Let $G$ be a discrete Abelian group and denote by $\widehat G$ the (compact) Pontryagin-Van Kampen dual of $G$. I was reading in a paper of Justin Peters that Fourier Transform induces a bijection between the following sets of functions:


(1) $L^1(G)^+\cap \mathcal P(G)$ (continuous, non-negative, positive-definite and absolutely integrable functions $G\to \mathbb C$);


(2) $L^1(\widehat G)^+\cap \mathcal P(\widehat G)$ (continuous, non-negative, positive-definite and absolutely integrable functions $\widehat G\to \mathbb C$).


Peters says that this is easy to deduce from Fourier Inversion Theorem but it does not seem so elementary to me. Can anyone help me? Is this true for any LCA group? 


Is there any analog if we start with a compact non-commutative group and we use the Tannaka-Krein duality?



A bijective correspondence induced by Fourier Transform
I try to write down in full detail the answers of Nik and BS (many thanks to both!). 


The references in the proof are to Folland's book "A Course in Abstract Harmonic Analysis" and Rudin's book "Fourier Analysis on Groups".




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Let $G$ be an LCA group, then {$\widehat\phi:\phi\in L^1(G)^+\cap \mathcal P(G) $}$=L^1(\widehat G)^+\cap \mathcal P(\widehat G)$.


*Proof*.
Let us start proving the inclusion {$\widehat\phi:\phi\in L^1(G)^+\cap \mathcal P(G)$}$\subseteq L^1(\widehat G)^+\cap \mathcal P(\widehat G)$. Indeed, consider $\phi \in L^1(G)^+\cap \mathcal P(G)$, then $\widehat\phi\in L^1(\widehat G)$ by the Fourier Inversion Theorem (see [Rudin, page 22, line 1]) and $\widehat\phi\geq 0$ by [Folland, Corollary 4.23]. Let now $\mu\_\phi$ be the non-negative and bounded (as $\phi\in L^1(G)^+$) regular measure defined on a generic Borel subset $E$ of $G$ by $\mu\_\phi(E)=\int\_{x\in E}\phi(x)d\mu(x)$ (here $\mu$ is a fixed Haar measure on $G$). One can show that 
$$\widehat\phi(\gamma)=\int\_{x\in G}\phi(x)\gamma(-x)d\mu(x)=\int\_{x\in G}\gamma(-x)d\mu\_\phi(x)\, .$$
By Bochner's Theorem (see [Rudin, page 19]), $\widehat\phi\in \mathcal P(\widehat G)$. 


On the other hand, let $\phi\in L^1(\widehat G)^+\cap \mathcal P(\widehat G)$ and $\psi$ be the function defined by $\psi(\gamma)=\phi(-\gamma)$ for all $\gamma\in \widehat G$. It is not difficult to see that $\psi\in L^1(\widehat G)^+\cap \mathcal P(\widehat G)$. By the first part of the proof, $\widehat\psi\in L^1( G)^+\cap \mathcal P( G)$ and, using Fourier Inversion Theorem (see [Folland, Theorem 4.32]), one obtains that $\widehat{\widehat\psi}=\phi$, which is therefore the Fourier transform of a function in $ L^1( G)^+\cap \mathcal P( G)$.\\\


