Poisson Kernel

P[x_,r_]:=(1-r^2)/(1-2r Cos[x]+r^2)

Manipulate[Plot[P[x,r],{x,-Pi,Pi},PlotRange->{{-Pi,Pi},{-1,200}}],{r,0,0.99,0.01}]

Plot[UnitBox[x-1],{x,-Pi,Pi}]

g5=1/(2Pi)Convolve[P[x,0.5]UnitBox[x/(2Pi)],UnitBox[x-1]UnitBox[x/(2Pi)],x,y];

g9=1/(2Pi)Convolve[P[x,0.9]UnitBox[x/(2Pi)],UnitBox[x-1]UnitBox[x/(2Pi)],x,y];

g99=1/(2Pi)Convolve[P[x,0.99]UnitBox[x/(2Pi)],UnitBox[x-1]UnitBox[x/(2Pi)],x,y];

Plot[{g5,g9,g99,UnitBox[y-1]},{y,-Pi,Pi},PlotRange->All]