Why is Fourier space not adequate for (theoretical or digital) filters?
We actually do implement long FIR filters doing that.  If it's running real-time or on an input that is indefinitely long, then the method for doing that is called &quot;fast convolution&quot;. Remember that the &quot;Fourier Transform&quot; that you are multiplying is the Discrete Fourier Transform and that has issues regarding circularity in the resulting convolution.
We deal with that problem using &quot;Overlap-Add&quot; or &quot;Overlap-Save&quot;.  The frequency response that you are multiplying by must be one of an adequately short FIR to prevent &quot;time aliasing&quot;.