| 释义 |
Orthogonal PolynomialsOrthogonal polynomials are classes of Polynomials  over a range  which obey anOrthogonality relation
 | (1) |
 is a Weighting Function and  is the Kronecker Delta. If  , then the Polynomialsare not only orthogonal, but orthonormal.
Orthogonal polynomials have very useful properties in the solution of mathematical and physical problems. Just asFourier 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 importantDifferential Equations. Orthogonal polynomials are especially easy to generate usingGram-Schmidt Orthonormalization. Abramowitz and Stegun (1972, pp. 774-775) give a table of common orthogonalpolynomials.
| Type | Interval | |  | | Chebyshev Polynomial of the First Kind |  |  |  | | Chebyshev Polynomial of the Second Kind | |  |  | | Hermite Polynomial |  |  |  | | Jacobi Polynomial |  |  |  | | Laguerre Polynomial |  |  | 1 | | Laguerre Polynomial (Associated) | |  |  | | Legendre Polynomial | | 1 |  | | Ultraspherical Polynomial | |  |  |
In the above table, the normalization constant is the value of
 | (2) |
 | (3) |
 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  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) |
 distinct Real Roots. If  ( ), these Roots lie in the interior of , with the exception of the greatest (least) Root which lies in only for
 | (5) |
The following decomposition into partial fractions holds
 | (6) |
 are the Roots of and
Another interesting property is obtained by letting be the orthonormal set of Polynomialsassociated with the distribution  on . Then the Convergents  of theContinued Fraction
 | (8) |
where  , 1, ...and
 | (11) |
References
Abramowitz, M. and Stegun, C. 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. 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. 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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