On the strong divergence of Hilbert transform approximations and a problem of Ul’yanov

This paper studies the approximation of the Hilbert transform f˜=Hf of continuous functions ff with continuous conjugate f˜ based on a finite number of samples. It is known that every sequence {HNf}N∈N which approximates f˜ from samples of ff diverges (weakly) with respect to the uniform norm. This paper conjectures that all of these approximation sequences even contain no convergent subsequence. A property which is termed strong divergence.The conjecture is supported by two results. First it is proven that the sequence of the sampled conjugate Fejér means diverges strongly. Second, it is shown that for every sample based approximation method {HN}N∈N there are functions ff such that ‖HNf‖∞ exceeds any given bound for any given number of consecutive indices NN.As an application, the later result is used to investigate a problem associated with a question of Ul’yanov on Fourier series which is related to the possibility to construct adaptive approximation methods to determine the Hilbert transform from sampled data. This paper shows that no such approximation method with a finite search horizon exists.