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