Carath\'eodory's Theorem and moduli of local connectivity

We give a constructive proof of the Carath\'eodory Theorem by means of the concept of a modulus of local connectivity and the extremal distance of the separating curves of an annulus

The isometry degree of a computable copy of ℓp\ell^p

When $p$ is a computable real so that $p \geq 1$, the isometry degree of a computable copy $\mathcal{B}$ of $\ell^p$ is defined to be the least powerful Turing degree that computes a linear isometry of $\ell^p$ onto $\mathcal{B}$. We show that this degree always exists and that when $p \neq 2$ these degrees are precisely the c.e. degrees

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

Computable copies of ℓp

Suppose p is a computable real so that p ≥ 1. It is shown that the halting set can compute a surjective linear isometry between any two computable copies of Rᵖ. It is also shown that this result is optimal in that when p /= 2 there are two computable copies of Rᵖ with the property that any oracle that computes a linear isometry of one onto the other must also compute the halting set. Thus, Rᵖ is ∆⁰-categorical and is computably categorical if and only if p = 2. It is also demonstrated that there is a computably categorical Banach space that is not a Hilbert space. These results hold in both the real and complex case