Chebyshev nodes

In numerical analysis, Chebyshev nodes are a set of specific real algebraic numbers, used as nodes for polynomial interpolation.

They are the projection of equispaced points on the unit circle onto the real interval

Both of these sets of numbers are commonly referred to as Chebyshev nodes in literature.

[1] Polynomial interpolants constructed from these nodes minimize the effect of Runge's phenomenon.

the Chebyshev nodes of the first kind in the open interval

These are the roots of the Chebyshev polynomials of the first kind with degree

For nodes over an arbitrary interval

Similarly, for a given positive integer

the Chebyshev nodes of the second kind in the closed interval

These are the roots of the Chebyshev polynomials of the second kind with degree

For nodes over an arbitrary interval

The Chebyshev nodes of the second kind are also referred to as Chebyshev-Lobatto points or Chebyshev extreme points.

[3] Note that the Chebyshev nodes of the second kind include the end points of the interval while the Chebyshev nodes of the first kind do not include the end points.

These formulas generate Chebyshev nodes which are sorted from greatest to least on the real interval.

Both kinds of nodes are always symmetric about the midpoint of the interval.

, both kinds of nodes will include the midpoint.

Geometrically, for both kinds of nodes, we first place

points on the upper half of the unit circle with equal spacing between them.

-axis are called Chebyshev nodes.

The Chebyshev nodes are important in approximation theory because they form a particularly good set of nodes for polynomial interpolation.

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 Tn, which are also monic.

[5]) Therefore, when the interpolation nodes xi are the roots of Tn, the error satisfies

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

Many applications for Chebyshev nodes, such as the design of equally terminated passive Chebyshev filters, cannot use Chebyshev nodes directly, due to the lack of a root at 0.

However, the Chebyshev nodes may be modified into a usable form by translating the roots down such that the lowest roots are moved to zero, thereby creating two roots at zero of the modified Chebyshev nodes.

[6] The even order modification translation is:

function is chosen to be the same as the sign of

Running all the nodes through the translation yields

The modified even order Chebyshev nodes now contains two nodes of zero, and is suitable for use in designing even order Chebyshev filters with equally terminated passive element networks.