eng
episciences.org
Logical Methods in Computer Science
1860-5974
2017-09-12
Volume 13, Issue 3
10.23638/LMCS-13(3:21)2017
3924
journal article
Complexity theory for spaces of integrable functions
Florian Steinberg
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.
https://lmcs.episciences.org/3924/pdf
Computer Science - Computational Complexity