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I am trying to find the original reference which introduced the definition of discrete Fourier transform as used today. When did this modern formulation (which includes the indexing from n to N-1) of DFT appear in the literature?

$$ \begin{aligned} X_{k} &=\sum_{n=0}^{N-1} x_{n} \ e^{-\frac{i 2 \pi}{N} k n} \\ &=\sum_{n=0}^{N-1} x_{n} \left[\cos \left(\frac{2 \pi}{N} k n\right)-i \ \sin \left(\frac{2 \pi}{N} k n\right)\right] \end{aligned} $$

Fourier original on Analytical Theory of Heat does not deal with discrete versions. History related articles credit Gauss well before Fourier. For example here, Gauss and the History of the Fast Fourier Transform, Archive for History of Exact Sciences , 1985, Vol. 34, No. 3 (1985), pp. 265-277 ([Link])1, shows a table, but the article just defines the DFT in "modern notation" for Gauss's Latin work and credits Gauss rather.

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You can go earlier than Gauss if you allow for a DFT involving only sines or only cosines: I quote from Gauss and the history of the fast Fourier transform

Alexis-Claude Clairaut (1713-1765) published in 1754 what we currently believe to be the earliest explicit formula for the DFT (the computation for series coefficients from equally spaced samples of the function), but it was restricted to a cosine Fourier series. Joseph Louis Lagrange (1736-1813 published a DFT-like formula for finite Fourier series containing only sines, in 1759 and in 1762.

The earliest explicit DFT formula containing both sines and cosines is due to Carl Friedrich Gauss (1777-1855) in "Theoria Interpolationis Methodo Nova Tractata". It was published only posthumously in 1866 [10], but was originally written, most likely, in 1805.

Here is Gauss's formula ( source)

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  • $\begingroup$ What do the primes indicate? Is he writing $(\alpha, \alpha', \alpha'', \alpha''', \dotsc)$ in place of $(\alpha_0, \alpha_1, \alpha_2, \alpha_3, \dotsc)$? $\endgroup$
    – LSpice
    Commented Feb 26, 2022 at 20:27
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    $\begingroup$ yes, you can see that the $m$-th term is denoted $\alpha^m$ and $\beta^m$. $\endgroup$
    – Carlo Beenakker
    Commented Feb 26, 2022 at 20:30
  • $\begingroup$ Thanks Carlo, I am also interested in the earliest appearance og modern discrete representation and notation. $\endgroup$
    – ACR
    Commented Feb 26, 2022 at 20:49
  • $\begingroup$ Ah, I see. I didn't zoom in and thought that the $m$s were triple primes. $\endgroup$
    – LSpice
    Commented Feb 26, 2022 at 20:49
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    $\begingroup$ @M.Farooq, re, your definition of "modern notation" seems compatible with this answer, except for the difference of indexing and the explicit appearance of the $\sum$ operator. Are you looking for the first place that the equation appears exactly as you have written it? $\endgroup$
    – LSpice
    Commented Feb 26, 2022 at 20:50

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