Chebyshev rational functions

For the Chebyshev rational functions used in the design of elliptic filters, see Elliptic rational functions.

Plot of the Chebyshev rational functions for n = 0, 1, 2, 3, 4 for 0.01 ≤ x ≤ 100, log scale.

In mathematics, the Chebyshev rational functions are a sequence of functions which are both rational and orthogonal. They are named after Pafnuty Chebyshev. A rational Chebyshev function of degree n is defined as:

$R_{n}(x)\ {\stackrel {\mathrm {def} }{=}}\ T_{n}\left({\frac {x-1}{x+1}}\right)$

where T_n(x) is a Chebyshev polynomial of the first kind.