Approximation in the closed unit ball

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by  Javad Mashreghi

Approximation in the closed unit ball
Javad Mashreghi⋆ and Thomas Ransford⋆⋆
D´epartement demath´ematiques et destatistique, Universit´e Laval,
7
1045 avenuedela M´edecine, Qu´ebec (Qu´ebec),Canada, G1V 0A6
1
[email protected]
0
2 [email protected]

n
a
J Abstract. In this expository article, we present a number of classic

9 theorems that serve to identify the closure in the sup-norm of various
1 sets of Blaschke products, inner functions and their quotients, as well
as the closure of the convex hulls of these sets. The results presented
]
V include theorems of Carath´eodory, Fisher, Helson–Sarason, Frostman,
Adamjan–Arov–Krein, Douglas–Rudin and Marshall. As an application
C
of some of these ideas, we obtain a simple proof of the Berger–Stampfli
h. spectral mapping theorem for thenumerical range of an operator.
t
a Keywords: unit ball, Blaschke product,inner function, convex hull
m
[

1 Introduction
1
v
8 Let X be a Banach space and let E be a subset of X. The convex hull of E,
3 denotedbyconv(E),isthesetofallelementsoftheformλ x +λ x + +λ x ,
1 1 2 2 n n
···
5 where x E and where 0 λ 1 with λ +λ + +λ = 1. The closed
j j 1 2 n
5 ∈ ≤ ≤ ···
unit ball of X is
0
B = x X : x 1 .
1. X { ∈ k k≤ }
0 In this survey, we consider some Banach spaces of functions on the open unit
7 disc D or on the unit circle T, e.g., L∞(T), (T), H∞ and the disc algebra ,
1 and explore the norm closure of some subsetsCof B and of their convex hullsA.
X
:
v The unimodular elements of the above function spaces enter naturally into
i our discussion. The unimodular elements of H∞, denoted by I, are a celebrated
X
family that are called inner functions. For other function spaces we use the
r
a notation UX to denote the family of unimodular elements of X, e.g.,
UC(T) := f (T): f(ζ) =1 for all ζ T .
{ ∈C | | ∈ }
TheprototypecandidatesforEarethesetoffiniteBlaschkeproducts(FBP),
the set of all Blaschke products (BP), the inner functions (I), the measurable
unimodularfunctions,aswellasthequotientsofpairsoffunctionsinthesefami-
lies.Wenowsummarizethemainresults.Theformalstatementsandattributions
will be detailed in the sections that follow.
⋆ Supported bya grant from NSERC
⋆⋆ Supported bygrants from NSERCand theCanada research chairs program.
2 Javad Mashreghi and Thomas Ransford
Finite Blaschke products are elements of the disc algebra . In particular,
whenconsideredasfunctionsonT,theyareelements of (T).IAnthisregard,for
C
these elements and their quotients, we shall see that:
FBP=FBP
conv(FBP)=B
A
FBP/FBP=UC(T)
conv(FBP/FBP)=BC(T).
Infinite Blaschke products are elements of the Hardy space H∞. In particu-
lar, when considered as functions on T, they are elements of L∞(T). For these
functions and their quotients, we shall see that:
BP=I=I
conv(BP)=conv(I)=B
H∞
BP/BP=I/I=UL∞(T)
conv(BP/BP)=conv(I/I)=BL∞(T).
Inallthe resultsabove,we considerthe normtopology.For the Hardy space
H∞, and thus a priori for the disc algebra , there is a weaker topology which
A
is obtained via semi-norms
p (f):=max f(z).
r
|z|≤r| |
This is referred as the topology of uniform convergence on compact subsets
(UCC) ofD.Naivelyspeaking,itis easierto convergeunderthe lattertopology.
Therefore,insomecaseswewillalsostudytheUCC-closureofasetoritsconvex
hull. We shall see that:
UCC
FBP =B .
H∞
Sinceontheonehand,FBPisthesmallestapproximatingsetinourdiscussion,
andonthe otherhandB isthe largestpossiblesetthatwecanapproximate,
H∞
this last result closes the door on any further investigation regarding the UCC-
closure.
2 Approximation on D by finite Blaschke products
Goal: FBP=FBP
Letf H∞,andsupposethatthereisasequenceoffiniteBlaschkeproducts
that conv∈erges uniformly on D to f. Then, by continuity, we also have uniform
convergence on D. Therefore f is necessarily a continuous function on D, and
moreover it is a unimodular function on T. It is an easy exercise to show that
this function is necessarily a finite Blaschke product. A slightly more general
version of this result is stated below.
Approximation in theclosed unit ball 3
Lemma 2.1 (Fatou [7]) Let f be holomorphic in the open unit disc D and
suppose that
lim f(z) =1.
|z|→1| |
Then f is a finite Blaschke product.
Proof. Since f isholomorphiconDand f tendsuniformly to1asweapproach
T, it has a finite number of zeros in D.|L|et B be the finite Blaschke product
formedwith the zerosoff.Then f/B andB/f areboth holomorphicin D,and
their moduli uniformly tend to 1 as we approach T. Hence, by the maximum
principle, f/B 1 and B/f 1 on D. Thus f/B is constant on D, and the
| | ≤ | | ≤
constant has to be unimodular.
⊓⊔
Lemma 2.1 immediately implies the following result.
Theorem 2.2 ThesetFBPoffiniteBlaschke productsis aclosedsubsetof B
A
(and hence also a closed subset of B ).
H∞
The following result is another simple consequence of Lemma 2.1. It will be
needed in later approximationresults in this article (see Theorem 5.2).
Corollary 2.3 Letf bemeromorphic intheopen unitdiscDandcontinuouson
the closed unit disc D (as a function into the Riemann sphere). Suppose that f
is unimodular on the unit circle T. Then f is the quotient of two finite Blaschke
products.
Proof. Since f is unimodular on T, meromorphic in D and continuous on D, it
has a finite number of poles in D. Let B be the finite Blaschke product with
2
zeros at the poles of f. Put B := B f. Then B satisfies the hypotheses of
1 2 1
Lemma 2.1, and so it is a finite Blaschke product. Thus f =B /B .
1 2
⊓⊔
3 Approximation on compact sets by finite Blaschke
products
UCC
Goal: FBP =B
H∞
If f is holomorphic on D and can be uniformly approximated on D by a
sequence of finite Blaschke products, we saw that, by Lemma 2.1, f is itself a
finite Blaschke product. A general element of B is far from being a finite
H∞
Blaschke product and cannot be approached uniformly on D by finite Blaschke
products. Nevertheless, a weaker type of convergence does hold. The following
result says that, if we equip H∞ with the topology of uniform convergence on
compact subsets of D, then the family of finite Blaschke products form a dense
subsetofB .Inacertainsense,thistheoremcircumscribesalltheotherresults
H∞
in this article.
4 Javad Mashreghi and Thomas Ransford
Theorem 3.1 (Carath´eodory) Let f B . Then there is a sequence of
H∞
∈
finite Blaschke products that converges uniformly to f on each compact subset
of D.
Proof. (This proof is taken from [10, page 5].) We construct a finite Blaschke
product B such that the first n+1 Taylor coefficients of f and B are equal.
n n
Then, by Schwarz’s lemma, we have
f(z) B (z) 2z n, (z D),
n
| − |≤ | | ∈
and thus the sequence (B ) converges uniformly to f on compact subsets of D.
n
Let c := f(0). As f lies in the unit ball, c D. If c = 1, then, by the
0 0 0
∈ | |
maximum principle, f is a unimodular constant, and the result is obvious. So
let us assume that c <1. Writing
0
| |
a z
τ (z):= − (a,z D),
a
1 az ∈
−
let us set
z+c
B (z):= τ (z)= 0 , (z D). (1)
0 − −c0 1+c z ∈
0
Clearly, B is a finite Blaschke product and its constant term is c .
0 0
The rest is by induction. Suppose that we can construct B for each ele-
n−1
ment of B . Set
H∞
τ (f(z))
g(z):= c0 , (z D). (2)
z ∈
By Schwarz’s lemma, g B . Hence, there is a finite Blaschke product B
H∞ n−1
∈
such that g B has a zeroof of orderat leastn at the origin.If B is a finite
n−1
Blaschke pr−oduct of degree n, and w D, then it is easy to verify directly that
∈
τ B and B τ are also finite Blaschke products of order n. Hence
w w
◦ ◦
B (z):=τ (zB (z)), (z D), (3)
n c0 n−1 ∈
is a finite Blaschke product. Since
f(z)=τ (zg(z)), (z D),
c0 ∈
we naturally expect that B does that job. To establish this conjecture, it is
n
enough to observe that
(1 c 2)(z z )
0 1 2
τ (z ) τ (z )= −| | − .
c0 2 − c0 1 (1 c z )(1 c z )
0 1 0 2
− −
Hence, thanks to the presence of the factor z(g(z) B (z)), the difference
n−1
−
f(z) B (z)=τ (zg(z)) τ (zB (z))
− n c0 − c0 n−1
is divisible by zn+1.
⊓⊔
Approximation in theclosed unit ball 5
Remark.Theequation(3)isperhapsabitmisleading,asifwehavearecursive
formula for the sequence (B ) . A safer way is to write the formula as
n n≥0
B (z):=τ (zB (z)), (n 1),
n,f c0 n−1,g ≥
whereg isrelatedtof via(2).LetuscomputeanexamplebyfindingB :=B
1 1,f
We know that
B (z)=τ (zB (z)).
1,f c0 0,g
Write f(z)=c +c z+ and observe that
0 1
···
τ (f(z)) c
g(z)= c0 = − 1 +O(z).
z 1 c 2
0
−| |
Then, by (1), we have
B0,g(z)= τ c1 (z),
− 1−|c0|2
and so we get
B1(z)=τc0(−zτ1−c|c10|2(z)).
One may directly verify that
B (z)=c +c z+O(z2),
1 0 1
as required.
4 Approximation on D by convex combinations of finite
Blaschke products
Goal: conv(FBP)=B
A
AswesawinSection2,ifafunctionf H∞ canbeuniformlyapproximated
by a sequence of finite Blaschke product∈s on D, then f is continuous on D.
The same result holds if we can approximate f by elements that are convex
combinations of finite Blaschke products. The only difference is that, in this
case,f isnotnecessarilyunimodularonT.Wecanjustsaythat f 1.More
∞
k k ≤
explicitly, the uniform limit of convex combinations of finite Blaschke products
is a continuous function in the closed unit ball of H∞. It is rather surprising
that the converse is also true.
Theorem 4.1 (Fisher [8]) Let f B , and let ε > 0. Then there are finite
A
∈
Blaschke products B and convex weights (λ ) such that
j j 1≤j≤n
λ B +λ B + +λ B f <ε.
1 1 2 2 n n ∞
k ··· − k
6 Javad Mashreghi and Thomas Ransford
Proof. For 0 t 1, let f (z) := f(tz), z D. Since f is continuous on D, we
t
≤ ≤ ∈
have
lim f f =0. (4)
t ∞
t→1k − k
ByTheorem3.1,thereisasequenceoffiniteBlaschkeproductsthatconverges
uniformlytof oncompactsubsetsofD.Basedonournotation,thismeansthat,
given ε>0 and t<1, there is a finite Blaschke product B such that
f B <ε/2.
t t ∞
k − k
Therefore, by (4), there is a finite Blaschke product B such that
f B <ε.
t ∞
k − k
If we can show that B itself is actually a convex combinationof finite Blaschke
t
products, the proof is done.
Firstly, note that (gh) =g h for all g and h, and that the family of convex
t t t
combinations of finite Blaschke products is closed under multiplication. Hence
it is enough only to consider a Blaschke factor
α z
B(z)= − .
1 αz
−
Secondly, it is easy to verify that
α tz t(1 α2) αt z α(1 t2)
B (z)= − = −| | − + | | − eiargα. (5)
t 1 αtz 1 α2t2 × 1 αtz 1 α2t2 ×
− −| | − −| |
The combination on the right side is almost good. More precisely, it is a combi-
nationof a Blaschkefactor anda unimodular constant(a special case ofa finite
Blaschke product), with positive coefficients, but the coefficients do not add up
to one. Indeed, we have
t(1 α2) α(1 t2) (1 t)(1 α)
1 −| | | | − = − −| | .
− 1 α2t2 − 1 α2t2 1+ αt
−| | −| | | |
But this obstacle is easy to overcome. We can simply add
(1 t)(1 α) (1 t)(1 α)
0= − −| | 1+ − −| | ( 1)
2(1+ αt) × 2(1+ αt) × −
| | | |
to both sides of (5) to obtain a convex combination of finite Blaschke products.
Of course, the factor 1 in the last identity can be replaced by any other finite
Blaschke product.
⊓⊔
In technical language, Theorem 4.1 says that the closed convex hull of finite
Blaschke products is precisely the closed unit ball of the disc algebra .
A
Approximation in theclosed unit ball 7
5 Approximation on T by quotients of finite Blaschke
products
Goal: FBP/FBP=UC(T)
If B and B are finite Blaschke products, then B /B is a continuous uni-
1 2 1 2
modular function on T. Helson and Sarason showed that the family of all such
quotients is uniformly dense in the set of continuous unimodular functions [11,
page 9].
To prove the Helson–Sarasontheorem, we need an auxiliary lemma.
Lemma 5.1 Let f UC(T). Then there exists g UC(T) such that either
∈ ∈
f(ζ)=g2(ζ)
or
f(ζ)=ζg2(ζ)
for all ζ T.
∈
Proof. Since f : T T is uniformly continuous, we can take N so big that
θ θ′ 2π/N imp−li→es f(eiθ) f(eiθ′) <2. Now, we divide T into N arcs
| − |≤ | − |
2(k 1)π 2kπ
T = eiθ : − θ , (1 k N).
k
(cid:26) N ≤ ≤ N (cid:27) ≤ ≤
Thenf(T )isaclosedarcinasemicircle,andthusthereisacontinuousfunction
k
φ (θ) on the interval 2(k−1)π θ 2kπ such that
k N ≤ ≤ N
f(eiθ)=exp(iφ (θ)), (eiθ T ).
k k
∈
These functions are uniquely defined up to additive multiples of 2π. We adjust
those additive constants so that
φ (2kπ/N)=φ (2kπ/N)
k k+1
for k = 1,2,...,N 1. Define φ(θ) := φ (θ) for 2(k−1)π θ 2kπ, k =
− k N ≤ ≤ N
1,2,...,N. Then we get a continuous function φ(θ) on [0,2π] such that
f(eiθ)=exp(iφ(θ)), (eiθ T).
∈
Since
exp(i(φ(2π) φ(0)))=f(e2πi)/f(e0i)=1,
−
φ(2π) φ(0) is an integer multiple of 2π. If (φ(2π) φ(0))/2π is even, then set
− −
g(eiθ):=exp(iφ(θ)/2),
and if (φ(2π) φ(0))/2π is odd, then set
−
g(eiθ):=exp(i(φ(θ) θ)/2).
−
Then g is a continuous unimodular function on T such that either f(eiθ) =
g2(eiθ) or f(eiθ)=eiθg2(eiθ) for all eiθ T.
∈ ⊓⊔
8 Javad Mashreghi and Thomas Ransford
Theorem 5.2 (Helson–Sarason [11]) Let f UC(T) and let ε > 0. Then
∈
there are finite Blaschke products B and B such that
1 2
B
1
f <ε.
(cid:13) − B (cid:13)
(cid:13) 2(cid:13)C(T)
(cid:13) (cid:13)
(cid:13) (cid:13)
Proof. According to Lemma 5.1,it is enoughto provethe resultfor unimodular
functions of the form f = g2 (note that b(eiθ) := eiθ is a Blaschke factor).
Without loss of generality, assume that ε<1.
ByWeierstrass’stheorem,thereisatrigonometricpolynomialp(z)suchthat
g p <ε.
k − kC(T)
The restrictionε<1 ensures that p has no zeros onT. Let p∗(z):=p(1/z), and
considerthequotientp/p∗.Sincepisagoodapproximationtog,weexpectthat
p/p∗ should be a good approximation to g/g∗ = g2 = f. More precisely, on the
unit circle T, we have
g p (g p)p∗+(p∗ g∗)p
= − − ,
g∗ − p∗ g∗p∗
which gives
f p/p∗ g p + p∗ g∗ 2ε.
| − |≤| − | | − |≤
Itisenoughnowtonotethatp/p∗ isameromorphicfunctionthatisunimodular
and continuous on T, and thus, according to Corollary 2.3, it is the quotient of
two finite Blaschke products.
⊓⊔
Ifwe allowapproximationby quotients ofgeneralBlaschkeproducts,then it
turns out that we can approximate a much larger class of functions. This is the
subject of the Douglas–Rudin theorem, to be established in Section 9 below.
6 Approximation on D by convex combination of
quotients of finite Blaschke products
Goal: conv(FBP/FBP)=BC(T)
The quotient of two finite Blaschke products is a continuous unimodular
function on T. Hence a convex combination of such fractions stays in the closed
unit ball of (T). As the first step in showing that this set is dense in BC(T), we
consider theClarger set of all unimodular elements of (T), and then pass to the
C
special subclass of quotients of finite Blaschke products.
Lemma 6.1 Letf BC(T) andletε>0.Then thereareuj UC(T) andconvex
∈ ∈
weights (λ ) such that
j 1≤j≤n
λ1u1+λ2u2+ +λnun f C(T) <ε.
k ··· − k
Approximation in theclosed unit ball 9
Proof. Let w D. Then, by the Cauchy integral formula,
∈
1 ζ+w 1 2π eiθ+w
w = dζ = dθ.
2πiZT ζ(1+wζ) 2π Z0 1+weiθ
In a sense, the integral on the right side is an infinite convex combination of
unimodular elements. We shall approximate it by a Riemann sum and thereby
obtain an ordinary finite convex combination. Since
eiθ +w eiθ′ +w 1+ w
| | θ θ′ ,
(cid:12)1+weiθ − 1+weiθ′(cid:12)≤ 1 w | − |
(cid:12) (cid:12) −| |
(cid:12) (cid:12)
for (cid:12) (cid:12)
1 N ei2kπ/N +w
∆ := w ,
N (cid:12) − N 1+wei2kπ/N(cid:12)
(cid:12) Xk=1 (cid:12)
(cid:12) (cid:12)
we obtain the estimation (cid:12) (cid:12)
1 2π eiθ+w 1 N ei2kπ/N +w
∆ = dθ
N (cid:12)2π Z 1+weiθ − N 1+wei2kπ/N(cid:12)
(cid:12) 0 Xk=1 (cid:12)
(cid:12) (cid:12)
(cid:12)1 N 2kπ/N eiθ +w ei2kπ/N +w(cid:12)
dθ
≤ 2π Z (cid:12)1+weiθ − 1+wei2kπ/N(cid:12)
kX=1 2(k−1)π/N (cid:12) (cid:12)
(cid:12) (cid:12)
1 N 2kπ/N (cid:12)1+ w 2π (cid:12)
| | dθ
≤ 2π Z 1 w N
kX=1 2(k−1)π/N −| |
1+ w 2π
= | | .
1 w N
−| |
As f 1,wehave f(eiθ) 1foralleiθ T.Hence,bytheestimateabove,
∞
k k ≤ | |≤ ∈
1 N ei2kπ/N +(1 ε)f(eiθ) 1+ (1 ε)f(eiθ) 2π
(1 ε)f(eiθ) − | − | .
(cid:12) − − N 1+(1 ε)f(eiθ)ei2kπ/N(cid:12)≤ 1 (1 ε)f(eiθ) N
(cid:12)(cid:12) kX=1 − (cid:12)(cid:12) −| − |
Thu(cid:12)s, for each eiθ T, (cid:12)
∈
1 N ei2kπ/N +(1 ε)f(eiθ) 4π
f(eiθ) − ε+ .
(cid:12) − N 1+(1 ε)f(eiθ)ei2kπ/N(cid:12)≤ εN
(cid:12)(cid:12) kX=1 − (cid:12)(cid:12)
But, each (cid:12) (cid:12)
ei2kπ/N +(1 ε)f(eiθ)
u (eiθ)= −
k
1+(1 ε)f(eiθ)ei2kπ/N
−
isinfactaunimodularcontinuousfunctiononT.Thus,givenε>0,itisenough
to choose N so large that 4π/(εN)<ε, to get
N
1
f(eiθ) u (eiθ) 2ε
(cid:12) − N k (cid:12)≤
(cid:12) kX=1 (cid:12)
(cid:12) (cid:12)
for all eiθ T. (cid:12) (cid:12)
∈ ⊓⊔
10 Javad Mashreghi and Thomas Ransford
InthelightofTheorem5.2,itisnoweasytopassfromanarbitraryunimod-
ular element to the quotient of two finite Blaschke products.
Theorem 6.2 Let f BC(T) and let ε > 0. Then there are finite Blaschke
∈
products B , 1 i,j n and convex weights (λ ) such that
ij j 1≤j≤n
≤ ≤
B B B
11 21 n1
λ +λ + +λ f <ε.
(cid:13) 1B 2B ··· nB − (cid:13)
(cid:13) 12 22 n2 (cid:13)C(T)
(cid:13) (cid:13)
(cid:13) (cid:13)
Proof. By Lemma 6.1, there are uj UC(T) and convex weights (λj)1≤j≤n such
∈
that
λ1u1+λ2u2+ +λnun f C(T) <ε/2.
k ··· − k
Foreachk,byTheorem5.2,therearefiniteBlaschkeproductsB andB such
k1 k2
that
u B /B <ε/2.
k k1 k2 ∞
k − k
Hence
n n n
f λ B /B f λ u + λ u B /B <ε.
(cid:13) − k k1 k2(cid:13) ≤(cid:13) − k k(cid:13) kk k− k1 k2k∞
(cid:13) kX=1 (cid:13)∞ (cid:13) Xk=1 (cid:13)∞ Xk=1
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
This completes the proof.
⊓⊔
7 Approximation on D by infinite Blaschke products
Goal: BP=I=I
We start to describe our approximation problem as in the beginning of Sec-
tion2.But,extracareis needed heresince we aredealingwith infinite Blaschke
products and they are notcontinuous onD. Let f H∞ andassume that there
is a sequence of infinite Blaschke products that co∈nverges uniformly on D to f.
First of all, we surely have f 1. But, we can say more. For each Blaschke
∞
k k ≤
product in the sequence, there is an exceptional set of Lebesgue measure zero
such that on the complement the product has radial limits. The union of all
these exceptional sets still has Lebesgue measure zero, and, at all points out-
side this union, each infinite Blaschke product has a radial limit. Therefore, the
function f itself must have a radial limit of modulus one almost everywhere. In
technical language, f is an inner function. Hence, in short, if we can uniformly
approximate an f H∞ by a sequence of infinite Blaschke products, then f is
∈
necessarily an inner function. Frostman showed that the converse is also true.
Let φ be an inner function for the open unit disc. Fix w D and consider
∈
φ =τ φ, i.e.,
w w
◦
w φ(z)
φ (z)= − , (z D).
w
1 wφ(z) ∈
−