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 spec Θ 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 spec Θ ⊂ 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) 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].

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