Computabilities of Fine-Continuous Functions

Abstract

We propose a sequential-based de finition of locally uniformly Fine-computable functions together with a definition of effective locally uniform convergence. This definition of computability makes some dis- continuous functions, which may diverge, computable. It is proved that a locally uniformly Fine-computable function can be approximated effectively locally uniformly by a Fine-computable sequence of binary step functions on the unit interval [0, 1) with respect to the Fine metric. We also introduce effective integrability for locally uniformly Fine-computable functions, and prove that Walsh-Fourier coefficients of an effectively integrable function f form a computable sequence of reals. It is also proved that S ₂ ⁿ f, where S _n f is the partial sum of the Walsh-Fourier series, Fine-converges effectively locally uniformly to f.

Acknowledgements

The Author would like to express his gratitude to the referees and the editors, according to whose advices this article has been revised, and also Yasugi and Tsujii for their discussions.