Editing Chebyshev polynomials (section)

{{Short description|Polynomial sequence}}
{{distinguish|discrete Chebyshev polynomials}}
{{Use American English|date = March 2019}}
{{use dmy dates|date=August 2020}}
[[File:Chebyshev Polynomials of the First Kind.svg|thumb|Plot of the first five {{mvar|T<sub>n</sub>}} Chebyshev polynomials (first kind)]]
[[File:Chebyshev Polynomials of the Second Kind.svg|thumb|Plot of the first five {{mvar|U<sub>n</sub>}} Chebyshev polynomials (second kind)]]

The '''Chebyshev polynomials''' are two sequences of [[orthogonal polynomials]] related to the [[trigonometric functions|cosine and sine functions]], notated as <math>T_n(x)</math> and <math>U_n(x)</math>. They can be defined in several equivalent ways, one of which starts with [[trigonometric functions]]:

The '''Chebyshev polynomials of the first kind''' <math>T_n</math> are defined by
<math display="block">T_n(\cos \theta) = \cos(n\theta).</math>

Similarly, the '''Chebyshev polynomials of the second kind''' <math>U_n</math> are defined by
<math display="block">U_n(\cos \theta) \sin \theta = \sin\big((n + 1)\theta\big).</math>

That these expressions define polynomials in <math>\cos\theta</math> is not obvious at first sight but can be shown using [[de Moivre's formula]] (see [[#Trigonometric definition|below]]).

The Chebyshev polynomials {{math| ''T<sub>n</sub>''}} are polynomials with the largest possible leading coefficient whose [[absolute value]] on the [[interval (mathematics)|interval]] {{closed-closed|−1, 1}} is bounded by 1. They are also the "extremal" polynomials for many other properties.<ref>{{cite book |last=Rivlin |first=Theodore J. |author-link=Theodore J. Rivlin |year=1974 |title=The Chebyshev Polynomials |edition=1st |chapter=Chapter &nbsp;2, Extremal properties |pages=56–123 |series=Pure and Applied Mathematics |publisher=Wiley-Interscience [John Wiley & Sons] |place=New York-London-Sydney |isbn=978-047172470-4}}</ref>

In 1952, [[Cornelius Lanczos]] showed that the Chebyshev polynomials are important in [[approximation theory]] for the solution of linear systems;<ref>{{ cite journal  | title=Solution of systems of linear equations by minimized iterations | year=1952 | pages=33 | journal=Journal of Research of the National Bureau of Standards | doi=10.6028/jres.049.006 | volume=49 | issue=1 | last1=Lanczos | first1= C. | doi-access=free }}</ref>  the [[root of a polynomial|roots]] of {{math|''T<sub>n</sub>''(''x'')}}, which are also called ''[[Chebyshev nodes]]'', are used as matching points for optimizing [[polynomial interpolation]]. The resulting interpolation polynomial minimizes the problem of [[Runge's phenomenon]] and provides an approximation that is close to the best polynomial approximation to a [[continuous function]] under the [[maximum norm]], also called the "[[minimax]]" criterion. This approximation leads directly to the method of [[Clenshaw–Curtis quadrature]].

These polynomials were named after [[Pafnuty Chebyshev]].<ref>Chebyshev first presented his eponymous polynomials in a paper read before the St. Petersburg Academy in 1853: {{pb}} {{cite journal |last=Chebyshev |first=P. L. |year=1854 |title=Théorie des mécanismes connus sous le nom de parallélogrammes |language=fr |journal=Mémoires des Savants étrangers présentés à l'Académie de Saint-Pétersbourg |volume=7 |pages=539–586 |url=https://archive.org/details/mmoiresprsentsla07impe/page/537/ }} Also published separately as {{cite book |last = Chebyshev |first= P. L. |year=1853 |title=Théorie des mécanismes connus sous le nom de parallélogrammes |place=St. Petersburg |publisher=Imprimerie de l'Académie Impériale des Sciences |doi=10.3931/E-RARA-120037 |doi-access=free }}</ref> The letter {{mvar|T}} is used because of the alternative [[transliteration]]s of the name ''Chebyshev'' as {{lang|fr|Tchebycheff}}, {{lang|fr|Tchebyshev}} (French) or {{lang|de|Tschebyschow}} (German).