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How do you use Young's Inequality, and all these convolution formulae to prove the following inequality $$||f*g||^2_{L^2(\mathbb{T})}\leq ||f*f||_{L^2(\mathbb{T})}||g*g||_{L^2(\mathbb{T})}$$ where $\mathbb{T}=[0,1)$ and $f,g\in L^2(\mathbb{T})$

Many thanks

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    $\begingroup$ Have you tried this. Since $f,g \in L^2 \Rightarrow f\ast g \in L^2$. Then use the plancherel theorem with the fact that $\hat{f\ast g}(n) = \hat{f}(n)\hat{g}(n)$. $\endgroup$ – random123 Sep 22 '14 at 6:55
  • $\begingroup$ We have not learned plancherel theorem yet, so I think there might be an alternative proof. $\endgroup$ – amathnerd Sep 22 '14 at 12:58

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