On the computability of Walsh functions

The Haar and the Walsh functions are proved to be computable with respect to the Fine-metric dF which is induced from the in-nite product = {0; 1}{1;2; :::} with the weighted product metric dC 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 ( ; dC) and on ([0; 1]; dF). On ( ; dC), computable real-valued cylinder functions, which include the Walsh functions, become computable and every computable function can be approximated e6ectively by a computable sequence of cylinder functions. The metric space ([0; 1]; dF) is separable but not complete nor e6ectively complete. We say that a function on [0; 1] is uniformly Fine-computable if it is sequentially computable and e6ectively uniformly continuous with respect to the metric dF. It is proved that a uniformly Fine-computable function is essentially a computable function on . It is also proved that Walsh–Fourier coe9cients 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 e6ectively uniformly to f. c © 2002 Elsevier Science B.V. All rights reserved.