Existence of the Bedrosian identity for Fourier multiplier operators

Abstract

Abstract The Hilbert transform H satisfies the Bedrosian identity H(f g) = $=$ f H g whenever the supports of the Fourier transforms of f,g ∈ $\in$ L 2 $L^{2}$ ( ℝ $\mathbb{R}$ ) are respectively contained in A = $=$ [-a,b] and B = $=$ ℝ $\mathbb{R}$ ∖ $\setminus$ (-b,a), where 0 ≤ $\leq$ a,b ≤ $\leq$ + ∞ $\infty$ . Attracted by this interesting result arising from the time-frequency analysis, we investigate the existence of such an identity for a general bounded Fourier multiplier operator on L 2 $L^{2}$ ( ℝ d $\mathbb{R}^{d}$ ) and for general support sets A and B. A geometric characterization of the support sets for the existence of the Bedrosian identity is established. Moreover, the support sets for the partial Hilbert transforms are all found. In particular, for the Hilbert transform to satisfy the Bedrosian identity, the support sets must be given as above.