Limits of calculating the finite Hilbert transform from discrete samples

This paper studies the problem of calculating the finite Hilbert transform f˜=Hf of functions f from the set B of continuous functions with a continuous conjugate f˜ based on discrete samples of f. It is shown that all sampling based linear approximations which satisfy three natural axioms diverge strongly on B in the uniform norm. More precisely, we consider sequences {HN}N∈N of linear approximation operators HN:B→B such that the calculation of HNf is based on discrete samples of f and which satisfies two additional natural axioms. We show that for all such sequences {HN}N∈N there always exists an f∈B such that limN→∞⁡‖HNf‖∞=+∞. Moreover, it is shown that on the subset B1/2 of all f∈B with finite Dirichlet energy an even larger class of sampling based approximation sequences diverges weakly, i.e. for all such sequences there always exists an f∈B1/2 such that limsupN→∞‖HNf‖∞=+∞.