Integral of Fine Computable functions and Walsh Fourier series

We define the effective integrability of Fine-computable functions and effectivize some fundamental limit theorems in the theory of Lebesgue integral such as Bounded Convergence Theorem and Dominated Convergence Theorem. It is also proved that the Walsh-Fourier coefficients of an effectively integra...

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Bibliographic Details
Published in:Electronic notes in theoretical computer science 2008-03, Vol.202, p.279-293
Main Authors: Mori, Takakazu, Yasugi, Mariko, Tsujii, Yoshiki
Format: Article
Language:English
Subjects:
effective integrability
Fine convergence
Fine-computable function
Walsh Fourier series
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Summary:We define the effective integrability of Fine-computable functions and effectivize some fundamental limit theorems in the theory of Lebesgue integral such as Bounded Convergence Theorem and Dominated Convergence Theorem. It is also proved that the Walsh-Fourier coefficients of an effectively integrable Fine-computable function form an E-computable sequence of reals and converge effectively to zero. The latter fact is the effectivization of Walsh-Riemann-Lebesgue Theorem. The article is closed with the effective version of Dirichlet's test.
ISSN:1571-0661
1571-0661
DOI:10.1016/j.entcs.2008.03.021