3. FUNCTION THEORETIC BACKGROUND 13
where cn are unimodular constants chosen so that cn
i−λn
i−λ¯n
0. If Λ is an infinite
sequence then the necessary and sufficient condition for the normal convergence of
the partial products of BΛ is that Λ satisfies the Blaschke condition
λn
1 + |λn|2
∞.
Every function F in N
+
has a unique factorization F = IH, where H is the
outer function with modulus equal to |F| a.e. on the boundary and I is an inner
function.
A mean type of a function f(z) in N +(C+) is defined as
τ = lim sup
y→∞
log |F (iy)|/y.
It is easy to see that every function in N
+(C+)
has a non-positive mean type which
is exactly the exponent a of the singular inner function
Sa(z)
=
eiaz
in the inner-
outer factorization of f(z) taken with a negative sign. Here and throughout the
text S denotes the singular inner function in C+, S(z) =
eiz.
Similar statements and formulas are true for the case of the unit disk, see for
instance [52, 84].
A special role in our notes will be played by meromorphic inner functions
(MIF) which are inner functions in the upper half-plane that can be extended
meromorphically to the whole plane. The above formulas imply that an inner
function is a MIF if and only if its Blaschke factor corresponds to a discrete sequence
Λ ⊂ C (a sequence without finite accumulation points) and the measure in the
singular factor is a point mass at infinity, i.e. Jμ = Sa = eiaz for some non-negative
a.
3.4. Model spaces and Clark theory. For each inner function Θ(z) one
may consider a model subspace
KΘ =
H2 ΘH2
of the Hardy space
H2(C+)
Here ‘ ’ stands for the orthogonal difference, i.e. KΘ
is the orthogonal complement of the space
ΘH2
= {Θf|f ∈
H2}
in
H2.
These
subspaces play an important role in complex and harmonic analysis, as well as in
operator theory, see [116].
Each inner function Θ(z) defines a positive harmonic function
1 + Θ(z)
1 − Θ(z)
and, by the Herglotz representation, a positive measure σ such that
(1.5)
1 + Θ(z)
1 − Θ(z)
= py +
1
π
ydσ(t)
(x − t)2 + y2
, z = x + iy,
for some p ≥ 0. The number p can be viewed as a point mass at infinity. The
measure σ is a singular Poisson-finite measure, supported on the set where non-
tangential limits of Θ are equal to 1. The measure σ + pδ∞ on
ˆ
R is called the Clark
measure for Θ(z).
Following standard notations, we will sometimes denote the Clark measure
defined in (1.5) by σ1. If α ∈ C, |α| = 1 then σα is the measure defined by (1.5)
with Θ replaced by ¯Θ. α
