Integral of Fine Computable functions and Walsh Fourier series

We de ne the e ective integrability of Fine-computable functions and e ectivize 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 coe cients of an e ectively integrable Fine-computable function form an E-computable sequence of reals and converge e ectively to zero. The latter fact is the e ectivization of Walsh-Riemann-Lebesgue Theorem. The article is closed with the e ective version of Dirichlet's test. keyword: Fine-computable function, Fine convergence, Walsh Fourier series, e ective integrability, Dirichlet's test

• Type: Journal
• Year: 2008
• Where: ENTCS
• Authors: Takakazu Mori, Mariko Yasugi, Yoshiki Tsujii
• Keywords: Convergence Theorem, Ective Integrability, ENTCS 2008, Fine-computable Function