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.


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


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


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