Using Poisson kernel p[x,y]

p[x_,y_]:=Sin[x]/(Cosh[y]-Cos[x])

p[x,y]

Here is what it looks like, both p[x,y] and p[Pi-x,y]

GraphicsRow[{Plot3D[p[x,y],{x,0,Pi},{y,-5,5},PlotRange->{0,5}],Plot3D[p[Pi-x,y],{x,0,Pi},{y,-5,5},PlotRange->{0,5}]}, ImageSize->Large]

intp[x_]:=Integrate[p[x,y],{y,-Infinity,Infinity}]

Check that the integral of p[x,y] is 2(Pi-x) as claimed:

FullSimplify[intp[x],Assumptions->0<x<Pi]

Choose a pair of boundary conditions for left and right boundaries:

sL[t_]:=DiracDelta[t-t0]+DiracDelta[t-(t0+7)];
sR[t_]:=DiracDelta[t-t1]+DiracDelta[t-(t1-5)];

And calculate S_x(y) as in (1), in closed form this is:

sxy[x_,y_,t0_,t1_]=1/(2 Pi) Integrate[p[x,t-y] sL[t],{t,-Infinity,Infinity},Assumptions->{t0 ∈ Reals,Pi>x>0}]+1/(2 Pi) Integrate[p[Pi-x,t-y] sR[t],{t,-Infinity,Infinity},Assumptions->{t1 ∈ Reals,Pi>x>0}]

This can be plotted - note the nice smooth morphing between nearby frequencies.... this plot takes frequencies 5 & 12 to 6 and 11. Note how the amplitude drops in the middle.

Plot3D[sxy[x,y,5,11],{x,0,Pi},{y,0,20},PlotRange->{0,1},MaxRecursion->5, PlotPoints->100]

FullSimplify[p[x,y]-p[Pi-x,y] - 2p[2 x,2 y]]

g[x_]:=x;

sxyG[x_,y_,t0_,t1_]=(Pi-g[x])/(2 Pi) Integrate[p[x,t-y] sL[t],{t,-Infinity,Infinity},Assumptions->{t0∈Reals,Pi>x>0}]+g[x]/(2 Pi) Integrate[p[Pi-x,t-y] sR[t],{t,-Infinity,Infinity},Assumptions->{t1∈Reals,Pi>x>0}]

Plot3D[sxyG[x,y,5,11],{x,0,Pi},{y,0,20},PlotRange->{0,2},MaxRecursion->5,PlotPoints->100,ImageSize->800]

the above figure is poiss1.png

FullSimplify[sxyG[x,y,5,11],Assumptions->{y∈Reals,Pi>x>0}]

FullSimplify[Factor[Expand[sxyG[x,y,5,11]]]]