Two possibilities:
a) First use inverse fft to get acceleration, then integrate.
b) Use integration property of Fourier Transform to get velocity and displacement, then use inverse transform. Just look it up on wikipedia.
In both cases however, you need to be careful about the data range. The Fourier transform is defined for tau = -inf --> t, whereas your integration probably runs from tau = 0 --> t. This will be encaptured in the initial state.
