which is orthogonal to $function with respect to the
scalar product
\[ = \frac{1}{\pi} \int_{-\pi}^{\pi}dx g(x) \cdot h(x)\]
whose norm is 1.
overlap:integrate((RESPONSE[1])*(LONCAPALIST[1]),x,-%pi,%pi)/%pi;
norm:integrate((RESPONSE[1])*(RESPONSE[1]),x,-%pi,%pi)/%pi;
is(overlap=0 and norm=1);
overlap: integrate((LONCAPALIST[1])*(RESPONSE[1]),x,-%pi,%pi)/%pi;
is(not overlap = 0);
norm: integrate((RESPONSE[1])*(RESPONSE[1]),x,-%pi,%pi)/%pi;
is(not norm = 1);
The function you have provided does not have a norm of one.
The function you have provided is not orthogonal.
Note that with respect to the above norm, $\cos(nx)$ is
perpendicular to $\sin(nx)$ and perpendicular to $\cos(mx)$
for $n\ne m$.