We identify spaces of radiation fields at null infinity and the event horizon that describe the image of the forward scattering map for fixed spherical harmonic solutions to the linear scalar wave equation on Schwarzschild. We show that for any fixed spherical harmonic mode solution, the radiation field at the event horizon determines the solution up to perturbations that decay exponentially in time, at a rate that is determined by the surface gravity. We construct examples of fixed spherical harmonic mode solutions with no radiation at null infinity and which have polynomial decay along the event horizon. Finally, we construct examples of polynomially decaying solutions to the linear scalar wave equation which are regular at the event horizon, have unbounded support in spherical harmonic modes, and induce no radiation at the event horizon.