1− z) < φ},where φ is some fixed angle, 0 < φ < pi2 . Let φ < ψ <pi2 and let Γ˜ denotethe cone corresponding to ψ.Blaschke Products and Nevanlinna–Pick Interpolation 47We assume the interpolation points zn are contained in Γ and a finitenumber of rotated copies of Γ. We also consider Γ˜ and its correspondingrotations.Let us assume that some extremal solution J of (∗) contains a singularinner functionI(z) = exp(−t1 + z1− z)as a factor, where t is a positive constant. Using Nevanlinna’s param-etrization formula, we shall obtain a contradiction.We fix a positive integer N and letΓN = {z : z ∈ Γ, 0 < 1− |z| < 2−N}andΓ˜N = {z : z ∈ Γ˜, 0 < 1− |z| < 2−N+1}.Let Π(z) denote the Blaschke productΠ(z) =∏ |zn|znzn − z1− znzand denote by ΠN the subproduct consisting of the factors with its zeroin ΓN . Consider a point z ∈ D such that the pseudohyperbolic distanceσ(z, zn) =|z − zn||1− znz|exceeds some positive constant a for all n. Then it is well known (see [3,p. 279]) that4∑ (1− |z|)(1− |zn|)|1− znz|2 ≤ log |Π(z)|−2≤(1 + 2 log1a)4∑ (1− |z|)(1− |zn|)|1− znz|2 .(1)We apply the second inequality in (1) with Π replaced by ΠN andz ∈ D located on ∂Γ˜N . The constant a in (1) depend on φ and ψ, butnothing else.Assume that Mk is the number of zeros of ΠN in {z : 2−k−1 ≤ |z| ≤2−k}. Since ∣∣∣∣ 11− znz∣∣∣∣ = ∣∣∣∣ z − zn1− znz∣∣∣∣ ∣∣∣∣ 1z − zn∣∣∣∣48 A. Strayand z ∈ ∂Γ˜N we have ∣∣∣∣ 1z − zn∣∣∣∣ ≤ C1− |z|with C depending only on φ and ψ. We conclude thatlog |ΠN (z)|−2 ≤ C1− |z|∞∑k=N2−kMk <1− |z|for given  if N is large. This gives us the lower bound|ΠN (z)| ≥ e−1−|z|on ∂Γ˜N . For the inner function I, assumed to be a factor in someextremal solution J of (∗), we have for a given small number λ that∣∣∣∣∣ I12ΠN∣∣∣∣∣ < λon Γ˜N provided N is large, forcing  to be as small as we wish.In the following we consider the reduced interpolation problem(∗)N f(zν) = wν , ν = 1, 2, . . . ,where zν denotes the zeros of ΠN . Let the solutions of (∗)N be parame-trized by the functions PN , QN , RN , and SN .Using the method of dual extremal problems as presented in the proofof Carleson lemma in [2], we can conclude that for given δ > 0, we canchoose λ so small that (∗)N has a solution satisfying(2) ||h||∞ < δand(3) |h(z)| ≤ δ|ΠN (z)|uniformly on ∂Γ˜N . Indeed we find h = gI12 where g interpolates JI−12at the zeros of ΠN .To obtain g, we just have to replace the Carleson contour Γ in [2]by ∂Γ˜N and observe that arc length restricted to ∂Γ˜N is a Carlesonmeasure.Let us recall that for the general problem (∗), the functions P , Q,R, and S admit analytic continuation across the unit circle where {zn}doesn’t cluster. Hence the same is true for the extremal solutions Iα.We may also represent these extremal solutions asJw =PN −QNwRN − SNw ,Blaschke Products and Nevanlinna–Pick Interpolation 49where w has to be a solution to the Nevanlinna–Pick problem(∗∗) PN (zµ)−QN (zµ)w(zµ)RN (zµ)− SN (zµ)w(zµ) = wµ,where {zµ} represents the points from {zn} not being a zero of ΠN .Since (∗) is indeterminate, so is also (∗∗) and evidently Ju is an ex-tremal solution to (∗) if and only if u is an extremal solution to (∗∗).From (2) it follows that RN belongs to the Hardy space H2 (see [6]).This will be shown to contradict certain lower bounds on |RN (z)| oncertain subsets of D.Now fix a point z on ∂Γ˜N . The radius ρN (z) of the Wertevorrat4N (z) corresponding to (∗)N satisfies(1− δ)|ΠN (z)| ≤ ρN (z) ≤ |ΠN (z)|since the Wertevorrat contains all values of the form h(z)+ζΠN (z) with|ζ| < 1− δ.For z ∈ ∂Γ˜N it follows from (2) and (3) that the Wertevorrat4N (z) isnearly concentric around the origin. It follows from Schwarz lemmaapplied to the mappingw → PN −QNwRN − SNwthat(4)∣∣∣∣c(z)− J(z)ρN (z)∣∣∣∣ = σ(u(z), a(z)) ≤ t < 1,where c(z) is the center of 4N (z) and a(z) = SN (z)RN (z) . Here our extremalsolution J to (∗) is written asJ =PN −QNuRN − SNu ,where u is some extremal solution to (∗∗). Since u is inner and analyticnear z = 1, we have the estimate(5) |u(z)| ≥ 1− c(1− |z|)near 1. From (4) and (5) we deduce(6)∣∣∣∣ SNRN (z)∣∣∣∣ ≥ 1− d(1− |z|),where d depends only on t in (4) and c in (5). We then use our lowerboundρN (z) =|ΠN (z)||RN (z)|2 − |SN (z)|2 ≥ (1− δ)|ΠN (z)|50 A. Strayto obtain the lower bound(7) |RN (z)|2 ≥ δd(1− |z|)uniformly on ∂Γ˜N . This is in contradiction with the Hardy–Littlewoodmaximal theorem since it is well known that RN belongs to the Hardyspace H2. The proof of Theorem 1 is complete.Proof of Theorem 2: Given the fact that PS −RQ = Π, we findIα − Iβ = ((eiα − eiβ)Π)((R− Seiα)(R− Seiβ))−1.Since ((R−Seiα)(R−Seiβ))−1 is known to be an outer function and eachIα is analytic near the unit circle except for the limit points of S, thismeans that two different extremal solutions of (∗) cannot have a commonsingular factor corresponding to a point mass on the unit circle.Proof of Theorem 3: Let M be a compact subset of the open unit disc Dwith zero logarithmic capacity, and consider a universal covering mapφ : D → D\M.It is well known that φ is an inner function. We shall assume that0 ∈M and from this it follows that φ must be a singular inner function.Let u denote a point in D\M and defineφu =φ− u1− uφ.We claim that φu is a Blaschke product. To see this, consider apoint z0 such that φ(z0) = u. We can assume z0 is an interior pointof some fundamental domain Ω for the covering φ. Since φ is univalenton Ω, there is a small pseudohyperbolic disc D0 around z0 such that(8) D0 ⊃ {z : |φ(z)− u| < } ∩ Ωprovided  is sufficiently small. If φu contained a singular inner fac-tor, there would be a Stolz angle Γ of opening close to pi, such thatφ(z) → u uniformly inside Γ as |z| → 1 inside Γ. But then φ ≈ u inlarge pseudohyperbolic discs inside Γ and since φ is automorphic, thiscontradicts (8).Let S= {zν} = φ−1{u}. Consider the Nevanlinna–Pick problem(∗ ∗ ∗) f(zν) = u, ν = 1, 2, . . . , f ∈ U.It is now evident that any solution f to (∗ ∗ ∗) will be of the formu+ wφu1 + uwφu, where w is an arbitrary analytic function bounded by onein modulus in D. We also see that the extremal solutions Iα to (∗ ∗ ∗)Blaschke Products and Nevanlinna–Pick Interpolation 51corresponds to w = eiα, 0 ≤ α < 2pi. In particular φ = I0. For theseextremal solutions it is evident that Iα is a singular inner function if andonly ifφ 6= u(1− eiα)1− |u|2eiαin the unit disc. Let |ζ0|=1. Consider the mappingΛ: ζ → u(1− ζ0ζ)1− |u|2ζζ0taking the unit circle onto the circle centered at u1+|u|2 and containing theorigin. If K is a compact subset of {ζ : |ζ| = 1} containing the point ζ0and of zero logarithmic capacity, we shall assume in the following thatM = Λ(K). It follows that Iα is a singular inner function wheneverα ∈ K. If α ∈ T\K, it follows that Iα is a Blaschke product. Thisfollows in the same way as we showed the φu is a Blaschke productwhenever u ∈ D\M . To complete the proof, we need only show that thezeros of φ−b1−bφ form an interpolating sequence if b ∈ D\M . The map φis automorphic with respect to a group of cover transformations T of Dand the zeros is just the complete orbit of one fixed zero. If B is one ofthe Blaschke products under consideration, we have to verify Carleson’sproduct condition at the zeros of B. This condition can be written as|B′(z)|(1− |z|2) ≥ δ > 0,where δ doesn’t depend on the zero considered. Since φ is automorphicone finds that|B′(z)|(1− |z|2) = |B′(Tz)|(1− |Tz|2)for z ∈ D and any cover transformation T , and so Carleson’s conditionfollows.This completes the proof of Theorem 3.There is a strong connection between the problem considered hereand the work by J. R. Akeroyd, D. Khavinson, and H. S. Shapiro in [1].We say that a sequence in D converges onesidedly to a point on theunit circle if it converges and stays on one side of a line through thelimitpoint dividing the disc into two parts. The result from [1] we havein mind is the following: Let B be a Blaschke product whose zero setis a finite union of sequences converging onesidedly. The ∗-invariantsubspace H2 	BH2 of H2 contains no singular inner functions.To relate this result to our work, we shall call a Nevanlinna–Pickproblem scaled if it can be solved by a function in H∞ of norm strictly52 A. Strayless than one. The problem is called semiscaled if after excluding finitelymany interpolation points, the reduced problem is scaled.We then haveTheorem 4. Let S = {zn} be a finite union of Blaschke sequences Si,1 ≤ i ≤ m converging onesidedly to m distinct points ζi on the unitcircle. Suppose the reduced Nevanlinna–Pick problem corresponding tosome Si is semiscaled. If the full problem has more than one solution,none of the extremal solutions can have a singular inner factor with asingularity at ζi.Let us first remark that Theorem 4 follows from the above cited resultfrom [1] in the special case where the interpolation problem is scaled.The reason is that, for scaled problems, P − Qζ ∈ H2 	 zBH2 for anyconstant ζ. Indeed, integrating over the unit circle gives∫(P −Qζ)zBhdθ =∫(Rζ − S)zh dθ = 0for any h ∈ H2. The formulasQ = −BR, P = −BSvalid on the unit circle T , dates back to Nevanlinna’s original paper onthe subject. These formulas also give that P , Q, R, and S all belong toH2 	 zBH2 in the scaled case. See [3] for details.Let us prove Theorem 4: We assume ζi = 1 and consider the reducedscaled interpolation problem corresponding to Si minus a finite numberof points from Si. The set of all solutions to the reduced problem isparametrized byPr −QrwRr − Srw .Let Π denote the Blaschke product corresponding to the reduced prob-lem. We assume that the zeros {zn} of Π are located above some non-vertical line trough 1. Consider the half circle C− in the lower half planedefined by1− |z|2 = |1− z|2.Using equation (1) above it follows easily that for z ∈ C−limz→1|Π(z)| = 1.Using Theorem 1 in [6], it follows that the radius ρr(z) of the corre-sponding Wertevorrat also tends to 1 along C−. Hence the center cz ofBlaschke Products and Nevanlinna–Pick Interpolation 53the Wertevorrat tends to 0 at the same time. Any extremal solution tothe full problem in Theorem 4 will have the formJ =Pr −QrIRr − SrI ,where I is an inner function analytic near z = 1. If J contains a singularinner factor exp(−t 1+z1−z ) we conclude that the pseudohyperbolic distanceσ(I(z),(Sr(z)Rr(z)))=|cz − J(z)|ρr(z)is strictly less than one as z → 1 along C−. But then we arrive at acontradiction like in the proof of Theorem 1, since the H2-function Rrmust satisfy|Rr(z)|2 ≥ d(1− |z|)−1uniformly on C− near z = 1.Final remarks. It is not clear if Theorem 4 holds without assuming thereis a semiscaled reduced problem. A solution seems to depend on moredetailed information about the subharmonic function (|R|2 − |S|2)−1.Let us also mention two consequenses of Theorem 1. It follows im-mediately that the extremal Blaschke products in that theorem are allindestructible. Another consequence is that the functionI(z) = exp(−1 + z1− z)can never be a minimal solution to any Nevanlinna–Pick problem in aStoltz angle terminating at z = 1.References[1] J. R. Akeroyd, D. Khavinson, and H. S. Shapiro, Weak com-pactness in certain star-shift invariant subspaces, J. Funct. Anal.202(1) (2003), 98–122. DOI: 10.1016/S0022-1236(03)00060-0.[2] P. L. Duren, “Theory of Hp spaces”, Pure and Applied Mathemat-ics 38, Academic Press, New York-London, 1970.[3] J. B. Garnett, “Bounded analytic functions”, Revised first edition,Graduate Texts in Mathematics 236, Springer, New York, 2007.[4] A. Stray, Minimal interpolation by Blaschke products, J. LondonMath. Soc. (2) 32(3) (1985), 488–496. DOI: 10.1112/jlms/s2-32.3.488.54 A. Stray[5] A. Stray, Minimal interpolation by Blaschke products. II, Bull.London Math. Soc. 20(4) (1988), 329–332. DOI: 10.1112/blms/20.4.329.[6] A. Stray, Interpolating sequences and the Nevanlinna Pick prob-lem, Publ. Mat. 35(2) (1991), 507–516. DOI: 10.5565/PUBLMAT−35291−14.Department of MathematicsUniversity of Bergen5020 BergenNorwayE-mail address: Arne.Stray@math.uib.noPrimera versio´ rebuda el 17 de setembre de 2013,darrera versio´ rebuda el 10 de desembre de 2013.

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Blaschke products and Nevanlinna-Pick interpolation

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