The Haar and the Walsh functions are proved to be computable with respect to the Fine-metric d F which is induced from the infinite product Ω={0,1}{ 1 , 2 , . .. } with the weighted product metric d C of the discrete metric on {0,1}. Although they are discontinuous functions on [0,1] with respect to the Euclidean metric, they are continuous functions on (Ω,d C ) and on ([0,1],d F ).On (Ω,d C ), computable real-valued cylinder functions, which include the Walsh functions, become computable and every computable function can be approximated effectively by a computable sequence of cylinder functions. The metric space ([0,1],d F ) is separable but not complete nor effectively complete. We say that a function on [0,1] is uniformly Fine-computable if it is sequentially computable and effectively uniformly continuous with respect to the metric d F . It is proved that a uniformly Fine-computable function is essentially a computable function on Ω.It is also proved that Walsh-Fourier coefficients of a uniformly Fine-computable function f form a computable sequence of reals and there exists a subsequence of the Walsh-Fourier series which Fine-converges effectively uniformly to f.