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I am looking for some sufficient conditions for an even, continuous, nonnegative, non increasing function $f(x)$ on $R$ such that

$$ \int_0^\infty \cos(xz) f(z) d z \ge 0 \qquad\text{for all $x\ge 0$.} \tag{1} $$

I have a such function $f$. It has a complicated form involving some special functions. But it is an even, continuous, nonnegative and non increasing function. The goal is to see if (1) is satisfied. A direct calculation is hard.

Examples for these functions include: $(1+x^2)^{-1}$, $(1+|x|)^{-1}$, $\exp(-|x|)$, etc. One counterexample is $(1+x^4)^{-1}$.

This question is related the characterization of the nonnegative definite functions on $R$.

Thanks for any hints or references!

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see On positivity of Fourier transforms; one sufficient condition is that $f''(x)>0$ for all $x>0$.

there are other sufficient conditions, see for example On the positivity of Fourier transforms

a necessary and sufficient condition is not known.

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  • $\begingroup$ Thanks Carlo Beenakker for the reference. As is mentioned in p.2 lines 9--14 of this paper, it is not clear for the bell-shaped functions. Unfortunately my function $f$ falls into this case. Another non-convex function which has positive Fourier-cosine transform is $(1+x^2)^{-1}$. $\endgroup$
    – Anand
    Commented Jan 13, 2015 at 23:50
  • $\begingroup$ Is there an analogous result in higher dimensional euclidean space? $\endgroup$
    – Guy Fsone
    Commented May 16, 2023 at 14:13

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