On the strong divergence of Hilbert transform approximations from sampled data

Abstract

It is known that every linear method which determines the Hilbert transform from the samples of the function diverges (weakly). This paper presents strong evidence that all such methods even diverge strongly. It is shown that the common approximation method derived from the conjugate Fej'er means diverges strongly, and that all reasonable approximation methods with a finite search horizon diverge strongly. Moreover, the paper discusses the relation between strong divergence and the existence of adaptive approximation methods.