Clear[f,F,a]
f[t_,a_]=HeavisidePi[t/a]/a
F[ω_,a_]=FourierTransform[f[t,a],t,ω];
Manipulate[
g1=Plot[f[t,a],{t,-5,5},PlotRange->{-1/5,1.5},PlotStyle->{Thick,Red}];
g2=Plot[F[ω,a],{ω,-5,5},PlotRange->{-1/5,1.5},PlotStyle->{Thick,Green}];
Show[g1,g2]
,{a,1/5,15}]

Clear[f,F,a]
f[t_,a_]=HeavisidePi[t-a]+HeavisidePi[t+a];
F[ω_,a_]=FourierTransform[f[t,a],t,ω];
Manipulate[
g1=Plot[f[t,a],{t,-5,5},PlotRange->{-1/5,1.5},PlotStyle->{Thick,Red}];
g2=Plot[F[ω,a]^2,{ω,-5,5},PlotRange->{-1/5,1.5},PlotStyle->{Thick,Green}];
Show[g1,g2]
,{a,0,5}]

Clear[f,F,a]
f[t_,a_]=HeavisideLambda[t/a]/a
F[ω_,a_]=FourierTransform[f[t,a],t,ω];
Manipulate[
g1=Plot[f[t,a],{t,-5,5},PlotRange->{0,1},PlotStyle->{Thick,Red}];
g2=Plot[F[ω,a],{ω,-5,5},PlotRange->{0,1},PlotStyle->{Thick,Green}];
Show[g1,g2]
,{a,1/5,5}]

Clear[f,F,a]
f[t_,σ_]=1/Sqrt[σ]Exp[-t^2/2/σ^2]
F[ω_,σ_]=FourierTransform[f[t,σ],t,ω,Assumptions->σ>0]
Manipulate[
g1=Plot[f[t,σ],{t,-5,5},PlotRange->{0,2.5},PlotStyle->{Thick,Red}];
g2=Plot[F[ω,σ],{ω,-5,5},PlotRange->{0,2.5},PlotStyle->{Thick,Green}];
Show[g1,g2]
,{σ,1/5,5}]

Plot[{HeavisideLambda[t],HeavisidePi[t]},{t,-2,4}]
U=Convolve[HeavisideLambda[τ],HeavisidePi[τ],τ,t]
Plot[U,{t,-2,3}]

Convolve[PDF[NormalDistribution[2,3],τ],PDF[NormalDistribution[4,2],τ],τ,t]

f[t_,σ_,Ω_]=PDF[NormalDistribution[0,σ],t]*Cos[Ω*t]
F[ω_,σ_,Ω_]=FourierTransform[f[t,σ,Ω],t,ω]//FullSimplify
Manipulate[
g1=Plot[f[x,σ,Ω],{x,-4,15},PlotRange->All];
g2=Plot[F[x,σ,Ω],{x,-4,15},PlotRange->All,PlotStyle->Red];
Show[g1,g2],{σ,0.1,10,0.1},{Ω,0.1,10,0.1}]