Chebyshev rational functions

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This article is not about the Chebyshev rational functions used in the design of elliptic filters. For these functions, see Elliptic rational functions.
Plot of the Chebyshev rational functions for n=0,1,2,3 and 4 for x between 0.01 and 100.
Plot of the Chebyshev rational functions for n=0,1,2,3 and 4 for x between 0.01 and 100.

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:

R_n(x)\ \stackrel{\mathrm{def}}{=}\ T_n\left(\frac{x-1}{x+1}\right)

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

Contents

[edit] Properties

Many properties can be derived from the properties of the Chebyshev polynomials of the first kind. Other properties are unique to the functions themselves.

[edit] Recursion

R_{n+1}(x)=2\,\frac{x-1}{x+1}R_n(x)-R_{n-1}(x)\quad\mathrm{for\,n\ge 1}

[edit] Differential equations

(x+1)^2R_n(x)=\frac{1}{n+1}\frac{d}{dx}\,R_{n+1}(x)-\frac{1}{n-1}\frac{d}{dx}\,R_{n-1}(x) \quad\mathrm{for\,n\ge 2}
(x+1)^2x\frac{d^2}{dx^2}\,R_n(x)+\frac{(3x+1)(x+1)}{2}\frac{d}{dx}\,R_n(x)+n^2R_{n}(x) = 0

[edit] Orthogonality

Plot of the absolute value of the seventh order (n=7) Chebyshev rational function for x between 0.01 and 100. Note that there are n zeroes arranged symmetrically about x=1 and if x0 is a zero, then 1/x0 is a zero as well. The maximum value between the zeros is unity. These properties hold for all orders.
Plot of the absolute value of the seventh order (n=7) Chebyshev rational function for x between 0.01 and 100. Note that there are n zeroes arranged symmetrically about x=1 and if x0 is a zero, then 1/x0 is a zero as well. The maximum value between the zeros is unity. These properties hold for all orders.

Defining:

\omega(x) \ \stackrel{\mathrm{def}}{=}\ \frac{1}{(x+1)\sqrt{x}}

The orthogonality of the Chebyshev rational functions may be written:

\int_{0}^\infty R_m(x)\,R_n(x)\,\omega(x)\,dx=\frac{\pi c_n}{2}\delta_{nm}

where cn equals 2 for n=0 and cn equals 1 for n \ge 1 and δnm is the Kronecker delta function.

[edit] Expansion of an arbitrary function

For an arbitrary function f(x)\in L_\omega^2 the orthogonality relationship can be used to expand f(x):

f(x)=\sum_{n=0}^\infty F_n R_n(x)

where

F_n=\frac{2}{c_n\pi}\int_{0}^\infty f(x)R_n(x)\omega(x)\,dx

[edit] Particular values

R_0(x)=1\,
R_1(x)=\frac{x-1}{x+1}\,
R_2(x)=\frac{x^2-6x+1}{(x+1)^2}\,
R_3(x)=\frac{x^3-15x^2+15x-1}{(x+1)^3}\,
R_4(x)=\frac{x^4-28x^3+70x^2-28x+1}{(x+1)^4}\,

[edit] References

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.