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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.

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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 a eulogy for Fejer.

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.

(Also of interest) A Chronology of Interpolation: From Ancient Astronomy to Modern Signal and Image Processing (paper) (website)

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    $\begingroup$ Here's some more on the history by a multi-national panel:: "Interpolation and sampling: E. T. Whittaker, K. Ogura, and their followers" by Butzer, Ferreira, Higgins, Saitoh, Schmeisser, and Stens $\endgroup$ – Tom Copeland Oct 8 '15 at 19:04
  • $\begingroup$ Also see "Five short stories about the cardinsl series" by Higgins $\endgroup$ – Tom Copeland Apr 10 '17 at 9:27
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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

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    $\begingroup$ A biography of Prof. Kotelnikov is available at ieeeghn.org/wiki/index.php/Vladimir_A._Kotelnikov and the English translation of his paper on sampling theorem is available at ict.open.ac.uk/classics/1.pdf $\endgroup$ – Papiro Dec 23 '12 at 11:28
  • $\begingroup$ Don't confuse the son J. M. (1929, and later in his book in 1935) with the father E.T. (1915) who is actually given the majority of the credit for a bandwidth analysis. (I guess you see what you look for.) If you're concerned about accreditation to Russians, you might want to confirm (or not) the chronology at the website linked to in my answer which notes Kotelnikov (1930), as well as other nationalities (Ogura, 1920). $\endgroup$ – Tom Copeland Jan 7 '13 at 5:18
  • $\begingroup$ You are right: the theorem was published by E.T. Whittaker many years before his son J.M. Whittaker included it in his book. $\endgroup$ – Alexandre Eremenko Jan 7 '13 at 14:20
  • $\begingroup$ @Eremenko : the paper you cite in the edit is precisely the one in my answer. Why the repetition? Luke in the cited paper states, "The first scientist to formulate the sampling theorem precisely and apply it to problems of communication engineering is probably V. A. Kotelnikov." (1933) $\endgroup$ – Tom Copeland Jan 9 '13 at 23:39
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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.

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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.

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