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I am thinking about this question:

Does a positive definite and radial function imply its Fourier transform is nonnegative?

I know that the converse is correct. That is, we can apply the inverse Fourier transform formula, and the definition of the positive definite function to show the double finite sum is positive. But for the above statement, I have no further idea. Someone told me considering the Bochner theorem and I searched it that can be stated as below:

  • Bochner's theorem: In order that a function $f:\mathbb{R}^d\rightarrow \mathbb{C}$ be positive definite and continuous, it is necessary and sufficient that it be the Fourier transform of a nonnegative finite-valued Borel measure on $\mathbb{R}^d$.

But I have no idea to prove the Fourier transform of $f$ is positive. Any suggestions would be welcome! Thank you!

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