Fourier Transform only defined in some interval

Question

$f(x)$ is defined over ($−\infty, \infty)$. fourier transfrom of $f(x)$ is $g(w)$ ≈ $1-w^2$ for small $w$ ($w<<1$) but $g(w)$ over full range of $w$ is not provided. Can we calculate the value of $$\int_{−\infty}^\infty x^2f(x) \,dx?$$

Answer 1

Yes because $\int_{-\infty}^{\infty}x^2 f(x)dx = - \frac{d^2}{d \omega^2}g(\omega)|_{\omega = 0}$. If we know $g(\omega)$ for small positive $\omega$ then we also know its second derivative and can safely take its limit.

This will work provided that we can interchange the integral and the derivative.