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.