Mono-components for decomposition of signals

This note further carries on the study of the eigenfunction problem: Find f(t)=P(t)ei(theta)(t) such that Hf=-if, P(t)>=0 and (theta)ʹ(t)>=0, a.e. where H is Hilbert transform. Functions satisfying the above conditions are called mono-components, that have been sought in time-frequency analysis. A systematic study for the particular case P(identical)1 with demonstrative results in relation to Mobius transform and Blaschke products has been pursued by a number of authors. In this note, as a key step, we characterize a fundamental class of solutions of the eigenfunction problem for the general case P>=0. The class of solutions is identical to a special class of starlike functions of one complex variable, called circular Hatoms. They are building blocks of circular mono-components. We first study the unit circle context, and then derive the counterpart results on the line. The parallel case of dual mono-components is also studied.