restart;
phi:=piecewise(x>-1 and x<1, 1-abs(x), 0);
plot(phi,x=-2..2);
phihat:=int(phi*exp(-2*Pi*I*g*x),x=-1..1);
simplify(phihat);
phihat2:=(sin(Pi*g)/(Pi*g))^2;
plot(phihat,g=-3..3);
phix:=unapply(phi,x);
Bn:=unapply(int(phix(x)*phix(x-n),x=-infinity..infinity),n);
phid:=sum(Bn(n)*exp(-2*Pi*I*n*g),n=-2..2);
phid2:=simplify(phid);
Phi:=1/sqrt(phid2);
phitildehat:=phihat2*Phi;
Cn:=Int(Phi*exp(2*Pi*I*n*g),g=0..1);
phitilde:=sum(Cn*phi(x-n),n=-infinity..infinity);
phitildeapp:=simplify(sum(evalf(Re(Cn)*phix(x-n)),n=-10..10));
plot(phitildeapp,x=-5..5);
phitildeappx:=unapply(phitildeapp,x);
Hk:=unapply(evalf(Int(phitildeappx(s)*sqrt(2)*phitildeappx(2*s-k),s=-25..25)),k);
Hkf:=k->evalf(Hk(k));
Hkf(2);
Gk:=k->(-1)^k*Hkf(1-k);
Gka:=[seq(Gk(k),k=-10..10)];
psi:=simplify(sum(Gka[k+11]*sqrt(2)*phitildeappx(2*x-k),k=-10..10));
plot(psi,x=-5..5);
psix:=unapply(psi,x);
int(psi*psi,x=-10..10);
int(phitildeapp*phitildeapp,x=-10..10);
int(phitildeappx(x)*psix(x),x=-10..10);
int(phitildeappx(x)*phitildeappx(x-1),x=-10..10);
int(psix(x)*psix(x-1),x=-10..10);