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Computable Cyclic Functions

Computable Cyclic Functions
Computable Cyclic Functions
This dissertation concerns computable analysis where the idea of a representation of a set is of central importance. The key ideas introduced are those commenting on the computable relationship between two newly constructed representations, a representation of integrable cyclic functions, and the continuous cyclic function representation. Also, the computable relationship of an absolutely convergent Fourier series representation is considered. It is observed that the representation of integrable cyclic functions gives rise to a much larger set of computable functions than obtained by the continuous cyclic function representation and that integration remains a computable operation, but that basic evaluation of the function is not computable. Many other representations are acknowledged enhancing the picture of the partial order structure on the space of representations of cyclic functions. The paper can also be seen as a foundation for the study of Fourier analysis in a computable universe and concludes with an investigation into the computability of the Fourier transform.
computable analysis
Horne, Ross
4847dccd-7a84-44b7-9402-341d942c9427
Horne, Ross
4847dccd-7a84-44b7-9402-341d942c9427

Horne, Ross (2005) Computable Cyclic Functions. Oxford University, Computing Laboritory, Masters Thesis.

Record type: Thesis (Masters)

Abstract

This dissertation concerns computable analysis where the idea of a representation of a set is of central importance. The key ideas introduced are those commenting on the computable relationship between two newly constructed representations, a representation of integrable cyclic functions, and the continuous cyclic function representation. Also, the computable relationship of an absolutely convergent Fourier series representation is considered. It is observed that the representation of integrable cyclic functions gives rise to a much larger set of computable functions than obtained by the continuous cyclic function representation and that integration remains a computable operation, but that basic evaluation of the function is not computable. Many other representations are acknowledged enhancing the picture of the partial order structure on the space of representations of cyclic functions. The paper can also be seen as a foundation for the study of Fourier analysis in a computable universe and concludes with an investigation into the computability of the Fourier transform.

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Accepted/In Press date: 18 May 2005
Additional Information: It is well known that evaluation of continuous functions is computable. The thesis shows that integration is computable in the larger class of integrable functions. Integrable functions are represented effectively using absolutely convergent sequences of step functions (step function which converge at a fixed rate). For cyclic functions, the representation of continuous functions computably lifts to the new representation (due to the compactness of the domain of a cyclic function). In the new representation, integration of a function is computable but evaluation of a function of the at a point is not computable. The representation is applied to Fourier analysis and Hilbert spaces. Thus the representation has applications in physics where measurement of functions using integration is required, but evaluate at a point is not required. The work is an extension of the Type-2 Theory of Effectivity, which is a foundation for exact computable analysis.
Keywords: computable analysis
Organisations: Web & Internet Science

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Local EPrints ID: 265017
URI: https://eprints.soton.ac.uk/id/eprint/265017
PURE UUID: 4f3efa91-2734-4d0e-9330-8950508637b4

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Date deposited: 08 Jan 2008 17:49
Last modified: 24 May 2018 16:34

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Contributors

Author: Ross Horne

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