Where does the Chebyshev polynomial notation come from? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T22:57:07Z http://mathoverflow.net/feeds/question/41658 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41658/where-does-the-chebyshev-polynomial-notation-come-from Where does the Chebyshev polynomial notation come from? alext87 2010-10-10T06:55:27Z 2010-10-11T18:21:57Z <p>The $k$th Chebyshev polynomial is denoted by $T_k$ where</p> <p>$T_k(x) = \cos(k\cos^{-1}(x))$</p> <p>I was wondering where this notation came from. It has been suggested that it comes from Tschebyscheff (the Russian name for Chebyshev) but does anyone know the first use of this notation or verify this is the reason?</p> http://mathoverflow.net/questions/41658/where-does-the-chebyshev-polynomial-notation-come-from/41717#41717 Answer by Andrey Rekalo for Where does the Chebyshev polynomial notation come from? Andrey Rekalo 2010-10-10T22:32:41Z 2010-10-11T18:21:57Z <p>Great Soviet mathematician <a href="http://en.wikipedia.org/wiki/Naum_Akhiezer" rel="nofollow">N.I. Akhiezer</a> mentions in his survey article <em>"Чебышевское направление в теории функций"</em> (<em>"Function Theory According to Chebyshev"</em>) that the notation $$T_n(x)=\frac{1}{2^{n-1}} \cos{(n\arccos x)}$$ was first introduced by S. Bernstein. </p> <p>I think that the first published paper on the Chebyshev polynomials by Bernstein was <em>"О наилучшем приближении непрерывных функций посредством <s>полиномов</s> многочленов данной степени"</em> which appeared in "Сообщения Харьковского математического общества", series 2, vol. 13 (<strong>1912</strong>), pp. 49-194. The paper is in Russian as you may guess.</p> <p>In his paper, Bernstein refers to the Chebyshev polynomials as <em>trigonometric polynomials</em> which probably might explain the letter <em>T</em> in the notation.</p> <hr> <p>English translation of Akhiezer's survey article is contained in <a href="http://www.amazon.com/Mathematics-19th-Century-Differential-Variations/dp/0817658459" rel="nofollow"><em>Mathematics of the 19th Century</em></a> edited by A.N. Kolmogorov. </p> <p><strong>Edit added.</strong> I don't know if there is an English translation of the original paper by Bernstein. <a href="http://www.archive.org/details/sovietmenofscien00turk" rel="nofollow">This source</a> refers to the paper as <em>"The optimum approximation to continuous functions by polynomials of a given power"</em>, Reports of the Kharkov Mathematical Society, Second Series, 1912, 13, #2-3. The original paper can be also found in volume 1 of S.N. Bernstein Collected Works (<em>С.Н. Бернштейн, Собрание сочинений (Том 1. Конструктивная теория функций [1905-1930])</em>, Москва, 1952). </p>