I'll write everything down from the start.

First, the notation:

$\displaystyle \alpha \in \mathbb{N}^n, \alpha=(\alpha_1, \ldots, \alpha_n)$

$\displaystyle |\alpha|=\alpha_1 + \ldots \alpha_n$

$\displaystyle x \in \mathbb{R}^n$

$\displaystyle x^{\alpha}=x_1^{\alpha_1} \ldots x_n^{\alpha_n}$

$\displaystyle \partial^{\alpha}=\partial_1^{\alpha_1} \ldots \partial_n^{\alpha_n}$, where

$\displaystyle \partial_i^{\alpha_i}=\frac{\partial^{\alpha_i}}{\ partial x_i ^{\alpha_i}}$

Then the theorem:

If $\displaystyle x^{\alpha}f(x)$ is summable for every $\displaystyle \alpha, |\alpha|\leq k$, then

$\displaystyle \partial^{\alpha} \hat{f}={[(-2 \pi i x)^{\alpha})f]}^{\wedge}$, where $\displaystyle \hat{f}$ denotes Fourier transform of f.

Now, the problem.

We're supposed to prove this using induction on $\displaystyle \alpha$, and I just can't do it. The definition of $\displaystyle \partial^{\alpha}f$ is somewhat confusing, and I would b really grateful if someone could help me out, just for the basis, and I will try to work from there.

Thank you!