There exists no always convergent algorithm for the calculation of spectral factorization, Wiener filter, and Hilbert transform

Spectral factorization, Wiener filtering, and many other important operations in information theory and signal processing can be lead back to a Hilbert transform and a Poisson integral. Whereas the Poisson integral causes generally no problems, the Hilbert transform has a much more complicated behavior. This paper investigates the possibility to calculate the Hilbert transform f̃ of a given continuous function f based on a finite set of sampling points of f. It shows that even if f is continuous, no linear approximation operator exists which approximates f arbitrary well from a finite number of sampling points of f, in general. Moreover, the paper characterizes the set of all functions for which such linear approximation operators exist and discusses some consequences for practical applications.