Assume you are given a non-negative, continuous, radial function $f\in L^q(\mathbb{R}^3)$ (for any $q\geq 1$).
Are there any conditions which would guarantee that the Fourier transform of $f$, that is $\hat{f}(p)$, is also non-negative?
[Math] Conditions for positivity of Fourier transform
fourier analysisharmonic-analysis
Related Solutions
No, such function doesn't need to be a Fourier transform of a finite measure (and, thereby, doesn't need to be a multiplier in $L^1$ or $L^\infty$). This is well-known and the most classical counterexample is just a smoothed Heaviside function $q$ that is $0$ on $(-\infty,1]$, $1$ on $[1,+\infty)$ and whatever you want (just keep it $C^\infty$) in between. One reason it is not a Fourier transform of a measure is that on one hand, such measure should simultaneously have and have not point masses by Wiener's formula $$ \sum_x|\mu(\{x\})|^2=\lim_{|I|\to+\infty}|I|^{-1}\int_I |\widehat \mu(y)|^2\ dy $$ (usually it is written for the interval $[-T,T]$, but the truth is that the limit can be taken over any sequence of intervals whose lengths tend to infnity). The existence of this limit is quite a restrictive condition on the absolute value of a bounded function that wants to be a Fourier transform of a finite measure.
The full description of Fourier transforms of measures seems beyond reach (not in the sense that it is hard to figure out what happens for each particular function or whether a given condition is necessary or sufficient, but in the sense that there are no easily checkable conditions that would be exactly equivalent to that property).
Consider functions of the form $s_k(x) = \Theta(x) x^k$, where $\Theta(x)$ is the Heaviside step function. Then any function of the kind you describe can be written as $f(x) = g(x) + \sum_{i,k} (A_{ik} s_k(x-y_i))$, where the constants $A_{i,k}$ and $y_i$ determine the magnitudes of some finite number of discontinuities in derivatives of order less than $r$ and their locations, and $g(x)$ is $C^r$ regular. Obviously the same decomposition will be true for the Fourier transform $\hat{f}(\xi)$. It is straightforward to compute the Fourier transforms $\hat{s}_k(\xi)$ and match them to the asymptotics of $\hat{f}(\xi)$ for large $\xi$. Essentially, they will decay as powers of $\xi$ and $\hat{s}_k(x)$ will always dominate $\hat{g}(\xi)$ for any $g(x)$ that is $C^r$ regular, with $r>k$.
Then, $\hat{f}(\xi) = \sum_k F_k(\xi) \xi^{-k-1}$, where each $F_k(\xi)$ is a finite linear combination of the form $\sum_i A_{ik} \exp(i\xi y_i)$. It should be straight forward to extract the locations $y_i$ and the jump magnitudes $A_{ik}$ if the $F_k(\xi)$ are known.
Best Answer
the three-dimensional Fourier transform $F(\vec{p})$, of the radial function $f(r)$ has Fourier transform
$$F(\vec{p})=\int_0^\infty dr\int_0^\pi d\theta \int_0^{2\pi}d\phi\;e^{ ipr\cos\theta}f(r) r^2\sin\theta$$ $$\qquad=\frac{4\pi}{p}\int_0^\infty rf(r)\sin(pr)\,dr,\;\;{\rm with}\;\;p=|\vec{p}|.$$
so you're asking when the Fourier-sine-transform $S(p)$ of $rf(r)$ will be (pointwise) positive for $p>0$. A sufficient condition is that $rf(r)$ is a decreasing function of $r>0$, see On positivity of Fourier transforms.
For a related question, see this MO post.
For the connection to Bochner's theorem: the OP's question amounts to finding a function that is both positive and positive-definite, since positive functions have positive-definite Fourier transforms.