Give an example of a function
  1. which is orthogonal to
    $function

    with respect to the scalar product \[ = \frac{1}{\pi} \int_{-\pi}^{\pi}dx g(x) \cdot h(x)\]
  2. 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 $.