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

This paper studies the approximation of the Hilbert transform (f) over tilde = Hf of continuous functions f with continuous conjugate (f) over tilde based on a finite number of samples. It is known that every sequence {H-N f}(N is an element of N) which approximates (f) over tilde from samples of f 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 Fejer means diverges strongly. Second, it is shown that for every sample based approximation method {H-N}(N is an element of N) there are functions f such that parallel to H-N f parallel to(infinity) exceeds any given bound for any given number of consecutive indices N. 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. (C) 2016 Elsevier Inc. All rights reserved.