 Orthogonal polynomials

In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.
The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the Legendre polynomials.
The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by P. L. Chebyshev and was pursued by A.A. Markov and T.J. Stieltjes. Some of the mathematicians who have worked on orthogonal polynomials include Gábor Szegő, Sergei Bernstein, Naum Akhiezer, Arthur Erdélyi, Yakov Geronimus, Wolfgang Hahn, Theodore Seio Chihara, Mourad Ismail, Waleed AlSalam, and Richard Askey.
Contents
Definition for 1variable case for a real measure
Given any nondecreasing function α on the real numbers, we can define the Lebesgue–Stieltjes integral
of a function f. If this integral is finite for all polynomials f, we can define an inner product on pairs of polynomials f and g by
This operation is a positive semidefinite inner product on the vector space of all polynomials, and is positive definite if the function α has an infinite number of points of growth. It induces a notion of orthogonality in the usual way, namely that two polynomials are orthogonal if their inner product is zero.
Then the sequence (P_{n})_{n=0}^{∞} of orthogonal polynomials is defined by the relations
In other words, obtained from the sequence of monomials 1, x, x^{2}, ... by the Gram–Schmidt process.
Usually the sequence is required to be orthonormal, namely,
however, other normalisations are sometimes used.
Absolutely continuous case
Sometimes we have
where
is a nonnegative function with support on some interval [x_{1},x_{2}] in the real line (where and are allowed). Such a W is called a weight function. Then the inner product is given by
However there are many examples of orthogonal polynomials where the measure dα(x) has points with nonzero measure where the function α is discontinuous, so cannot be given by a weight function W as above.
Examples of orthogonal polynomials
The most commonly used orthogonal polynomials are orthogonal for a measure with support in a real interval. This includes:
 The classical orthogonal polynomials (Jacobi polynomials, Laguerre polynomials, Hermite polynomials, and their special cases ultraspherical polynomials, Chebyshev polynomials and Legendre polynomials).
 The Wilson polynomials, which generalize the Jacobi polynomials. They include many orthogonal polynomials as special cases, such as the Meixner–Pollaczek polynomials, the continuous Hahn polynomials, the continuous dual Hahn polynomials, and the classical polynomials, described by the Askey scheme
 The Askey–Wilson polynomials introduce an extra parameter q into the Wilson polynomials.
Discrete orthogonal polynomials are orthogonal with respect to some discrete measure. Sometimes the measure has finite support, in which case the family of orthogonal polynomials is finite, rather than an infinite sequence. The Racah polynomials are examples of discrete orthogonal polynomials, and include as special cases the Hahn polynomials and dual Hahn polynomials, which in turn include as special cases the Meixner polynomials, Krawtchouk polynomials, and Charlier polynomials.
Sieved orthogonal polynomials, such as the sieved ultraspherical polynomials, sieved Jacobi polynomials, and sieved Pollaczek polynomials, have modified recurrence relations.
One can also consider orthogonal polynomials for some curve in the complex plane. The most important case (other than real intervals) is when the curve is the unit circle, giving orthogonal polynomials on the unit circle, such as the Rogers–Szegő polynomials.
There are some families of orthogonal polynomials that are orthogonal on plane regions such as triangles or disks. They can sometimes be written in terms of Jacobi polynomials. For example, Zernike polynomials are orthogonal on the unit disk.
Properties
Orthogonal polynomials of one variable defined by a nonnegative measure on the real line have the following properties.
Relation to moments
The orthogonal polynomials P_{n} can be expressed in terms of the moments
as follows:
where the constants c_{n} are arbitrary (depend on the normalisation of P_{n}).
Recurrent relation
The polynomials P_{n} satisfy a recurrent relation of the form
See Favard's theorem for a converse result.
Christoffel–Darboux formula
Main article: Christoffel–Darboux formulaZeros
If the measure dα is supported on an interval [a, b], all the zeros of P_{n} lie in [a, b]. Moreover, the zeros have the following interlacing property: if m>n, there is a zero of P_{m} between any two zeros of P_{n}.
Multivariate orthogonal polynomials
The Macdonald polynomials are orthogonal polynomials in several variables, depending on the choice of an affine root system. They include many other families of multivariable orthogonal polynomials as special cases, including the Jack polynomials, the Hall–Littlewood polynomials, the Heckman–Opdam polynomials, and the Koornwinder polynomials. The Askey–Wilson polynomials are the special case of Macdonald polynomials for a certain nonreduced root system of rank 1.
See also
 Appell sequence
 Askey scheme of hypergeometric orthogonal polynomials
 Polynomial sequences of binomial type
 Biorthogonal polynomials
 Generalized Fourier series
 Secondary measure
 Sheffer sequence
 Umbral calculus
References
 Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 22", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, pp. 773, ISBN 9780486612720, MR0167642, http://www.math.sfu.ca/~cbm/aands/page_773.htm.
 Chihara, Theodore Seio (1978). An Introduction to Orthogonal Polynomials. Gordon and Breach, New York. ISBN 0677041500.
 Chihara, Theodore Seio (2001), 45 years of orthogonal polynomials: a view from the wings, "Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999)", Journal of Computational and Applied Mathematics 133 (1): 13–21, doi:10.1016/S03770427(00)006324, ISSN 03770427, MR1858267
 Foncannon, J. J. (2008), "Review of Classical and quantum orthogonal polynomials in one variable by Mourad Ismail", The Mathematical Intelligencer (Springer New York) 30: 54–60, doi:10.1007/BF02985757, ISSN 03436993
 Ismail, Mourad E. H. (2005). Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge: Cambridge Univ. Press. ISBN 0521782015. http://www.cambridge.org/us/catalogue/catalogue.asp?isbn=9780521782012.
 Jackson, Dunham (2004) [1941]. Fourier Series and Orthogonal Polynomials. New York: Dover. ISBN 0486438082.
 Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Orthogonal Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 9780521192255, MR2723248, http://dlmf.nist.gov/18
 Suetin, P. K. (2001), "Orthogonal polynomials", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 9781556080104, http://eom.springer.de/O/o070340.htm
 Szegő, Gábor (1939), Orthogonal Polynomials, Colloquium Publications, XXIII, American Mathematical Society, ISBN 9780821810231, MR0372517, http://books.google.com/books?id=3hcW8HBh7gsC
 Totik, Vilmos (2005). "Orthogonal Polynomials". Surveys in Approximation Theory 1: 70–125. arXiv:math.CA/0512424.
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