# Fourier transforms of smooth, compactly supported functions

1. May 31, 2009

### chub

1. The problem statement, all variables and given/known data

$$f, \hat{f} \in C_c^\infty(\mathbb{R}^n)$$

2. Relevant equations

$$\hat{f} = \int_{\mathbb{R}^n} f(x) e^{-2\pi i \xi \cdot x} \,dx$$
$$\check{f} = \int_{\mathbb{R}^n} \hat{f}(x) e^{2\pi i \xi \cdot x} \,d\xi$$

3. The attempt at a solution

As $$C_c^\infty \subset \mathcal{S}$$ (the Schwarz space) we know that the Fourier transformation is invertible, and that
$$f = (\hat{f})\check{} = (\check{f})\hat{}$$
in other words
$$f = \int_{\mathbb{R}^n} \hat{f} e^{2 \pi i \xi \cdot x} \,d\xi = \int_{\mathbb{R}^n} \check{f} e^{-2 \pi i \xi \cdot x} \,dx$$

Somehow these must be zero. I am familiar with the idea that the smoother f is, the faster its transform must decay at infinity; and vice versa. Since $$C^\infty, C_c$$ are the ultimate of both this must somehow indicate that the situation is "too good to be true" in a nontrivial way. But I do not know how to implement this. Advice?