# 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:

- $(1): \quad \hat f$, $\hat g$ are Schwarz functions.

- $(2): \quad \widehat {\left({\hat f}\right)} \left({x}\right) = f \left({-x}\right)$ for all $x \in \R$.

- $(3): \quad$ If $f * g$ is the convolution of $f$ and $g$, then:

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