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==Historical background==
The sampling theorem was implied by the work of [[Harry Nyquist]] in 1928,<ref>{{cite journal | last=Nyquist |first=Harry | author-link =Harry Nyquist | title =Certain topics in telegraph transmission theory | journal =Transactions of the AIEE | volume =47 | issue =2 | pages =617–644 | date =April 1928 | doi=10.1109/t-aiee.1928.5055024| bibcode =1928TAIEE..47..617N }} [http://www.ieee.org/publications_standards/publications/proceedings/nyquist.pdf Reprint as classic paper] in: ''Proceedings of the IEEE'', Vol. 90, No. 2, February 2002. {{webarchive|url=https://web.archive.org/web/20130926031230/http://www.ieee.org/publications_standards/publications/proceedings/nyquist.pdf |date=2013-09-26 }}</ref> in which he showed that up to <math>2B</math> independent pulse samples could be sent through a system of bandwidth <math>B</math>; but he did not explicitly consider the problem of sampling and reconstruction of continuous signals. About the same time, [[Karl Küpfmüller]] showed a similar result<ref>{{cite journal |first=Karl |last=Küpfmüller |title=Über die Dynamik der selbsttätigen Verstärkungsregler |journal=Elektrische Nachrichtentechnik |volume=5 |issue=11 |pages=459–467 |year=1928 |language=de}} [http://ict.open.ac.uk/classics/2.pdf (English translation 2005)] {{Webarchive|url=https://web.archive.org/web/20190521021624/http://ict.open.ac.uk/classics/2.pdf |date=2019-05-21 }}.</ref> and discussed the sinc-function impulse response of a band-limiting filter, via its integral, the step-response [[sine integral]]; this bandlimiting and reconstruction filter that is so central to the sampling theorem is sometimes referred to as a ''Küpfmüller filter'' (but seldom so in English).
The sampling theorem, essentially a [[duality (mathematics)|dual]] of Nyquist's result, was proved by [[Claude E. Shannon]].<ref name="Shannon49"/> The mathematician [[E. T. Whittaker]] published similar results in 1915,<ref>{{cite journal |last=Whittaker |first=E. T. |author-link=E. T. Whittaker |title=On the Functions Which are Represented by the Expansions of the Interpolation Theory |journal=Proceedings of the Royal Society of Edinburgh |volume=35 |pages=181–194 |date=1915 |doi=10.1017/s0370164600017806|url=https://zenodo.org/record/1428702 }} ({{lang|de|"Theorie der Kardinalfunktionen"}}).</ref> J. M. Whittaker in 1935,<ref>{{cite book | last=Whittaker |first=J. M. | author-link =J. M. Whittaker | title =Interpolatory Function Theory | url=https://archive.org/details/in.ernet.dli.2015.219870 | publisher =Cambridge University Press | date =1935 | ___location =Cambridge, England}}</ref> and [[Dennis Gabor|Gabor]] in 1946 ("Theory of communication").
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In Russian literature it is known as the Kotelnikov's theorem, named after [[Vladimir Kotelnikov]], who discovered it in 1933.<ref>Kotelnikov VA, ''On the transmission capacity of "ether" and wire in electrocommunications'', [http://ict.open.ac.uk/classics/1.pdf (English translation, PDF)] {{Webarchive|url=https://web.archive.org/web/20210301042517/http://ict.open.ac.uk/classics/1.pdf |date=2021-03-01 }}, Izd. Red. Upr. Svyazzi RKKA (1933), Reprint in ''[http://www.ieeta.pt/~pjf/MSTMA/ Modern Sampling Theory: Mathematics and Applications]'', Editors: J. J. Benedetto und PJSG Ferreira, Birkhauser (Boston) 2000, {{ISBN|0-8176-4023-1}}.</ref>
===Why Nyquist?===
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