§18.38 Mathematical Applications
Contents
- §18.38(i) Classical OP’s: Numerical Analysis
- §18.38(ii) Classical OP’s: Other Applications
- §18.38(iii) Other OP’s
§18.38(i) Classical OP’s: Numerical Analysis
¶ Approximation Theory
The scaled Chebyshev polynomial ,
, enjoys
the “minimax” property on the interval
, that is,
has the least maximum value among all monic
polynomials of degree
. In consequence, expansions of functions that are
infinitely differentiable on
in series of Chebyshev polynomials
usually converge extremely rapidly. For these results and applications in
approximation theory see §3.11(ii) and
Mason and Handscomb (2003, Chapter 3), Cheney (1982, p. 108), and
Rivlin (1969, p. 31).
¶ Quadrature
Classical OP’s play a fundamental role in Gaussian quadrature. If the nodes in
a quadrature formula with a positive weight function are chosen to be the
zeros of the th degree OP with the same weight function, and the interval of
orthogonality is the same as the integration range, then the weights in the
quadrature formula can be chosen in such a way that the formula is exact
for all polynomials of degree not exceeding
. See §3.5(v).
¶ Differential Equations
Linear ordinary differential equations can be solved directly in series of Chebyshev polynomials (or other OP’s) by a method originated by Clenshaw (1957). This process has been generalized to spectral methods for solving partial differential equations. For further information see Mason and Handscomb (2003, Chapters 10 and 11), Gottlieb and Orszag (1977, pp. 7–19), and Guo (1998, pp. 120–151).
§18.38(ii) Classical OP’s: Other Applications
¶ Integrable Systems
The Toda equation provides an important model of a completely integrable
system. It has elegant structures, including -soliton solutions, Lax pairs,
and Bäcklund transformations. While the Toda equation is an important model
of nonlinear systems, the special functions of mathematical physics are usually
regarded as solutions to linear equations. However, by using Hirota’s technique
of bilinear formalism of soliton theory, Nakamura (1996) shows
that a wide class of exact solutions of the Toda equation can be expressed in
terms of various special functions, and in particular classical OP’s. For
instance,
![V_{n}(x)=\ifrac{2n\mathop{H_{{n+1}}\/}\nolimits\!\left(x\right)\mathop{H_{{n-1}}\/}\nolimits\!\left(x\right)}{(\mathop{H_{{n}}\/}\nolimits\!\left(x\right))^{2}},](https://webarchive.library.unt.edu/web/20130304033535im_/http://dlmf.nist.gov/18/38/E1.png)
with as in §18.3, satisfies the Toda equation
![(\ifrac{{d}^{2}}{{dx}^{2}})\mathop{\ln\/}\nolimits V_{n}(x)=V_{{n+1}}(x)+V_{{n-1}}(x)-2V_{n}(x),](https://webarchive.library.unt.edu/web/20130304033535im_/http://dlmf.nist.gov/18/38/E2.png)
![n=1,2,\dots](https://webarchive.library.unt.edu/web/20130304033535im_/http://dlmf.nist.gov/18/38/m16.png)
¶ Complex Function Theory
The Askey–Gasper inequality
![\sum _{{m=0}}^{n}\mathop{P^{{(\alpha,0)}}_{{m}}\/}\nolimits\!\left(x\right)\geq 0,](https://webarchive.library.unt.edu/web/20130304033535im_/http://dlmf.nist.gov/18/38/E3.png)
![-1\leq x\leq 1](https://webarchive.library.unt.edu/web/20130304033535im_/http://dlmf.nist.gov/18/38/m1.png)
![\alpha>-1](https://webarchive.library.unt.edu/web/20130304033535im_/http://dlmf.nist.gov/18/38/m6.png)
![n=0,1,\dots](https://webarchive.library.unt.edu/web/20130304033535im_/http://dlmf.nist.gov/18/38/m15.png)
was used in de Branges’ proof of the long-standing Bieberbach conjecture concerning univalent functions on the unit disk in the complex plane. See de Branges (1985).
¶ Zonal Spherical Harmonics
¶ Random Matrix Theory
¶ Radon Transform
See Deans (1983, Chapters 4, 7).