Fine Computable Functions and Effective Fine Convergence

In this article, we discuss the Fine computability and the effective Fine convergence for functions on [0, 1) with respect to the Fine metric as the beginning of the effective Walsh-Fourier analysis. First we treat classically the Fine continuity and the Fine convergence. Next, we prove that Fine computability does not depend on the choice of an effective separating set. Subsequently, we propose a notion of effective Fine convergence for a sequence of functions. We prove that the limit of an effectively Fine continuous sequence of functions and the limit of a Fine computable sequence of functions under this effective Fine convergence is effectively Fine continuous and Fine computable respectively. We also investigate some properties of Fine computable functions through examples. Especially, we extend the result of Brattka, which asserts the existence of a Fine computable but not locally uniformly Fine continuous function. Finally, we treat other examples of Fine computable functions.