Erratum to: The History of Research on Percutaneous, Transapical, and Sutureless Aortic Valve Replacement
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Inner Functions
The theory of Hardy spaces is a well established part of analytic function theory. Inner functions constitute a special family in this category. Therefore, it is natural to start with several topics on Hardy spaces and apply them in our discussions. However, we are not in a position to study this theory in detail and we assume that our readers have an elementary familiarity with this subject. In this chapter, we briefly mention, mostly without proof, the main theorems that we need in the study of inner functions. For a detailed study of this topic, we refer to [33].
1.1 The Poisson Integral of a Measure Let μ be a complex Borel measure on the unit circle T. Then the Poisson integral of μ on the open unit disc D is defined by the formula 1 − |z|2 Pμ (z) = dμ(ζ), (z ∈ D). 2 T |z − ζ| If dμ(eiθ ) = u(eiθ ) dθ/2π, where u ∈ L1 (T), instead of Pμ we write Pu . It is easy to verify that h = Pμ is a harmonic function on D. Moreover, using Fubini’s theorem and the identity 1 2π
0
2π
1 − |z|2 dθ = 1, |z − eiθ |2
(z ∈ D),
(1.1)
we see that 2π 2π 1 1 − r2 1 |h(reiθ )| dθ ≤ dθ d|μ|(ζ) = μ, 2π 0 2π 0 |reiθ − ζ|2 T
1 J. Mashreghi, Derivatives of Inner Functions, Fields Institute Monographs 31, DOI 10.1007/978-1-4614-5611-7 1, © Springer Science+Business Media New York 2013
2
1 Inner Functions
where μ is the total variation of the measure μ on T. Hence, h fulfills the growth restriction 2π |h(reiθ )| dθ < ∞. (1.2) sup 0≤r
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