Clear[n,t,w,x]

omega=0.5

h[n_]:=omega/Pi Sinc[n omega]

draw[J_]:=Plot[Sum[h[n]Exp[-I n t],{n,-J/2,J/2}],{t,-Pi,Pi},PlotRange->All]

g1=draw[100]

g2=Plot[If[Abs[x]<omega,1,0],{x,-Pi,Pi}]

w[x_]:=(1-Cos[2Pi x])/2

g3=Plot[w[x],{x,0,1}]

draw2[J_]:=Plot[Sum[w[n/J-1/2]h[n]Exp[-I n t],{n,-J/2,J/2}],{t,-Pi,Pi},PlotRange->All]

g4=draw2[100]