Harmonic Properties of Schwarz Functions

Theorem
Let $f,g : \R \to \C$ be Schwarz functions.

Let $\hat f$, $\hat g$ be the Fourier transforms of $f$ and $g$ respectively.

Then:

1. $\hat f$, $\hat g$ are Schwarz functions.

2. $\hat{\hat{f}}(x) = f(-x)$ for all $x \in \R$.

3. If $f*g$ is the convolution of $f$ and $g$, then:


 * $\widehat{f*g} = \hat f \hat g$