We consider the inverse resonance problem in scattering theory in one
dimension. The signal is in form of Fourier transform, which has various
sorts of representation theorem via its zero set. We are interested at
the question if certain bound states information are disregarded, then
how much more information on the potential V is needed to recover the
potential? If partial knowledge of the potential function is given,
certain amount of zeros or bound states can be removed to locally
recover a representation theorem of the Fourier transform. Once the
representation form is recovered, we compare to conclude the inverse
uniqueness.