History of the Sampling Theorem - MathOverflow

History of the Sampling Theorem

2012-05-17T02:12:43Z

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.

Answer by Tom Copeland for History of the Sampling Theorem

2012-05-17T08:40:42Z

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)

Answer by Alexandre Eremenko for History of the Sampling Theorem

2012-12-23T01:13:30Z

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

Answer by Margaret Friedland for History of the Sampling Theorem

2013-01-07T15:59:45Z

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.