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:
Many properties can be derived from the properties of the Chebyshev polynomials of the first kind. Other properties are unique to the functions themselves.
Recursion
Differential equations
Orthogonality
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.
Expansion of an arbitrary function
For an arbitrary function f(x) ∈ L2 ω the orthogonality relationship can be used to expand f(x):