14 1. MATHEMATICAL SHAPES OF UNCERTAINTY

Conversely, for every positive Poisson-finite singular measure σ and a number

p ≥ 0, there exists an inner function Θ(z) satisfying (1.5).

Every function f ∈ KΘ has non-tangential boundary values σ-a.e. and can be

recovered from these values via the formula

(1.6) f(z) =

p

2πi

(1 − Θ(z)) f(t)(1 − Θ(t))dt +

1 − Θ(z)

2πi

f(t)

t − z

dσ(t)

see [122]. If the Clark measure does not have a point mass at infinity, the formula

is simplified to

(1.7) f(z) =

1

2πi

(1 − Θ(z))Kfσ

where Kfσ stands for the Cauchy integral

(1.8) Kfσ(z) =

f(t)

t − z

dσ(t).

This gives an isometry of L2(σ) onto KΘ. The Clark measure σ1 has a point mass

at infinity if and only if 1 − Θ(t) ∈ L2(R).

Similar formulas can be written for any σα corresponding to Θ. For any α, |α| =

1 and any f ∈ KΘ, f has non-tangential boundary values σα-a.e. on R. Those

boundary values can be used in (1.6) or (1.7) to recover f.

In the case of meromorphic Θ(z) (MIF), every function f ∈ KΘ also has a

meromorphic extension in C, and it is given by the formula (1.6). The corresponding

Clark measure is discrete with masses at the points of the set {Θ = 1} given by

σ({x}) =

2π

|Θ (x)|

.

Each meromorphic inner function Θ(z) can be written as Θ(t) =

eiφ(t)

on

R, where φ(t) is a real analytic and strictly increasing function. The function

φ(t) = arg Θ(t) is a continuous branch of the argument of Θ(z).

For any inner function Θ in the upper half-plane we denote by specΘ the closure

of the set {Θ = 1}, the set of points on the line where the non-tangential limit of Θ

is equal to 1, plus the infinite point if the corresponding Clark measure has a point

mass at infinity, i.e. if p in (1.5) is positive. If specΘ ⊂ R, then p in (1.5) is 0.

We call a sequence of real points discrete if it has no finite accumulation points.

Note that {Θ = 1} is discrete if and only if Θ is meromorphic.

If Λ ⊂ R

(ˆ)

R is a given discrete sequence, one can easily construct a meromor-

phic inner function Θ satisfying {Θ = 1} = Λ by considering a positive Poisson-

finite measure concentrated on Λ and then choosing Θ to satisfy (1.5). One can

prescribe the derivatives of Θ at Λ with a proper choice of pointmasses.

The same construction shows that an arbitrary continuous growing function

γ on R can be approximated, up to a bounded function, by the argument of a

meromorphic inner function. If Λ = {γ = 2πn} then Θ constructed as above with

{Θ = 1} = Λ satisfies |γ − arg Θ| 2π on R. Furthermore, if Γ = {γ = (2n + 1)π}

one can easily construct Θ so that {Θ = 1} = Λ and {Θ = −1} = Γ and achieve

an even better approximation |γ − arg Θ| π. The last construction is discussed

in sections 12.5 and 9.1 of chapter 7 in connection with the two spectra problem.

We will return to Clark theory in chapter 7. For more information and further

references see [126].