The Bedrosian identity for Hp functions

We establish a necessary and sufficient condition for f∈Hp(R), g∈Hq(R) with p−1+q−1⩽1 to satisfy the Bedrosian identity H(fg)=fHg, where H denotes the Hilbert transform. As applications, we prove the Bedrosian theorem for this identity, and give a characterization of f satisfying the identity when g is a finite linear combination of complex sinusoids. We also show that if f is of low Fourier frequency then it is necessary for g to have high Fourier frequency in order to satisfy the Bedrosian identity.