In terms of forward and backward shift invariant subspaces, we characterize functions in Hardy spaces, or, analytic signals in the terminology of signal analysis, through multiplications between analytic and conjugate analytic signals. As applications, we give some necessary and sufficient conditions for solutions of the Bedrosian equation H(fg)=f(Hg) when f or g is a bandlimited signal. We also solve the band preserving problem by means of the shift invariant subspace method, which establishes some necessary and sufficient conditions on the functions f that make fg have bandwidth within that of the function g.