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. $\hat f$, $\hat g$ 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:
- $\widehat{f*g} = \hat f \hat g$
Proof
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