Although the Hilbert transform and the spectral factorization are of central importance in signal processing,
both operations can generally not be calculated in closed form. Therefore, algorithmic solutions are prevalent which provide an approximation of the true solution. Then it is important to effectively control the approximation error of these approximate solutions. This paper characterizes for both operations precisely those signal spaces of differentiable functions for which such an effective control of the approximation error is possible. In other words, the paper provides a precise characterization of signal spaces of smooth functions on which these two operations are computable on Turing machines.