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Mono‐components for decomposition of signals
Author(s) -
Qian Tao
Publication year - 2006
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.721
Subject(s) - mathematics , eigenfunction , context (archaeology) , pure mathematics , class (philosophy) , hilbert transform , hilbert space , unit circle , mathematical analysis , combinatorics , algebra over a field , discrete mathematics , eigenvalues and eigenvectors , computer science , biology , paleontology , physics , spectral density , statistics , quantum mechanics , artificial intelligence
This note further carries on the study of the eigenfunction problem: Find f ( t )=ρ( t )e iθ( t ) such that Hf =−i f , ρ( t )⩾0 and θ′( 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 ρ≡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 ρ⩾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. Copyright © 2006 John Wiley & Sons, Ltd.