Solution
= Solution
Nonnegativity and unit mass give, by the <triangle inequality> for the <Fourier transform>, $|\widehat f(\xi)|\leq\int f=1=\widehat f(0)$. The finite <second moment> implies a finite first absolute moment, so differentiating under the Fourier <integral> is justified and
$$
\boxed{\|\widehat f\|_\infty\leq1,\qquad\widehat f(0)=1,\qquad\nabla\widehat f(0)=-i\int vf(v)\,dv=0}.
$$
The exponent convention has no $2\pi$, so no such factor occurs in this <derivative>.