World Library  
Flag as Inappropriate
Email this Article

Chebyshev rational functions

Article Id: WHEBN0006112758
Reproduction Date:

Title: Chebyshev rational functions  
Author: World Heritage Encyclopedia
Language: English
Subject: Rational functions, Elliptic rational functions, Continued fraction, WikiProject Mathematics/List of mathematics articles (C)
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

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.
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. 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)

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

Contents

  • Properties 1
    • Recursion 1.1
    • Differential equations 1.2
    • Orthogonality 1.3
    • Expansion of an arbitrary function 1.4
  • Particular values 2
  • Partial fraction expansion 3
  • References 4

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

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

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.

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 c_n equals 2 for n = 0 and c_n equals 1 for n \ge 1 and \delta_{nm} is the Kronecker delta function.

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.

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}\,
R_n(x)=\frac{1}{(x+1)^n}\sum_{m=0}^{n} (-1)^m{2n \choose 2m}x^{n-m}\,

Partial fraction expansion

R_n(x)=\sum_{m=0}^{n} \frac{(m!)^2}{(2m)!}{n+m-1 \choose m}{n \choose m}\frac{(-4)^m}{(x+1)^m}

References

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.