Analytic signals and harmonic measures

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.