Characterizations of Mono-Components: the Blaschke and Starlike Types

Abstract

Since the last decade, motivated by attempts of positive frequency decomposition of signals, complex periodic functions \(s(e^{it})=\rho (t)e^{i\theta (t)}\) satisfying the conditions $$\begin{aligned} {H}(\rho (t)\cos \theta (t))=\rho (t)\sin \theta (t),\quad \rho (t)\ge 0,\ \theta '(t)\ge 0,\;a.e., \end{aligned}$$ have been sought, where H is the circular Hilbert transform and the phase derivative \(\theta '(t)\) is suitably defined and interpreted as instantaneous frequency of the signal \(\rho (t)\cos \theta (t)\). Functions satisfying the above conditions are called mono-components. Mono-components have been found to form a large pool and used to decompose and analyze signals. This note in a great extent concludes the study of seeking for mono-components through characterizing two classes of mono-components of which one is phrased as the Blaschke type and the other the starlike type. The Blaschke type mono-components are of the form \(\rho (t)\cos \theta (t)\), where \(\rho (t)\) is a real-valued (generalized) amplitude functions and \(e^{i\theta (t)}\) is the boundary limit of a finite or infinite Blaschke product. For the starlike type mono-components, we assume the condition \(\int _{0}^{2\pi }\theta '(t)dt=n\pi \), where n is a positive integer. It shows that such class of mono-components is identical with the class consisting of products between p-starlike and boundary \((n-2p) \)-starlike functions. The results of this paper explore connections between harmonic analysis, complex analysis, and signal analysis.