# Bochner nonnegativity theorem for Laplace-Beltrami eigenfunctions?

Suppose $\psi_0,\psi_1,\ldots\in L_2(M)$ are the Laplace-Beltrami eigenfunctions of a Riemannian manifold $(M,g)$. Furthermore, take $f(x):=\sum_{i=0}^\infty a_i\psi_i(x)$ to be a function on $M$ with coefficients $\{a_i\}_{i=0}^\infty$. Is there a necessary/sufficient condition on the $a_i$'s guaranteeing that $f(x)\geq0$ for all $x\in M$?

My inspiration here is the Bochner theorem from harmonic analysis, whose proof seems to depend on having stronger structure (Lie group) than just a general manifold $(M,g)$.

Sufficient conditions do exist. For instance, the kernel of the associated heat semigroup $$k_t(x,y) = \sum_{i} e^{-t \lambda_i} \psi_i(x) \psi_i(y),$$ where $\lambda_i$ is the eigenvalue associated with $\psi_i$, is positive. Therefore, all functions of the form $$f(x) = \int_{M \times \mathbb{R_+}} k_t(x,y) \, d \mu(y,t) = \sum_{i = 1}^\infty \bigg( \int_{M \times \mathbb{R_+}} \psi_i(y) e^{- t \, \lambda_i} \, d \mu(y,t) \bigg) \, \psi_i(x)$$ are also positive provided that $\mu$ is a finite positive measure. I ignore whether those are all the positive functions.