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For a positive function $f(x)$, its auto correlation function $A(x)=\int_{-\infty}^{\infty} f(s)f(s+x) ds>0$ and is positive-definite (its Fourier transform $\mathcal{F}(A(x))>0$).

Now for the opposite, given a positive-definite function $A(x)>0$, is there a positive function $f(x)$ such at $A(x)=\int_{-\infty}^{\infty} f(s)f(s+x) ds$?

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Don't you need some assumptions on $f$ so that $A$ and its Fourier transform exist? –  Robert Israel Mar 6 '13 at 9:16
    
Let's say they are in Schwartz space. –  user19848 Mar 6 '13 at 19:24
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1 Answer

This is called the "phase retrieval problem" which is quite famous, especially in applied mathematics, crystallography, acoustics, and so on. The term comes from the fact that

$$\hat A(\xi)=|\hat f (\xi)|^2 >0.$$

which is equivalent to the problem of recontructing $f$ from $|\hat f|$ or equivalently, of recontructing $\hat f$ from $|\hat f|$. In general this doesn't have a unique solution as you can reconstruct $\hat f$ in the form

$$\hat f(\xi)= [\hat A(\xi)]^\frac{1}{2} e^{i\phi(\xi)}$$

for some arbitrary real phase $\phi$. Put differently, any modulation of $\hat f$ gives the same $|\hat f|$. Thus you lose all phase information hence the "phase retrieval" term. I don't know the details of the full solution to this (due to Rosenblatt) but I'm guessing the solution is not unique in general, but unique up to "some invariances". Also, in the special case you are considering (probability measures) maybe this answer is simpler although probably still non-unique as any translate of the density (so modulation of the its Fourier transform), gives the same answer.

See also this paper of Jaming and Kolountzakis where similar problems are considered; in particular you can ask the same question for the $k$-correlation function (instead of the $2$-correlation=auto correlation that you consider here).

Hope this helps.

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