Chebyshev nodes

Zeros of the first 50 Chebyshev polynomials of the first kind

For a given positive integer n the Chebyshev nodes in the interval (−1, 1) are

${\displaystyle x_{k}=\cos \left({\frac {2k-1}{2n}}\pi \right),\quad k=1,\ldots ,n.}$

These are the roots of the Chebyshev polynomial of the first kind of degree n. For nodes over an arbitrary interval [a, b] an affine transformation can be used:

${\displaystyle x_{k}={\frac {1}{2}}(a+b)+{\frac {1}{2}}(b-a)\cos \left({\frac {2k-1}{2n}}\pi \right),\quad k=1,\ldots ,n.}$

The Chebyshev nodes are important in approximation theory because they form a particularly good set of nodes for polynomial interpolation. Given a function ƒ on the interval ${\displaystyle [-1,+1]}$ and ${\displaystyle n}$ points ${\displaystyle x_{1},x_{2},\ldots ,x_{n},}$ in that interval, the interpolation polynomial is that unique polynomial ${\displaystyle P_{n-1}}$ of degree at most ${\displaystyle n-1}$ which has value ${\displaystyle f(x_{i})}$ at each point ${\displaystyle x_{i}}$. The interpolation error at ${\displaystyle x}$ is

${\displaystyle f(x)-P_{n-1}(x)={\frac {f^{(n)}(\xi )}{n!}}\prod _{i=1}^{n}(x-x_{i})}$

for some ${\displaystyle \xi }$ (depending on x) in [−1, 1].[3] So it is logical to try to minimize

${\displaystyle \max _{x\in [-1,1]}\left|\prod _{i=1}^{n}(x-x_{i})\right|.}$

This product is a monic polynomial of degree n. It may be shown that the maximum absolute value (maximum norm) of any such polynomial is bounded from below by 2¹⁻ⁿ. This bound is attained by the scaled Chebyshev polynomials 2¹⁻ⁿ T_n, which are also monic. (Recall that |T_n(x)| ≤ 1 for x ∈ [−1, 1].[4]) Therefore, when the interpolation nodes x_i are the roots of T_n, the error satisfies

${\displaystyle \left|f(x)-P_{n-1}(x)\right|\leq {\frac {1}{2^{n-1}n!}}\max _{\xi \in [-1,1]}\left|f^{(n)}(\xi )\right|.}$

For an arbitrary interval [a, b] a change of variable shows that

${\displaystyle \left|f(x)-P_{n-1}(x)\right|\leq {\frac {1}{2^{n-1}n!}}\left({\frac {b-a}{2}}\right)^{n}\max _{\xi \in [a,b]}\left|f^{(n)}(\xi )\right|.}$

• Stewart, Gilbert W. (1996), Afternotes on Numerical Analysis, SIAM, ISBN 978-0-89871-362-6.

• Burden, Richard L.; Faires, J. Douglas: Numerical Analysis, 8th ed., pages 503–512, ISBN 0-534-39200-8.