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Characterization of boundary values of functions in hardy spaces with applications in signal analysis
Qian T.
2005
Source PublicationJournal of Integral Equations and Applications
ISSN8973962
Volume17Issue: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. 

KeywordAnalytic Signal Phase Amplitude Hilbert Transform Hardy Spaces Blaschke Products M¨obius Transform Nevanlina Class Inner Function
DOI10.1216/jiea/1181075323
URLView the original
Language英语
The Source to ArticleScopus
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Document TypeJournal article
CollectionDEPARTMENT OF MATHEMATICS
AffiliationFaculty of Science and Technology, University of Macau, P.O. Box 3001, Macau
First Author AffilicationFaculty 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.
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