In this paper we develop a
constructive theory of continuous almost periodic functions. We expose those aspects
of the standard theory that are not constructive and give constructive substitutes.
For example, it is not true in constructive mathematics that each trigonometric
polynomial is almost periodic. A trigonometric polynomial is almost periodic if and
only if its exponents are rationally discrete. We obtain a constructive proof of Bohr’s
fundamental theorem that leads to a computational method for uniform
approximations to continuous almost periodic functions by trigonometric
polynomials.
