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**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.

### MSC:

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.) |

### Keywords:

analytic signal; instantaneous frequency; Hilbert transform; Möbius transform; mono-component; empirical mode decomposition; intrinsic mode functions; HHT (Hilbert-Huang transform); starlike functions
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\textit{T. Qian}, Math. Methods Appl. Sci. 29, No. 10, 1187--1198 (2006; Zbl 1104.94005)

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