Derivatives of fourier transform

[mnb96]

Can anyone explain me how to prove the following identity?

$$\frac{\partial \hat{f}}{\partial x}(0,0) = \int \int x^2f(x,y)dxdy$$

where $$\hat{f}$$ denotes the Fourier Transform of f(x,y) ?

 

[mnb96]

Well,
I figured it out by myself. It was actually not hard, but unfortunately I noticed that in my first post the statement of the problem was wrong. I will show the solution for one dimension: the 2-dimensional case is trivial.

$$\frac{d}{d\omega}F\{f\} = \int_{-\infty}^{\infty}\frac{\partial}{\partial \omega}f(x)e^{-i\omega x}dx = -i\int_{-\infty}^{\infty}xf(x)e^{-i\omega x}dx$$

The first step was Leibniz integral rule; now, iterating this process n-times yields:

$$\frac{d^n}{d\omega^n}F\{f\} = (-i)^{n}F\{x^{n}f\}$$

We have just to set $$n=2$$ and $$\omega=0$$ in order to obtain:

$$\frac{d^2}{d\omega^2}F\{f\} = -\int_{-\infty}^{\infty}x^{2}f(x)dx$$

That was it: the second derivative (with opposite sign) of the Fourier transform of f at 0, is equivalent to the second moment of f.