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].
