The development of Fast Fourier transform is attributed to Cooley & Tukey, both have written a lot about it is historical development. Both Cooley and Tukey call it a re-discovery rather. However, I am searching for early publications which showed how people made the conversions in discrete frequency axis from time to frequency and vice versa.

For example, if we have $N$ data points, the FFT output also consists of $N$ data points. The specific question is:

Who came up with the equation or the proof that the frequency spacing in discrete Fourier transform is $= k/T$, where $k =0,1,2,3,..., N-1$ and $T$ is the total duration of time for which the signal was acquired? Further searching of older literature I found Cooley's paper of 1967 Application of the fast Fourier transform to computation of Fourier integrals, Fourier series, and convolution integrals. In this paper Cooley shows a theorem as shown in the snippet of his original 1967 paper.

Does anyone know who originated this theorem for discrete data set or was it very well known before the 1960s? I would appreciate a earlier reference, if it exists about the following theorem. Thanks.