Boundary Values and Inner Functions
Someday, I'll do a post that doesn't draw heavily from "Introduction to Model Spaces and their Operators" by Garcia, Mashreghi, and Ross, but today is not that day. I wish I had studied this book as a grad student: it's so well written! Anyway, let's talk about inner functions! First, we need to establish the world in which we live. Today, this is going to be the Banach space of analytic functions bounded on the unit disk, \(H^\infty(\mathbb{D})\). In a complex analysis class, you probably did the exercise where you express the Poisson kernel in a couple of different ways: as a "difference quotient," as the real part of a rational function analytic on the disk, as a geometric series, and in terms of the modulus and argument of the complex variable \(z=re^{i\theta}\): Since the Poisson kernel is the real part of an analytic function, it's harmonic, and taking derivatives, this lets us see that the Poisson integral of a measure \(\mu\) is (comple...