Fourier Transform of Derivative of Tempered Distribution

Theorem
Let $T \in \map {\SS'} \R$ be a tempered distribution.

Let $\xi \in \R$ be a real number.

Let the hat denote the Fourier transform.

Then in the distributional sense it holds that:


 * $\hat {\paren{T'} } = 2 \pi i \xi \hat T$

Proof
Let $\phi \in \map \SS \R$ be a Schwartz test function.

Then: