Harmonic Properties of Schwarz Functions

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


Proof