Chebyshev 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:

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

Plot of the Chebyshev rational functions for

n = 0, 1, 2, 3, 4

for

0.01 \leq x \leq 100

, log scale.

where $T n (x)$ is a Chebyshev polynomial of the first kind.

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.

Recursion

R_{n+1}(x)=2\left({\frac {x-1}{x+1}}\right)R_{n}(x)-R_{n-1}(x)\quad {\text{for}}\,n\geq 1

Differential equations

(x+1)^{2}R_{n}(x)={\frac {1}{n+1}}{\frac {\mathrm {d} }{\mathrm {d} x}}R_{n+1}(x)-{\frac {1}{n-1}}{\frac {\mathrm {d} }{\mathrm {d} x}}R_{n-1}(x)\quad {\text{for }}n\geq 2

(x+1)^{2}x{\frac {\mathrm {d} ^{2}}{\mathrm {d} x^{2}}}R_{n}(x)+{\frac {(3x+1)(x+1)}{2}}{\frac {\mathrm {d} }{\mathrm {d} x}}R_{n}(x)+n^{2}R_{n}(x)=0

Orthogonality

Plot of the absolute value of the seventh-order (

n = 7

) Chebyshev rational function for

0.01 \leq x \leq 100

. Note that there are

n

zeroes arranged symmetrically about

x = 1

and if

x 0

is a zero, then

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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)\,\mathrm {d} x={\frac {\pi c_{n}}{2}}\delta _{nm}

where $c n = 2$ for $n = 0$ and $c n = 1$ for $n \geq 1$ ; $δ nm$ is the Kronecker delta function.

Expansion of an arbitrary function

For an arbitrary function $f (x) \in L 2 ω$ 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)\,\mathrm {d} x.

Particular values

{\begin{aligned}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}}}\\R_{n}(x)&=(x+1)^{-n}\sum _{m=0}^{n}(-1)^{m}{\binom {2n}{2m}}x^{n-m}\end{aligned}}

Partial fraction expansion

R_{n}(x)=\sum _{m=0}^{n}{\frac {(m!)^{2}}{(2m)!}}{\binom {n+m-1}{m}}{\binom {n}{m}}{\frac {(-4)^{m}}{(x+1)^{m}}}

References

Guo, Ben-Yu; Shen, Jie; Wang, Zhong-Qing (2002). "Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval" (PDF). Int. J. Numer. Methods Eng. 53 (1): 65–84. Bibcode:2002IJNME..53...65G. CiteSeerX 10.1.1.121.6069. doi:10.1002/nme.392. S2CID 9208244. Retrieved 2006-07-25.

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