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 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. = 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 = H2 ΘH2 of the Hardy space H2(C+) Here stands for the orthogonal difference, i.e. 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 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.
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