Complexity theory for spaces of integrable functions

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 Arzelà-Ascoli and Fréchet-Kolmogorov Theorems are presented and used to argue that these representations fulfill a minimality condition. To provide tight bounds for the Fréchet-Kolmogorov Theorem, a form of exponential time computability of the norm of L p is proven.

Computer Science for Continuous Data

“…5) Space of probability measures [24,44,57] and subspace of Haar measures on compact groups [51]. 6) Spaces of continuous [63, §6.1], of smooth [28], and of integrable functions [29,60]; equipped with operations like (anti or weak) derivative, or trace. 7) Differential geometry, that is, the hyperspace of closed (smooth) manifolds equipped with (smooth) tensor fields on them.…”

Computing Haar Measures

“…Moreover, Item (c) remains true for f ∈ C ∞ 0 [0; 1]: the class of smooth (infinitely often differentiable) f : [0; 1] → R such that f (0) = 0 = f (1); cmp. [8,28].…”

“…integrating continuous real functions f : G → R. For the arguably most important additive groups G = [0; 1) d mod 1 with Lebesgue measure λ d , this amounts to Euclidean/Riemann integration -whose complexity had been shown to characterize the discrete class #P 1 [13,Theorem 5.32] cmp. [8,28]: indicating that standard quadrature methods, although taking runtime exponential in n to achieve guaranteed absolute output error 2 −n , are likely optimal. And Section 6 extends this numerical characterization of #P 1 to the arguably next-most important compact metric groups:…”