Let $f: {\bf Z} \to {\bf R}$ be a finitely supported function on the integers ${\bf Z}$. I am interested in knowing when there exists a finitely supported non-negative function $g: {\bf Z} \to [0,+\infty)$ (not identically zero) such that the convolution $f \* g$ is also non-negative. In other words, is there a convex combination of translates of $f$ that is non-negative?


From the Laplace transform identity
$$ \sum\_n f\*g(n) e^{nt} = (\sum\_n f(n) e^{nt}) (\sum\_n g(n) e^{nt})$$
 there is the obvious necessary condition that the Laplace transform of $f$ must be everywhere positive:
$$ \sum\_n f(n) e^{nt} > 0 \hbox{ for all } t \in {\bf R}.$$
But is this condition also sufficient? That is, if a function $f$ has positive Laplace transform, can it always be averaged to be non-negative?


A possibly more general necessary condition is that for any positive function $h: {\bf Z} \to (0,+\infty)$, one has
$$ \sum\_n f(n) h(n-m) > 0$$
for at least one integer $m$, since if $\sum\_n f(n) h(n-m) \leq 0$ for all $m$ then $\sum\_n f\*g(n) h(n) \leq 0$ for all non-negative $g$, and then one could not make $f\*g$ non-negative. Duality suggests that this more general necessary condition is essentially sufficient. The Laplace transform condition corresponds to the special case when $h(n) = e^{nt}$, but I don't know if this case already is strong enough to cover all the others.


EDIT: an equivalent question is to ask when a polynomial of one variable can be written as the quotient of two other polynomials with non-negative coefficients. The obvious necessary condition is that the polynomial has to be positive on $(0,+\infty)$; is this also sufficient?



When can a function be made positive by averaging?
Here is the divisibility theorem for polynomials (and thus for Laurent polynomials in several variables), chapter 3 in *Positive polynomials, convex integral polytopes, and a random walk problem* SLN 1282 (1986 or so) [either this, or *Positive polynomials and product type actions of compact groups* in Memoirs AMS 320; I am out of town, so do not have access to the references]. This can of course be translated back via Fourier transforms, if you want. 


Let $Q$ be a polynomial in $d$ variables, let $K$ be its Newton polytope (the convex hull of its exponents with nonzero coefficients). We can reduce to the case that $K$ contains interior in $R^d$ (by changing variables). For each face of $K$, $F$, define $Q\_F$ to be the subpolynomial obtained from $Q$ by throwing away all the terms whose exponents don't belong to $F$. Then (assuming $Q$ has at least one positive value on the positive orthant) $Q$ can be multiplied by a polynomial (and additionally we can assume it has no negative coefficients) so that the product has no negative coefficients if and only if (a) $Q$ is strictly positive on the strictly positive orthant and (b) for every face $F$ of dimension one or more, $Q\_F$ is strictly positive on the positive orthant. 


From this there is an easy corollary, that if merely $Q$ is strictly positive on the positive orthant, then an arbitrarily small perturbation with Newton polytope equalling $K$ and with positive coefficients will render it of this form. 


A fine variation is given in *Deciding eventual positivity of polynomials,* Ergodic theory and dynamical systems
(1986) 6, 57-79, which determines, given $P$ with no negative coefficients, when there exists $n$ such that $P^nQ $ has no negative coefficients. This generalizes the results of Polya (simplex) and Meissner (cube).


A variation occurs in *Iterated multiplication of characters of compact connected Lie groups*
Journal of Algebra (1995) 173, 67-96, which deals with random walks arising from characters of compact groups, rather than just characters of tori (Laurent polynomials). There are others, including convolution with general measures, etc (write to me, this answer is getting too long). 


See also [Reference request: one of Poincare's theorems about positive functions](https://mathoverflow.net/questions/245469/reference-request-one-of-poincares-theorems-about-positive-functions/245470#245470)


