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I was looking for an earliest reference or the name of the mathematician who showed calculating the derivatives is possible in the Fourier domain?

The Fourier transform of the derivative is (Wikipedia) $$ \mathcal{F}(f')(\xi)=2\pi i\xi\cdot\mathcal{F}(f)(\xi). $$

Thank you.

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    $\begingroup$ Has to be fourier since he invented this to solve a differential eqn $\endgroup$ – Piyush Grover Feb 9 '20 at 17:54
  • $\begingroup$ To compute $f'$ this way, you have to first compute a Fourier transform and then compute an inverse Fourier transform, which is quite a bit of work to do. Ignoring historical firstness, what references do you have for computing derivatives this way? $\endgroup$ – Wojowu Feb 9 '20 at 18:13
  • $\begingroup$ Wojowu, I am a chemist by training. Our instruments generate data as time vs. signal. In that case, this is the only way to determine the derivative using FT method as far as I know and one can include smoothing before inverse FT. Is there a better way to calculate derivatives of the data $using$ FT methods? $\endgroup$ – M. Farooq Feb 9 '20 at 18:24
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Despite this being complicated by Fourier’s use of sine and cosine transforms, I think @PiyushGrover is right: Théorie analytique de la chaleur, Art. 419:

$$ \begin{align} \frac{d^{2i}}{dx^{2i}}fx =\pm&\int d\alpha f\alpha\int dp\,p^{2i}\cos.(px-p\alpha)\\ \frac{d^{(2i+1)}}{dx^{2i+1}}fx =\mp\frac1{2\pi}&\int d\alpha f\alpha\int dp\,p^{2i+1}\sin.(px-p\alpha) \end{align} $$

and Art. 422 “Expression générale de la fluxion d’ordre $i$”.

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