Let's say I have a set of complex functions $\{\phi_1,\phi_2,...\}$ defined on $0\leq x \leq1$, and they are not orthogonal i.e. $\int_0^1\phi_m^*(x) \phi_n(x)dx \neq \delta_{mn}$. Is it possible to find another set of functions $\{\psi_1,\psi_2,...\}$ that satisfy the orthogonality property i.e. $\int_0^1\psi_m^*(x) \phi_n(x)dx = \delta_{mn}$? Alternatively is it possible to find a weight function that makes the first set of functions orthogonal $\int_0^1 w(x) \phi_m^*(x) \phi_n(x)dx = \delta_{mn}$. For concreteness I'm looking at the functions $\phi_n(x)=\cosh((\alpha_n+i\omega_n)x)$, where $\omega_n=(2n+1)\frac{\pi}{2}$ and $\tanh(\alpha_n)=1-\frac{J_1(0.002\omega_n)}{0.002\omega_n}$.
I'm pretty sure this can be done with a modification of the Gram-Schmidt process, but I'm hoping to find a non iterative solution.
See the Gram-Schmidt process:
https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process