Chebyshev rational functions

In mathematics, the Chebyshev rational functions are a sequence of functions which are both rational and orthogonal.

A rational Chebyshev function of degree n is defined as: where Tn(x) is a Chebyshev polynomial of the first kind.

Other properties are unique to the functions themselves.

Defining: The orthogonality of the Chebyshev rational functions may be written: where cn = 2 for n = 0 and cn = 1 for n ≥ 1; δnm is the Kronecker delta function.

For an arbitrary function f(x) ∈ L2ω the orthogonality relationship can be used to expand f(x): where

Plot of the Chebyshev rational functions for n = 0, 1, 2, 3, 4 for 0.01 ≤ x ≤ 100 , log scale.
Plot of the absolute value of the seventh-order ( n = 7 ) Chebyshev rational function for 0.01 ≤ x ≤ 100 . Note that there are n zeroes arranged symmetrically about x = 1 and if x 0 is a zero, then ⁠ 1 / x 0 ⁠ is a zero as well. The maximum value between the zeros is unity. These properties hold for all orders.