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… (More)
An effective sequence of unifomities on a set and its limit are defined. By taking the diagonal of the limit space, we can express the uniform computability of the Rademacher function system, which is the basis of Walsh-Fourier analysis.
Contents Abstract We will speculate on some computational properties of the system of Rademacher functions f n g. The n-th Rademacher function n is a step function on the interval [0; 1), jumping at nitely many dyadic rationals of size 1 2 n and assuming values f1; 01g alternatingly.
y Abstract We will speculate on the theory of the eeective sequence of unifomities on a set and its eeective limit as a methodology to vest a sequence o f r eal functions which h a ve diierent points of discontinuities with some notion of computability. Some model examples which explain the necessity of such a methodology are presented.