On the calculation of the Hilbert transform from interpolated data

This correspondence studies the calculation of the Hilbert transform of continuous functions f with continuous conjugate f from a finite set of sampling points. It shows that there exists no linear operator which approximates f arbitrary well in the uniform norm from a finite number of sampling points for all possible continuous function f with continuous conjugate f. However for smooth functions such linear approximation operators exist and sufficient conditions on the smoothness of the functions are presented. The correspondence also examines the robustness of the calculation of the Hilbert transform from interpolated data and it gives explicit error bounds. It is shown that for a large class of algorithms the error grows at least proportional to the logarithm of the number of sampling points.