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When does Fourier transform not preserve positive definiteness?
In this question: Rotation invariant operator is positive definite, the answer uses the fact that if $f\in L^1[0,1]$, then
$
\hat{f}(u)=\int_0^1 f(t) e^{ -2\pi iut}dt
$
is a "real-valued" function (the Fourier transform is also defined over the reals).
Is it just a function that is the sum of complex-valued functions (as the partial Fourier transforms of each term in the sum are also real-valued)?
In which case is this equality true for any $f\in L^1[0,1]$?
A:
Consider $g$ defined as $ 0b46394aab
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