Uniformly Computable Aspects of Inner Functions

AbstractThe theory of inner functions plays an important role in the study of bounded analytic functions. Inner functions are also very useful in applied mathematics. Two foundational results in this theory are Frostman's Theorem and the Factorization Theorem. We give a uniformly computable version of Frostman's Theorem. We then claim that the Factorization Theorem is not uniformly computably true. We then claim that for an inner function u, the Blaschke sum of u provides the exact amount of information necessary to compute the factorization of u. Along the way, we discuss some uniform computability results for Blaschke products. These results play a key role in the analysis of factorization. We also give some computability results concerning zeros and singularities of analytic functions. We use Type-Two Effectivity as our foundation