Characterizing the Hilbert transform by the Bedrosian theorem

It is proved that a bounded linear translation invariant operator on L2(Rd) satisfies the Bedrosian theorem if and only if it is a linear combination of the compositions of the partial Hilbert transforms and the identity operator. This observation justifies a definition of multidimensional analytic signals in the papers [T. Bulow, G. Sommer, Hypercomplex signals—a novel extension of the analytic signal to the multidimensional case, IEEE Trans. Signal Process. 49 (2001) 2844–2852] and [S.L. Hahn, Multidimensional complex signals with single-orthant spectra, Proc. IEEE 80 (1992) 1287–1300].