On the Approximability of the Hilbert Transform

It was recently shown that on a large class of important Sobolev-like Banach spaces there exist no linear methods which are able to approximate the Hilbert transform from samples of the given function. This implies that there exists no linear algorithm for calculating the Hilbert transform which can be implemented on a digital computer and which converges for all functions from the corresponding Banach spaces. The present paper develops a much more general framework which includes also non-linear approximation methods. Algorithms within this framework have to satisfy only an axiom which guarantees the computability of the algorithm on a digital computer based on given samples of the function. Then the paper investigates whether there exists an algorithm within this general framework which converges to the Hilbert transform for all functions in the Sobolev-like Banach spaces. It is shown that non-linear methods give actually no improvement over linear methods.