On approximations for functions in the space of uniformly convergent Fourier series

This paper studies the possibility of approximating functions in the space of all uniformly convergent symmetric and non-symmetric Fourier series from finitely many samples of the given function. It is shown that no matter what approximation method is chosen, there always exists a residual subset such that the approximation method diverges for all functions from this subset. This general result implies that there exists no method to effectively calculate the Fourier series expansion on a digital computer for all functions from the space of uniformly convergent Fourier series. In particular, there exists no Turing computable approximation method in these spaces.