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It is well known that if $f(x),f'(x) \in L_{PC}^1\left( { - \infty ,\infty } \right)$ then $F\left\{ {f'(x)} \right\} = i\omega F\left\{ {f(x)} \right\}$. The proof itself is rather straightforward using integration by parts.

How would one go about proving that if $f(x),f'(x) \in L_{PC}^2\left( { - \infty ,\infty } \right)$ then the same formula takes hold (in the $L2$ sense) ?

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    What is $L^1_{PC}$? Integrable , piece-wise constant function? – 2017-02-10
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    Sorry for not making it clear, it's the space of all piecewise continuous absolutely integrable functions on the real line – 2017-02-10
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    $L^1 \cap L^2$ is dense in $L^2$. That's how we define the Fourier transform on $L^2$, then we use the Plancherel theorem for showing it is well-defined, bounded and unitary. So if $f,f' \in L^2$ then everything works well, by considering a sequence $f_n \in L^1 \cap L^2$ such that $f_n \to f, f_n' \to f'$ in $L^2$ – 2017-02-10
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    Is there any chance you can elaborate? I'm trying to follow but I fail to fill in the gaps in the proof – 2017-02-11

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