History of the Sampling Theorem In January, 1949, Shannon publishes the paper Communication in the Presence of Noise, Proc. IRE, Vol. 37, no. 1, pp. 10-21, available here, which establishes the Information Theory. In this paper, the sampling theorem is presented. 
Any references about the history of the sampling theorem, its connection with Fourier theory, would be appreciated. 
EDIT1: In C.E. Shannon, Mathematical Theory of Communication, Bell System Technical Journal, Vol. 27, pp. 379–423, 623–656, July, October, 1948, available here, the Sampling Theorem is Theorem 13. 
 A: As a start to a more comprehensive search, some notes on interpolation using the Dirichlet and Fejer kernels, close cousins of the sinc kernel, can be found in "Leopold Fejer: In Memoriam 1880-1959" by Gabor Szego.
And, you yourself in your answer to MO-Q58325 present a link to a paper by J. de Seguier, published in 1892, that has a Dirichlet kernel interpolation formula and a series that looks suspiciously like a sinc interpolation with the bandwidth $\omega$.
Edit: In the old days, the sinc function was referred to as the cardinal interpolation function and sinc function interpolations as cardinal series. Here is an article (1927) by J. M. Whittaker (son of E. T.:) The "Fourier" Theory of the Cardinal Function in which you can find the nascent Whittaker-Shannon sampling theorem, but E. T. Whittaker published an earlier one in 1915 as discussed by H. D. Luke in The Origins of the Sampling Theorem.
Of interest also: "A Chronology of Interpolation: From Ancient Astronomy to Modern Signal and Image Processing" by Erik Meijering (paper) (website).
A: In Russian literature the sampling theorem is attributed to Kotelnikov. There is a recent
English translation of his paper:
Kotelʹnikov, V. A.
On the transmission capacity of the "ether'' and wire in electrocommunications. 
Translated from the Russian by V. E. Katsnelson. Appl. Numer. Harmon. Anal., Modern sampling theory, 27–45, Birkhäuser Boston, Boston, MA, 2001. 
In England, this theorem is sometimes credited to J. M. Whittaker: it is in his book Interpolatory function theory
(1935), but actually his father E. T. Whittaker published it in 1915.
The theorem in a rudimentary form can be found in Cauchy, and even Lagrange..
EDIT.
Here is a paper with the history which seems complete:
http://www.hit.bme.hu/~papay/edu/Conv/pdf/origins.pdf
A: The  recent article
Ferreira, Paulo J. S. G.; Higgins, Rowland
The establishment of sampling as a scientific principle—a striking case of multiple discovery.
Notices Amer. Math. Soc. 58 (2011), no. 10, 1446–1450
discusses the mathematical work of Kotel'nikov, Shannon and Someya, and the contributions of Nyquist and Raabe, which had more of an engineerig character. 
A: Some background on Kotelnikov's contribution is given in A G Vitushkin: Half a Century As One Day in the book Bolibruch, Osipov, Sinai (Editors): Mathematical Events of the Twentieth Century and also in the Russian original of this book.
