Vakmanʼs problem and the extension of Hilbert transform

Abstract

To determine the instantaneous amplitude and frequency of a nonstationary signal, it is equivalent to determine the imaginary operator ℑ. Vakman argued that ℑ must be the Hilbert transform if the demodulation is subject to certain fundamental physical conditions. But the proof provided by Vakman lacks rigor. To rigorously prove Vakmans statements, we construct a weighted space Lwp(R) that includes LTp, the p-th integrable periodic function space, and ^Lp(R), the p-th integrable function space on R. On Lwp(R) an extension of the classical Hilbert transforms H and HËœ<inf>T</inf> is defined and a rigorous Vakmans theory is established on this space. © 2012 Elsevier Inc.