Characterization of boundary values of functions in hardy spaces with applications in signal analysis | |
Qian T. | |
2005 | |
Source Publication | Journal of Integral Equations and Applications |
ISSN | 8973962 |
Volume | 17Issue:2Pages:159 |
Abstract | In time-frequency analysis Hilbert transformation is used to define analytic signals based on which meaningful instantaneous amplitude and instantaneous frequency are defined. In relation to this background we study characteristic properties of the real-valued measurable functions ρ(t) and θ(t), t ρ R, such that H(ρ(·) cos θ(·))(t) = ρ(t) sin θ(t), ρ(t)≥ 0, where H is the Hilbert transformation on the line. A weaker form of this equation is H(ρ(·)c(·))(t) = ρ(t)s(t), c2 + s2 = 1, ρ(t) ≥ 0. We prove that a characterization of a triple (ρ, c, s) satisfying the equation with ρ ρ Lp(R), 1 ≤ p ≤ 8, is that ρ(c + is) is the boundary value of an analytic function in the Hardy space Hp(C+) in the upper-half complex plane C+. We will be dealing with parameterized and non-parameterized solutions combined with the cases ρ ≡ and ρ ρ ≠ 1. The counterpart theory in the unit disc is formulated first. The upper-half complex plane case is solved by converting it to the unit disc through Cayley transform. Examples in relation to signal analysis are constructed. |
Keyword | Analytic Signal Phase Amplitude Hilbert Transform Hardy Spaces Blaschke Products M¨obius Transform Nevanlina Class Inner Function |
DOI | 10.1216/jiea/1181075323 |
URL | View the original |
Language | 英语 |
The Source to Article | Scopus |
Fulltext Access | |
Citation statistics | |
Document Type | Journal article |
Collection | DEPARTMENT OF MATHEMATICS |
Affiliation | Faculty of Science and Technology, University of Macau, P.O. Box 3001, Macau |
First Author Affilication | Faculty of Science and Technology |
Recommended Citation GB/T 7714 | Qian T.. Characterization of boundary values of functions in hardy spaces with applications in signal analysis[J]. Journal of Integral Equations and Applications,2005,17(2):159. |
APA | Qian T..(2005).Characterization of boundary values of functions in hardy spaces with applications in signal analysis.Journal of Integral Equations and Applications,17(2),159. |
MLA | Qian T.."Characterization of boundary values of functions in hardy spaces with applications in signal analysis".Journal of Integral Equations and Applications 17.2(2005):159. |
Items in the repository are protected by copyright, with all rights reserved, unless otherwise indicated.
Edit Comment