Complex Inner Product and Orthogonalization

Question

Orthogonalize[{E^x, E^(w^1*x), E^(w^2*x), E^(w^3*x)}, 
  Integrate[#1 *Conjugate[#2], {x, -Pi, Pi}] &]

I am trying to orthogonalize the first 4 expression using my own custom inner product which is the integral of F(x)[G(x)]* from -pi to pi. I want the second argument of the product (G(x)) to be the complex conjugate but I cannot figure out how to do that. Any tips?

Answer 1

I think the inner-product function needs to be linear in the second argument, meaning you need to reverse the order of the parameters in your inner-product function. If you set w=a+I*b and reverse your inner-product function:

F = Integrate[ComplexExpand[Conjugate[#1]]*#2, {x, -Pi, Pi}] &

then this simplified example works:

S = Orthogonalize[{E^x, E^(w^1*x)}, F]

Test:

F[S[[1]], S[[1]]]
(* 1 *)
F[S[[1]], S[[2]]]
(* 0 *)

With your four terms it should work too, but will take a long time.

Keep in mind that the manual on Orthogonalize says that "The $e_i$ can be any expressions for which $f$ always yields real results." Your example does not satisfy this condition (some of the overlap integrals are complex-valued); what I wrote above about linearity in the second argument may or may not work properly.