Limits of calculating the finite Hilbert transform from discrete samples

Abstract

This paper studies the problem of calculating the finite Hilbert transform f ˜ = H f 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 { H N } N ∈ N of linear approximation operators H N : B → B such that the calculation of H N f is based on discrete samples of f and which satisfies two additional natural axioms. We show that for all such sequences { H N } N ∈ N there always exists an f ∈ B such that lim N → ∞ ⁡ ‖ H N f ‖ ∞ = + ∞ . Moreover, it is shown that on the subset B 1 / 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 ∈ B 1 / 2 such that lim sup N → ∞ ‖ H N f ‖ ∞ = + ∞ .