Fourier Transform of a box function

<< Calculus`FourierTransform`

f[x_]  =  UnitStep[1 - x] + UnitStep[
          1 + x]

Plot[f[x], {x, -2, 2}]

F[k_]  =  FourierTransform[f[x], x, k]

where did the dirac delta come from?

f[x_]  =  
      UnitStep[1 - x]  +  UnitStep[1 + x]  - 1
Plot[f[x], {x, -2, 2}]

F[k_]  =  FourierTransform[f[x], x, k]

Plot[F[k], {k, -40, 40}, PlotRange -> All]

Plot[F[k]^2, {k, -40, 40}, 
      PlotRange -> All]
Plot[F[k]^2, {k, -40, 40}, 
      PlotRange -> {All, {0,  .3}}]

where have you seen this before?  think optics.

Fourier Transform of a Gaussian

g[x_, s_]  =  Exp[-x^2/s^2]

Plot[g[x, 1], {x, -3, 3}, PlotRange -> All]

Plot[g[x,  .1], {x, -3, 3}, PlotRange -> All]

G[k_, s_]  =  FourierTransform[g[x, s], x, k]

Plot[G[k, 1], {k, -3, 3}, PlotRange -> All]

Plot[G[k,  .1], {k, -10, 10}, PlotRange -> All]