It's well known that some $\phi(x^{\mu})$ fulfilling a basic set of criteria (on a manifold M of arbitrary dimension lets say dim=1 here) may be represented as a fourier series of orthogonal functions.
$$\phi(x)=\sum_{n=0}^{\infty}a_{n}\psi_{n}(x)$$
Where the $\psi_{n}$ 's are a set of complete orthogonal basis weighted by their coefficients (the $a_{n}$ 's), which may be chosen normalized such that:
$$1=\intop_{M}\mid\psi_n\mid^{2}dx$$
(sure if we want we can take the limit as the period of $\phi(x^{\mu}$) goes $\rightarrow\infty$ and make it a fourier transform in a particular case; however I'm interested in finite unbounded spaces)
This is all standard, I'm wondering about positive definite functions.
Specifically, lets say $\phi(x)$ is positive definite, and well-behaved as above. Can we always represent it then in a form:
$$\phi(x)=\sum_{n=0}^{\infty}\mid a_{n}\mid^{2}\mid\psi_n(x)\mid^{2}$$
???
I should probably add that There will be a condition such that ( I think it's pretty evident):
$$Constant=\intop_{M}\phi(x)dx$$