On the spectra of computable bounded analytic functions
Brian Rudolph Zilli · 2024
In this dissertation, we investigate the arithmetical complexity of the spectra of computable bounded analytic functions on the unit disc. We begin with some background theorems of classical complex analysis and an introduction to computable complex analysis. In particular, we discuss notions of computability for complex numbers, functions of a single complex variable, and sequences of complex numbers. We also introduce the Blaschke product, a canonical construction of a bounded analytic function with a given zero sequence. Additionally, we define the spectrum of a bounded analytic function and present a computability notion for closed subsets of the unit circle in the complex plane. We then discuss results of Matheson and McNicholl and McNicholl characterizing relationship between the computability of bounded analytic functions on the unit disc (Blaschke products, in particular) and the computability of their zero sequences. Next, we use these results to classify the arithmetical complexity of spectra of computable bounded analytic functions on the unit disc. Namely, we show that all such spectra are Sigma-0-3-closed subsets of the unit circle and construct a function with Sigma-0-3-complete spectrum. We then construct a Sigma-0-2-closed set which is not the spectrum of any such function. We conclude this line of inquiry with a proof that there is no such Pi-0-2-closed set; that is, we prove that every Pi-0-2-closed subset of the circle is the spectrum of a computable bounded analytic function. We then discuss uniform Frostman Blaschke products and prove an effectivization of a theorem of Matheson concerning the spectra of such functions. Namely, we show that every nonempty, computably closed, and nowhere dense subset of the unit circle is the spectrum of a computable, uniform Frostman Blaschke product and that this function may be constructed with Frostman constant arbitrarily close to one.