Editing Chebyshev polynomials (section)

==As a basis set==
[[Image:ChebyshevExpansion.png|thumb|right|240px|The non-smooth function (top) {{math|1=''y'' = −''x''<sup>3</sup>''H''(−''x'')}}, where {{mvar|H}} is the [[Heaviside step function]], and (bottom) the 5th partial sum of its Chebyshev expansion. The 7th sum is indistinguishable from the original function at the resolution of the graph.]]

In the appropriate [[Sobolev space]], the set of Chebyshev polynomials form an [[Hilbert space#Orthonormal bases|orthonormal basis]], so that a function in the same space can, on {{math|−1 ≤ ''x'' ≤ 1}}, be expressed via the expansion:<ref name=boyd>{{cite book|title = Chebyshev and Fourier Spectral Methods|first = John P.|last = Boyd|isbn = 0-486-41183-4|edition = second|year = 2001|publisher = Dover|url = http://www-personal.umich.edu/~jpboyd/aaabook_9500may00.pdf|access-date = 2009-03-19|archive-url = https://web.archive.org/web/20100331183829/http://www-personal.umich.edu/~jpboyd/aaabook_9500may00.pdf|archive-date = 2010-03-31|url-status = dead}}</ref>
<math display="block">f(x) = \sum_{n = 0}^\infty a_n T_n(x).</math>

Furthermore, as mentioned previously, the Chebyshev polynomials form an [[orthogonal]] basis which (among other things) implies that the coefficients {{math|''a''<sub>''n''</sub>}} can be determined easily through the application of an [[inner product]]. This sum is called a '''Chebyshev series''' or a '''Chebyshev expansion'''.

Since a Chebyshev series is related to a [[Fourier cosine series]] through a change of variables, all of the theorems, identities, etc. that apply to [[Fourier series]] have a Chebyshev counterpart.<ref name=boyd/> These attributes include:

* The Chebyshev polynomials form a [[Complete metric space|complete]] orthogonal system.
* The Chebyshev series converges to {{math|''f''(''x'')}} if the function is [[piecewise]] [[Smooth function|smooth]] and [[Continuous function|continuous]]. The smoothness requirement can be relaxed in most cases{{snd}} as long as there are a finite number of discontinuities in {{math|''f''(''x'')}} and its derivatives.
* At a discontinuity, the series will converge to the average of the right and left limits.

The abundance of the theorems and identities inherited from [[Fourier series]] make the Chebyshev polynomials important tools in [[numeric analysis]]; for example they are the most popular general purpose basis functions used in the [[spectral method]],<ref name=boyd/> often in favor of trigonometric series due to generally faster convergence for continuous functions ([[Gibbs' phenomenon]] is still a problem).

The [[Chebfun]] software package supports function manipulation based on their expansion in the Chebysev basis.

===Example 1===
Consider the Chebyshev expansion of {{math|log(1&thinsp;+&nbsp;''x'')}}. One can express:
<math display="block"> \log(1+x) = \sum_{n = 0}^\infty a_n T_n(x)~. </math>

One can find the coefficients {{math|''a<sub>n</sub>''}} either through the application of an inner product or by the discrete orthogonality condition. For the inner product:
<math display="block">\int_{-1}^{+1}\,\frac{T_m(x)\,\log(1 + x)}{\sqrt{1-x^2}}\,\mathrm{d}x = \sum_{n=0}^{\infty}a_n\int_{-1}^{+1}\frac{T_m(x)\,T_n(x)}{\sqrt{1-x^2}}\,\mathrm{d}x,</math>
which gives:
<math display="block">a_n = \begin{cases}
-\log 2            & \text{ for }~ n = 0, \\
\frac{-2(-1)^n}{n} & \text{ for }~ n > 0.
\end{cases}</math>

Alternatively, when the inner product of the function being approximated cannot be evaluated, the discrete orthogonality condition gives an often useful result for ''approximate'' coefficients:
<math display="block">a_n \approx \frac{\,2-\delta_{0n}\,}{N}\,\sum_{k=0}^{N-1}T_n(x_k)\,\log(1+x_k),</math>
where {{mvar|δ<sub>ij</sub>}} is the [[Kronecker delta]] function and the {{mvar|x<sub>k</sub>}} are the {{mvar|N}} Gauss–Chebyshev zeros of {{math|''T''<sub>''N''&thinsp;</sub>(''x'')}}:
<math display="block"> x_k = \cos\left(\frac{\pi\left(k+\tfrac{1}{2}\right)}{N}\right) .</math>
For any {{mvar|N}}, these approximate coefficients provide an exact approximation to the function at {{mvar|x<sub>k</sub>}} with a controlled error between those points. The exact coefficients are obtained with {{math|1=''N'' = ∞}}, thus representing the function exactly at all points in {{closed-closed|−1,1}}. The rate of convergence depends on the function and its smoothness.

This allows us to compute the approximate coefficients {{mvar|a<sub>n</sub>}} very efficiently through the [[discrete cosine transform]]:
<math display="block">a_n \approx \frac{2-\delta_{0n}}{N}\sum_{k=0}^{N-1}\cos\left(\frac{n\pi\left(\,k+\tfrac{1}{2}\right)}{N}\right)\log(1+x_k).</math>

===Example 2===
To provide another example:
<math display="block">\begin{align}
\left(1-x^2\right)^\alpha &= -\frac{1}{\sqrt{\pi}} \, \frac{\Gamma\left(\tfrac{1}{2} + \alpha\right)}{\Gamma(\alpha+1)} + 2^{1-2\alpha}\,\sum_{n=0} \left(-1\right)^n \, {2 \alpha \choose \alpha-n}\,T_{2n}(x) \\[1ex]
 &= 2^{-2\alpha}\,\sum_{n=0} \left(-1\right)^n \, {2\alpha+1 \choose \alpha-n}\,U_{2n}(x).
\end{align}</math>

===Partial sums===
The partial sums of:
<math display="block">f(x) = \sum_{n = 0}^\infty a_n T_n(x)</math>
are very useful in the [[approximation theory|approximation]] of various functions and in the solution of [[differential equation]]s (see [[spectral method]]). Two common methods for determining the coefficients {{mvar|a<sub>n</sub>}} are through the use of the [[inner product]] as in [[Galerkin's method]] and through the use of [[collocation method|collocation]] which is related to [[interpolation]].

As an interpolant, the {{mvar|N}} coefficients of the {{math|(''N''&nbsp;−&thinsp;1)}}st partial sum are usually obtained on the Chebyshev–Gauss–Lobatto<ref>{{Cite web |url=http://www.scottsarra.org/chebyApprox/chebyshevApprox.html |title=Chebyshev Interpolation: An Interactive Tour |access-date=2016-06-02 |archive-url=https://web.archive.org/web/20170318214311/http://www.scottsarra.org/chebyApprox/chebyshevApprox.html |archive-date=2017-03-18 |url-status=dead }}</ref> points (or Lobatto grid), which results in minimum error and avoids [[Runge's phenomenon]] associated with a uniform grid. This collection of points corresponds to the extrema of the highest order polynomial in the sum, plus the endpoints and is given by:
<math display="block">x_k = -\cos\left(\frac{k \pi}{N - 1}\right); \qquad k = 0, 1, \dots, N - 1.</math>

===Polynomial in Chebyshev form===
An arbitrary polynomial of degree {{mvar|N}} can be written in terms of the Chebyshev polynomials of the first kind.{{sfn|Mason|Handscomb|2002}} Such a polynomial {{math|''p''(''x'')}} is of the form:
<math display="block">p(x) = \sum_{n=0}^N a_n T_n(x).</math>

Polynomials in Chebyshev form can be evaluated using the [[Clenshaw algorithm]].