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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$ L\sp{p}$-computability in recursive analysis


Authors: Marian Boykan Pour-El and Ian Richards
Journal: Proc. Amer. Math. Soc. 92 (1984), 93-97
MSC: Primary 03F60; Secondary 03D80, 46E99, 46R05
MathSciNet review: 749899
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Abstract: $ {L^p}$-computability is defined in terms of effective approximation; e.g. a function $ f \in {L^p}[0,1]$ is called $ {L^P}$-computable if $ f$ is the effective limit in $ {L^p}$-norm of a computable sequence of polynomials. Other families of functions can replace the polynomials; see below. In this paper we investigate conditions which are not based on approximation. For $ p > 1$, we show that $ f$ is $ {L^p}$-computable if and only if (a) the sequence of Fourier coefficients of $ f$ is computable, and (b) the $ {L^p}$-norm of $ f$ is a computable real. We show that this fails for $ p = 1$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0749899-0
PII: S 0002-9939(1984)0749899-0
Keywords: $ {L^p}$-functions, computability
Article copyright: © Copyright 1984 American Mathematical Society