Editing Chebyshev polynomials (section)

===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>