12 1. MATHEMATICAL SHAPES OF UNCERTAINTY

3.2. Schwarz integrals, Outer functions, Nevanlinna classes. If h ∈

LΠ1

(where Π is the Poisson measure on the real line dΠ(x) = dx/(1 +

x2))

then its

Schwarz integral is

Sh(z) =

1

πi

1

t − z

−

t

1 + t2

h(t)dt.

If h is a real-valued function then real and the imaginary parts of Sh are the Poisson

and the conjugate Poisson integrals of h:

Sh = Ph + iQh,

Ph(x + iy) =

1

π

y

(t − x)2 + y2

h(t)dt.

Qh(z) =

1

π

x − t

(x − t)2 + y2

+

t

1 + t2

h(t)dt.

Outer functions in the upper half-plane are holomorphic functions of the form

H =

eSh,

h ∈

LΠ.1

A similar definition can be given for the unit disk.

The Nevanlinna class N (C+) is the class of all functions of the form f/g where

f, g ∈

H∞(C+).

A slightly smaller class is the Smirnov class N

+(C+)

consisting of

all functions of the form f/g where f, g ∈

H∞(C+)

and g is outer. All

Hp-functions

are contained in N

+.

The definitions for N (D) and N

+(D)

are analogous.

Note, that since all non-trivial

H∞-functions

have non-tangential boundary

values almost everywhere with respect to the Lebesgue measure on the boundary,

which are not equal to zero except possibly on zero sets, so do all the functions from

N and N +. Moreover, Hp(C+) = Lp(R)∩N +, in the sense that a function f ∈ N +

belongs to Hp if and only if its boundary values fall into Lp on the boundary.

3.3. Inner/outer factorizations. Inner functions are functions from

H∞

whose boundary values are equal to 1 by absolute value a.e. on the boundary.

Every inner function I can be represented as a product

I = BΛJμ,

where BΛ is the Blaschke product corresponding to the sequence Λ = {λn} ⊂ C+

of zeros of I and Jμ is a singular inner function corresponding to a positive finite

singular measure μ on the boundary of the domain. In the case of C+, μ is a

measure on

ˆ,

R μ = ν + cδ∞ where ν is Poisson-finite on R, i.e.

dν(x)

1 + x2

∞,

and c ≥ 0 is the mass at infinity. The singular function Jμ is defined as

Jμ =

e−Sμ

=

e−Sν+icz,

where Sμ is the Schwarz integral of μ. The Blaschke Product BΛ for Λ = {λn} is

defined as

BΛ = cn

z − λn

z −

¯

λ

n

,