Fourier filtering of overlapping amplitude-modulation sidebands

The modulation property of the Fourier transform adds shifted copies of the message spectrum. If those copies overlap, their complex amplitudes add before taking a magnitude; a frequency filter then acts on this sum. For a real even triangular message spectrum proportional to $(1-|\omega |{)}_{+}$ and carrier frequency $1/2<\Omega <1$, the translated spectra overlap over $|\omega |<1-\Omega$. A filter retaining $|\omega |<\Omega$ therefore leaves a central plateau and sloping outer sections, rather than two disjoint triangular bands. Spectral Dirac delta lines at the cutoff need an explicitly specified filter gain: multiplying a distribution by a discontinuous mask is not defined by pointwise reasoning alone.