Perhaps the phenomenon you are asking about is: why is the definition of a positive-definite function natural?
One answer is that positive-definite functions are exactly coefficients of group representations, in the following sense. If $\pi : \mathbb{R}\to U(H)$ is a unitary representation of $\mathbb{R}$ on some Hilbert space $H$, and $h\in H$ is a vector, then the function $$t\mapsto \langle \pi (t) h, h\rangle$$ is positive-definite. Conversely, given a positive-definite function $\phi$, there exists a Hilbert space $H$, a vector $h\in H$ and a unitary representation $\pi$ of $\mathbb{R}$ on $H$, for which $\phi(t)=\langle \pi(t)h,h\rangle$.
Indeed, the $n\times n$ matrix occurring in the definition of a positive definite function is nothing more than the Gramm matrix of inner products $\langle \pi (t_i) h, \pi (t_j) h\rangle$; and positivity of this matrix is just a reflection of the fact that the inner product of $H$, restricted to the linear span of $\pi(t_i)h: i=1,\dots,n$ is positive-definite.
The Fourier transform goes from the functions on the group to functions on the space of irreducible unitary representations of the group, and thus switches positivity and complete positivity.
To expand on Douglas Zare comment, the problem is that the identity $\hat{f''}(\xi)=-\xi^2\hat{f}(\xi)$ holds only for regular functions (say Schwartz functions). By duality you can extend this to distributions. So the identity holds for your function $f$, but you have to consider its distributional second derivative $f''$, which is $g$ plus some Dirac deltas and derivates of Dirac deltas supported on $\{-1,+1\}$. These corrections account for the non zero value at the origin of $\hat{g}$.
Best Answer
Sure. Let $f$ be a smooth real-valued function supported on $A \subseteq [0,2\pi]$. Then, denoting convolution by $*$, we have:
From 1, if the support of $f$ is a small interval, the support of $f*f$ will be a slightly bigger interval. Clearly, the support of $f$ can be chosen appropriately so that $f*f$ will vanish on any specified interval.