Bochner's Theorem essentially provides necessary/sufficient conditions for when something is the Fourier transform of a nonnegative measure on a compact abelian group. I'm looking for a similar result on $SO(3)$, which sadly is not abelian.
Suppose I have a function $f:SO(3)\rightarrow[0,\infty)$. By the Peter-Weyl Theorem, I can write $f(g)=\sum_k d_k\mathrm{Tr}[\hat f(k)\rho_k(g)],$ where $$\hat f(k):=\int_{SO(3)}f(g)\rho_k^\ast(g)\,dg.$$ In this expression $\rho_k(g)$ is the $k$-th unitary irreducible representation of $SO(3)$, the Wigner-$D$ matrix $W^{(k)}(\alpha,\beta,\gamma)$, and $d_k=2k+1.$
Is there a necessary/sufficient condition on the $\hat f(k)$'s that guarantees $f(g)\geq0$ for all $g\in SO(3)$?
There are a few related works, e.g. this paper and parts of this book, but I haven't been able to assemble a meaningful condition for $SO(3)$ out of these.
I don't mind some imprecision about function spaces (whether $f$ needs to be in $L_2$, $C^1$, $C^\infty$, or something else) or just a "formal" argument. In fact, even a condition assuming $\hat f(k)=0$ when $|k|\geq k_0$ is OK as a starting point.
There is a Bochner's theorem for compact groups (Bochner, Hilbert distances and positive definite functions, Annals of Math. 42 (1941) 647-656). It asserts that a function $f\colon G \to \mathbb{C}$ on a compact group $G$ is of positive type (meaning that $(f(y^{-1}x))_{x,y \in U}$ is a positive semidefinite matrix for every finite set $U \subseteq G$) if and only if the Fourier coefficient matrices of its Fourier expansion are positive semidefinite matrices.
In a sense, the proper extension of Bochner's theorem to non-Abelian groups is therefore in terms of positive type functions and positive semidefinite Fourier coefficient matrices. Of course, when the group is Abelian, the matrices are $1 \times 1$ (they are numbers), and so being positive semidefinite is the same as being a nonnegative number. Hence in this case Bochner's theorem is about functions which are Fourier transforms of nonnegative measures, but for general compact groups this is not the case.