Chebyshev rational functions
- This article is not about the Chebyshev rational functions used in the design of elliptic filters. For those functions, see Elliptic rational functions.
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:
where is a Chebyshev polynomial of the first kind.
Many properties can be derived from the properties of the Chebyshev polynomials of the first kind. Other properties are unique to the functions themselves.
The orthogonality of the Chebyshev rational functions may be written:
where equals 2 for n = 0 and equals 1 for and is the Kronecker delta function.
Expansion of an arbitrary function
For an arbitrary function the orthogonality relationship can be used to expand :
Partial fraction expansion
- Ben-Yu, Guo; Jie, Shen; Zhong-Quing, Wang (2002). "Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval" (PDF). Int. J. Numer. Meth. Engng 53: 65–84. doi:10.1002/nme.392. Retrieved 2006-07-25.