For the closed surface of some (pseudo)Riemannian manifold M , we're interested in a function $\rho$ which satisfies:
$$m=\int_{\partial M}\rho(x,y,z) \, d\mathrm{vol}$$
Where $dvol$ is the volume element on $\partial M$ and $m$ (a constant) and $\rho$ are positive definite quantities ($\rho,m\geq0$ ). In attempting to represent $\rho$ by a fourier series of weighted orthogonal basis,
$$\rho(x,y,z)=\Phi(x,y,z)=\sum_{n=0}^{\infty}a_{n}\phi_{n}(x,y,z)$$ it appears more natural (due to the positive definiteness of $\rho$ ) to specify instead:
$$\rho(x,y,z)=\Phi^{\star}\Phi$$
Where the star denotes the complex conjugate of the general Fourier series. I get that without a further condition uniqueness of $\Phi$ will be lost. In trying to look up a procedure for this sort of thing, I came across Fredholm theory which seems very relevant. Can anyone point me towards a book on this sort of thing. Thank you!