Article ID Journal Published Year Pages File Type
4624173 Journal of Mathematical Analysis and Applications 2006 11 Pages PDF
Abstract

We prove that a sufficient and necessary condition for HeiΘ(s)=−ieiΘ(s), where H is Hilbert transformation, Θ is a continuous and strictly increasing function with |Θ(R)|=2π, is that dΘ(s) is a harmonic measure on the line. The counterpart result for the periodic case is also established. The study is motivated by, and has significant impact to time–frequency analysis, especially to aspects of analytic signals inducing instantaneous amplitude and frequency. As a by-product we introduce the theory of Hardy-space-preserving weighted trigonometric series and Fourier transformations induced by harmonic measures in the respective contexts.

Related Topics
Physical Sciences and Engineering Mathematics Analysis