Show that f and fourier are in S(R)

Question

I am dealing with fourier tranforms and have come to some problems when I need to show the following things:

How can I show that if $f\in S(R)$, then $\hat{f}\in S(R)$?

And how will properties change if $f\in S(R^2)$ rather than $f \in S(R)$?

Answer 1

To start you off:

Show that for every $f\in S(R)$ and for every $n\in \mathbb{N}$, $\hat{f}$ is infinitely differentiable and

$$\widehat{((-2\pi i x)^{n} f)} = \hat{f}^{(n)}.$$

and that

$$\widehat{f^{(n)}}(y) = (2\pi i y)^{n}\hat{f}(y)$$

for every $f\in S$ and $n\in \mathbb{N}$. Use these to show that for every $f\in S$ and every $m,n \geq 0$,

$$y^{m}\hat{f}^{(n)}(y)$$

is a Fourier transform of a function in $S$ which is bounded.