On algorithms for calculating the hilbert transform

Corollary 8.11 of the previous chapter shows that there exists no robust linear approximation method for causal and stable transfer functions ƒ ∈ D(А) which are defined only on a finite set of discrete sampling points. It was discussed that these convergence problems of the approximation methods are a consequence of the fact that the approximation operator is only defined on a finite, discrete sampling set. However, the sampling of the given data is essential in practical applications, since nowadays numerical calculations are (almost) exclusively carried out on digital computers and such digital computers can process only a finite number of input data. for these reasons, the present chapter will discuss the consequences of the sampling of the given data for the behavior of certain numerical algorithms, a little bit more.