Characterization of Boundary Values of Functions in Hardy Spaces with Applications in Signal Analysis

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 ∈ 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), c + s = 1, ρ(t) ≥ 0. We prove that a characterization of a triple (ρ, c, s) satisfying the equation with ρ ∈ Lp(R), 1 ≤ p ≤ ∞, 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 ≡ 1 and 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.