Complexity theory for spaces of integrable functions

TL;DR

This paper develops and analyzes second-order representations for spaces of integrable functions, enabling polynomial-time computability of integration and other operations, with implications for complexity analysis in analysis.

Contribution

It introduces a new representation for Lp-spaces and Sobolev spaces that ensures polynomial-time computability of integration and basic operations, extending prior work on continuous functions.

Findings

Representation for integrable functions is polynomial-time computable for integration.

Modified representations are continuous and equivalent to standard metric space representations.

Polynomial-time computability extends to differentiation and Sobolev embeddings.

Abstract

This paper investigates second-order representations in the sense of Kawamura and Cook for spaces of integrable functions that regularly show up in analysis. It builds upon prior work about the space of continuous functions on the unit interval: Kawamura and Cook introduced a representation inducing the right complexity classes and proved that it is the weakest second-order representation such that evaluation is polynomial-time computable. The first part of this paper provides a similar representation for the space of integrable functions on a bounded subset of Euclidean space: The weakest representation rendering integration over boxes is polynomial-time computable. In contrast to the representation of continuous functions, however, this representation turns out to be discontinuous with respect to both the norm and the weak topology. The second part modifies the representation to be continuous and generalizes it to Lp-spaces. The arising representations are proven to be computably equivalent to the standard representations of these spaces as metric spaces and to still render integration polynomial-time computable. The family is extended to cover Sobolev spaces on the unit interval, where less basic operations like differentiation and some Sobolev embeddings are shown to be polynomial-time computable. Finally as a further justification quantitative versions of the Arzel\`a-Ascoli and Fr\'echet-Kolmogorov Theorems are presented and used to argue that these representations fulfill a minimality condition. To provide tight bounds for the Fr\'echet-Kolmogorov Theorem, a form of exponential time computability of the norm of Lp is proven.