Characterizations of Mono-Components: the Blaschke and Starlike Types

摘要

Since the last decade, motivated by attempts of positive frequency decomposition of signals, complex periodic functions s(ei t) =.(t) ei.(t) satisfying the conditions H(.(t) cos.(t)) =.(t) sin.(t),.(t) = 0,. (t) = 0, a. e., have been sought, where H is the circular Hilbert transform and the phase derivative. (t) is suitably defined and interpreted as instantaneous frequency of the signal.(t) cos.(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 monocomponents 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.(t) cos.(t), where.(t) is a real- valued (generalized) amplitude functions and ei.(t) is the boundary limit of a finite or infinite Blaschke product. For the starlike type mono- components, we assume the condition 2p 0. (t) dt = np, where n is a positive integer. It shows that such class of monocomponents 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.