A counterexample in Hardy spaces with an application to systems theory
Summary: We know from elementary calculus that if \(f\) is a differentiable function on \((0, \infty)\) with \(f(\infty) = 0\), then \(\lim_{x \to + \infty} xf(x) = - \lim_{z \to + \infty} x^2 f'(x)\) in the following sense: if the second limit exists, then so does the first and they are equal. The existence of the first limit does not imply the existence of the second, in general. It was hoped that the existence of the first limit, together with the assumption that \(f\) belongs to the Hardy class \(H^2\), does imply the existence of the second limit. This would have simplified certain results in systems theory. We show by counterexample that this is not the case. More precisely, for each \(q > 1\) (including \(q = \infty)\) we construct a function \(f\) in the Hardy space \(H^q\) on the right half-plane such that \(\lim_{x \to + \infty} xf(x) = 0\) and \(\limsup_{x \to + \infty} x^2 |f'(x) |= \infty\). A special Blaschke product plays a crucial role in this construction.
scientific article; zbMATH DE number 856791