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I am looking for some sufficient conditions for an even, continuous, nonnegative, non-increasing, non-convex function to be non-negative definite. In other words $$ \int_0^\infty f(x)\cos(x\omega) \, ...

I am wondering how to prove the below Fourier transform is non-negative? I did much simulation and it seems to be non-negative. $$\int_0^\inf (be^{-at^p}-ae^{-bt^p})\cos(tx)dt, 0<a<b, \frac{1}{2}...

I am interested in a specific density (positive function) and would like to prove that the tail of its characteristic function (Fourier transform) is positive ($>0$). Here is the density $f(x)=c_\...

I wonder how to prove that $$\int_0^\infty\exp(-x^p)\cos(tx)\,dt\geq 0, \quad \frac{1}{2}<p<1.$$ This conclusion is used in the answer to another question here Looking for sufficient conditions ...

I am looking for $\delta>0$, such that $$ \delta \int_{-\infty}^{\infty} \exp(its) { \Gamma\{2(it+1)/3\}\over \Gamma\{(it+1)/2\} }dt \le \\ \int_{-\infty}^{\infty} \exp(its) { \Gamma (it+1)\over \...