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- W2963508424 abstract "This paper investigates second-order representations in the sense of Kawamuraand 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 unitinterval: Kawamura and Cook introduced a representation inducing the rightcomplexity classes and proved that it is the weakest second-orderrepresentation such that evaluation is polynomial-time computable. The firstpart of this paper provides a similar representation for the space ofintegrable functions on a bounded subset of Euclidean space: The weakestrepresentation rendering integration over boxes is polynomial-time computable.In contrast to the representation of continuous functions, however, thisrepresentation turns out to be discontinuous with respect to both the norm andthe weak topology. The second part modifies the representation to be continuousand generalizes it to Lp-spaces. The arising representations are proven to becomputably equivalent to the standard representations of these spaces as metricspaces and to still render integration polynomial-time computable. The familyis extended to cover Sobolev spaces on the unit interval, where less basicoperations like differentiation and some Sobolev embeddings are shown to bepolynomial-time computable. Finally as a further justification quantitativeversions of the Arzel`a-Ascoli and Fr'echet-Kolmogorov Theorems are presentedand used to argue that these representations fulfill a minimality condition. Toprovide tight bounds for the Fr'echet-Kolmogorov Theorem, a form ofexponential time computability of the norm of Lp is proven." @default.
- W2963508424 created "2019-07-30" @default.
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- W2963508424 date "2017-09-12" @default.
- W2963508424 modified "2023-09-27" @default.
- W2963508424 title "Complexity theory for spaces of integrable functions" @default.
- W2963508424 hasPublicationYear "2017" @default.
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