Orthogonal Polynomials
المؤلف:
Abramowitz, M. and Stegun, I. A
المصدر:
"Orthogonal Polynomials." Ch. 22 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover,
الجزء والصفحة:
...
13-2-2019
6629
Orthogonal Polynomials
Orthogonal polynomials are classes of polynomials ![<span style=]()
{p_n(x)}" src="http://mathworld.wolfram.com/images/equations/OrthogonalPolynomials/Inline1.gif" style="height:15px; width:43px" /> defined over a range ![[a,b]](http://mathworld.wolfram.com/images/equations/OrthogonalPolynomials/Inline2.gif)
that obey an orthogonalityrelation
 |
(1)
|
where 
is a weighting function and 
is the Kronecker delta. If 
, then the polynomials are not only orthogonal, but orthonormal.
Orthogonal polynomials have very useful properties in the solution of mathematical and physical problems. Just as Fourier series provide a convenient method of expanding a periodic function in a series of linearly independent terms, orthogonal polynomials provide a natural way to solve, expand, and interpret solutions to many types of important differential equations. Orthogonal polynomials are especially easy to generate using Gram-Schmidt orthonormalization.
A table of common orthogonal polynomials is given below, where 
is the weighting function and
![c_n=int_a^bw(x)[p_n(x)]^2dx](http://mathworld.wolfram.com/images/equations/OrthogonalPolynomials/NumberedEquation2.gif) |
(2)
|
(Abramowitz and Stegun 1972, pp. 774-775).
| polynomial |
interval |
 |
 |
| Chebyshev polynomial of the first kind |
![[-1,1]](http://mathworld.wolfram.com/images/equations/OrthogonalPolynomials/Inline9.gif) |
 |
![<span style=]() {pi for n=0; 1/2pi otherwise" src="http://mathworld.wolfram.com/images/equations/OrthogonalPolynomials/Inline11.gif" style="height:50px; width:106px" /> |
| Chebyshev polynomial of the second kind |
![[-1,1]](http://mathworld.wolfram.com/images/equations/OrthogonalPolynomials/Inline12.gif) |
 |
 |
| Gegenbauer polynomial |
![[-1,1]](http://mathworld.wolfram.com/images/equations/OrthogonalPolynomials/Inline15.gif) |
 |
 |
| Hermite polynomial |
 |
 |
 |
| Jacobi polynomial |
 |
 |
 |
| Laguerre polynomial |
 |
 |
1 |
| generalized Laguerre polynomial |
 |
 |
 |
| Legendre polynomial |
![[-1,1]](http://mathworld.wolfram.com/images/equations/OrthogonalPolynomials/Inline29.gif) |
1 |
 |
In the above table,
 |
(3)
|
where 
is a gamma function.
The roots of orthogonal polynomials possess many rather surprising and useful properties. For instance, let 
be the roots of the 
with 
and 
. Then each interval ![[x_nu,x_(nu+1)]](http://mathworld.wolfram.com/images/equations/OrthogonalPolynomials/Inline36.gif)
for 
, 1, ..., 
contains exactly one root of 
. Between two roots of 
there is at least one root of 
for 
.
Let 
be an arbitrary real constant, then the polynomial
 |
(4)
|
has 
distinct real roots. If 
(
), these roots lie in the interior of ![[a,b]](http://mathworld.wolfram.com/images/equations/OrthogonalPolynomials/Inline47.gif)
, with the exception of the greatest (least) root which lies in ![[a,b]](http://mathworld.wolfram.com/images/equations/OrthogonalPolynomials/Inline48.gif)
only for
 |
(5)
|
The following decomposition into partial fractions holds
 |
(6)
|
where ![<span style=]()
{xi_nu}" src="http://mathworld.wolfram.com/images/equations/OrthogonalPolynomials/Inline49.gif" style="height:15px; width:23px" /> are the roots of 
and
Another interesting property is obtained by letting ![<span style=]()
{p_n(x)}" src="http://mathworld.wolfram.com/images/equations/OrthogonalPolynomials/Inline57.gif" style="height:15px; width:43px" /> be the orthonormal set of polynomials associated with the distribution 
on ![[a,b]](http://mathworld.wolfram.com/images/equations/OrthogonalPolynomials/Inline59.gif)
. Then the convergents 
of the continued fraction
 |
(9)
|
are given by
where 
, 1, ... and
 |
(13)
|
Furthermore, the roots of the orthogonal polynomials 
associated with the distribution 
on the interval ![[a,b]](http://mathworld.wolfram.com/images/equations/OrthogonalPolynomials/Inline73.gif)
are real and distinct and are located in the interior of the interval ![[a,b]](http://mathworld.wolfram.com/images/equations/OrthogonalPolynomials/Inline74.gif)
.
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). "Orthogonal Polynomials." Ch. 22 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771-802, 1972.
Arfken, G. "Orthogonal Polynomials." Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 520-521, 1985.
Chihara, T. S. An Introduction to Orthogonal Polynomials. New York: Gordon and Breach, 1978.
Gautschi, W.; Golub, G. H.; and Opfer, G. (Eds.) Applications and Computation of Orthogonal Polynomials, Conference at the Mathematical Research Institute Oberwolfach, Germany, March 22-28, 1998. Basel, Switzerland: Birkhäuser, 1999.
Iyanaga, S. and Kawada, Y. (Eds.). "Systems of Orthogonal Functions." Appendix A, Table 20 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1477, 1980.
Koekoek, R. and Swarttouw, R. F. The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its 
-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, 1-168, 1998.
Nikiforov, A. F.; Uvarov, V. B.; and Suslov, S. S. Classical Orthogonal Polynomials of a Discrete Variable. New York: Springer-Verlag, 1992.
Sansone, G. Orthogonal Functions. New York: Dover, 1991.
Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 44-47 and 54-55, 1975.
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