Suppose
$f \in C^{\infty}(\mathbb{R})$ with compact support, $f(0) = 1$ and derivatives satisfying $f^{(n)}(0) = 0$ for all $n = 1,2, \dots$.
Consider \begin{align*} K(u) = \frac{1}{2 \pi} \int_{- \infty }^{\infty} e^{i x u} f(x) dx \end{align*}
Show that \begin{align*} \int_{- \infty}^{\infty} K(u) du = 1 \end{align*} and \begin{align*} \int_{- \infty}^{\infty} u^j K(u) du = 0 \qquad \forall j = 1,2,\dots \end{align*}
I know that compact support implies that $f$ vanishes at $\pm \infty$ but I don't see how the other assumptions about $f$ and its derivatives are involved.
Any ideas? Thanks
By the fourier inversion formula,
$f(x) = \int_{\infty}^\infty K(u)e^{ixu} du$ so that $1 = f(0) = \int_{\infty}^\infty K(u)$
Moreover, we have that $u^j K(u) = \frac{1}{(2\pi)^n} f^{(j)}(u)$ and the second part of the problem follow from this fact and the restrictions of the derivative of $f$ at zero.