This does not make sense, to me at least. If you have exactly and only two components in the Fourier domain (and the rest are zero) then you should have the sum of two sine waves in the time domain. You should not have two spikes in the time domain. So there's a problem with your description.
OK, ignoring that for the moment, if you do have two spikes in the time domain, and you zero out the one at time 0, then you'll have just one spike left. If that spike is narrow, basically a delta function, then you'll get a constant in the Fourier domain. If the spike widens out so that it is a rect function, then you'll get a sinc function in the Fourier domain. The wider the time domain rect gets, the narrower the sinc in the Fourier domain, and the narrower the time domain rect gets, the wider the sinc in the Fourier domain gets, until you get to the extreme of an infinitely narrow rect (delta function) and an infinitely wide sinc (which is a constant) like I mentioned earlier.
So from all the inaccuracies in your description, I don't really know what the situation is or what you want to filter out. Perhaps you can attach code or screenshots (use the image and paperclip icons).
