History of the Sampling Theorem - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T06:59:33Z http://mathoverflow.net/feeds/question/97180 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97180/history-of-the-sampling-theorem History of the Sampling Theorem PaPiro 2012-05-17T02:12:43Z 2013-01-07T15:59:45Z <p>In January, 1949, Shannon publishes the paper <em>Communication in the Presence of Noise</em>, Proc. IRE, Vol. 37, no. 1, pp. 10-21, available <a href="http://www.stanford.edu/class/ee104/shannonpaper.pdf" rel="nofollow">here</a>, which establishes the Information Theory. In this paper, the sampling theorem is presented. </p> <p>Any references about the <strong>history</strong> of the sampling theorem, its connection with Fourier theory, would be appreciated. </p> <p>EDIT1: In C.E. Shannon, <em>Mathematical Theory of Communication</em>, Bell System Technical Journal, Vol. 27, pp. 379–423, 623–656, July, October, 1948, available <a href="http://cm.bell-labs.com/cm/ms/what/shannonday/shannon1948.pdf" rel="nofollow">here</a>, the Sampling Theorem is <strong>Theorem 13</strong>. </p> http://mathoverflow.net/questions/97180/history-of-the-sampling-theorem/97202#97202 Answer by Tom Copeland for History of the Sampling Theorem Tom Copeland 2012-05-17T08:40:42Z 2012-09-27T20:44:08Z <p>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 <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.bams/1183523685" rel="nofollow">eulogy for Fejer</a>. </p> <p>And, you yourself in your answer to <a href="http://mathoverflow.net/questions/58325/complex-fourier-coefficients-introduced-by/93375#93375" rel="nofollow">MO-Q58325</a> 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 $\omega$.</p> <p>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.:) <a href="http://journals.cambridge.org/download.php?file=%252FPEM%252FPEM2_1_03%252FS0013091500013511a.pdf&amp;code=726feb41c839d875f111c69b82140018" rel="nofollow">The "Fourier" Theory of the Cardinal Function</a> 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 <a href="http://www.hit.bme.hu/~papay/edu/Conv/pdf/origins.pdf" rel="nofollow">The Origins of the Sampling Theorem</a>. </p> <p>(Also of interest) A Chronology of Interpolation: From Ancient Astronomy to Modern Signal and Image Processing (<a href="http://bigwww.epfl.ch/publications/meijering0201.pdf" rel="nofollow">paper</a>) (<a href="http://www.imagescience.org/meijering/research/chronology/" rel="nofollow">website</a>) </p> http://mathoverflow.net/questions/97180/history-of-the-sampling-theorem/117059#117059 Answer by Alexandre Eremenko for History of the Sampling Theorem Alexandre Eremenko 2012-12-23T01:13:30Z 2013-01-07T14:23:18Z <p>In Russian literature the sampling theorem is attributed to Kotelnikov. There is a recent English translation of his paper:</p> <p>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. </p> <p>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.</p> <p>The theorem in a rudimentary form can be found in Cauchy, and even Lagrange..</p> <p>EDIT.</p> <p>Here is a paper with the history which seems complete: <a href="http://www.hit.bme.hu/~papay/edu/Conv/pdf/origins.pdf" rel="nofollow">http://www.hit.bme.hu/~papay/edu/Conv/pdf/origins.pdf</a></p> http://mathoverflow.net/questions/97180/history-of-the-sampling-theorem/118289#118289 Answer by Margaret Friedland for History of the Sampling Theorem Margaret Friedland 2013-01-07T15:59:45Z 2013-01-07T15:59:45Z <p>The recent article</p> <p>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</p> <p>discusses the mathematical work of Kotel'nikov, Shannon and Someya, and the contributions of Nyquist and Raabe, which had more of an engineerig character. </p>