Editing Chebyshev nodes (section)

==Approximation==

The Chebyshev nodes are important in [[approximation theory]] because they form a particularly good set of nodes for [[polynomial interpolation]]. Given a function {{math|''f''}} on the interval <math>[-1,+1]</math> and <math>n</math> points <math>x_1,  x_2, \ldots , x_n,</math> in that interval, the interpolation polynomial is that unique polynomial <math>P_{n-1}</math> of degree at most <math>n - 1</math> which has value <math>f(x_i)</math> at each point <math>x_i</math>. The interpolation error at <math>x</math> is
<math display="block">f(x) - P_{n-1}(x) = \frac{f^{(n)}(\xi)}{n!} \prod_{i=1}^n (x-x_i) </math>
for some <math>\xi</math> (depending on {{mvar|x}}) in {{closed-closed|−1, 1}}.<ref>{{harvnb|Stewart|1996|loc= (20.3)}}</ref> So it is logical to try to minimize
<math display="block">\max_{x \in [-1,1]} \biggl| \prod_{i=1}^n (x-x_i) \biggr|. </math>

This product is a ''[[monic polynomial|monic]]'' polynomial of degree {{mvar|n}}.  It may be shown that the maximum absolute value (maximum norm) of any such polynomial is bounded from below by {{math|2<sup>1−''n''</sup>}}.  This bound is attained by the scaled Chebyshev polynomials {{math|2<sup>1−''n''</sup> ''T''<sub>''n''</sub>}}, which are also monic. (Recall that {{math|{{!}}''T''<sub>''n''</sub>(''x''){{!}} ≤ 1}} for {{math|''x'' ∈ [−1, 1]}}.<ref>{{harvnb|Stewart|1996|loc= Lecture 20, §14}}</ref>) Therefore, when the interpolation nodes {{math|''x''<sub>''i''</sub>}} are the roots of {{math|''T''<sub>''n''</sub>}}, the error satisfies
<math display="block">\left|f(x) - P_{n-1}(x)\right| \le \frac{1}{2^{n - 1}n!} \max_{\xi \in [-1,1]} \left| f^{(n)} (\xi) \right|.</math>
For an arbitrary interval [''a'', ''b''] a change of variable shows that
<math display="block">\left|f(x) - P_{n-1}(x)\right| \le \frac{1}{2^{n - 1}n!} \left(\frac{b-a}{2}\right)^n \max_{\xi \in [a,b]} \left|f^{(n)} (\xi)\right|.</math>