Analytic computable structure theory and $L^p$ spaces

Computable Banach Spaces via Domain Theory 1

This paper extends the domain-theoretic approach to computable analysis to complete metric spaces and Banach spaces. We employ the domain of formal balls to deene a computability theory for complete metric spaces. For Banach spaces, the domain specialises to the domain of closed balls, ordered by reversed inclusion. We characterise computable linear operators as those which map computable seque...

Enumerations in computable structure theory

Enumeration Reducibility and Computable Structure Theory

In classical computability theory the main underlying structure is that of the natural numbers or equivalently a structure consisting of some constructive objects, such as words in a finite alphabet. In the 1960’s computability theorists saw it as a challenge to extend the notion of computable to arbitrary structure. The resulting subfield of computability theory is commonly referred to as comp...

Strength and Weakness in Computable Structure Theory

We survey the current results about degrees of categoricity and the degrees that are low for isomorphism as well as the proof techniques used in the constructions of elements of each of these classes. We conclude with an analysis of these classes, what we may deduce about them given the sorts of proof techniques used in each case, and a discussion of future lines of inquiry.

Lp Computable Functions and Fourier Series

This paper studies how well computable functions can be approximated by their Fourier series. To this end, we equip the space of L-computable functions (computable Lebesgue integrable functions) with a size notion, by introducing L-computable Baire categories. We show that L-computable Baire categories satisfy the following three basic properties. Singleton sets {f} (where f is L-computable) ar...