Mono-components for decomposition of signals. (English) Zbl 1104.94005

Summary: This note further carries on the study of the eigenfunction problem: Find \(f(t)=\rho(t)\mathrm e^{i\theta(t)}\) such that \(Hf=-if\), \(\rho (t)\geqslant 0\) and \(\theta^{\prime}(t) \geqslant 0\), 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 \(\rho \equiv 1\) with demonstrative results in relation to Möbius 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 \(\rho \geqslant 0\). The class of solutions is identical to a special class of starlike functions of one complex variable, called circular H-atoms. 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.


94A12 Signal theory (characterization, reconstruction, filtering, etc.)
44A15 Special integral transforms (Legendre, Hilbert, etc.)
31A20 Boundary behavior (theorems of Fatou type, etc.) of harmonic functions in two dimensions
47A75 Eigenvalue problems for linear operators
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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