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| In [[mathematics]], an '''orthogonal polynomial sequence''' is a family of [[polynomial]]s
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| such that any two different polynomials in the sequence are [[orthogonality|orthogonal]] to each other under some [[inner product]].
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| 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 [[Gegenbauer polynomials]], the [[Chebyshev polynomials]], and the [[Legendre polynomials]].
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| The field of orthogonal polynomials developed in the late 19th century from a study of [[continued fraction]]s by [[Pafnuty Chebyshev|P. L. Chebyshev]] and was pursued by [[Andrey Markov|A.A. Markov]] and [[Thomas Joannes Stieltjes|T.J. Stieltjes]]. Some of the mathematicians who have worked on orthogonal polynomials include [[Gábor Szegő]], [[Sergei Natanovich Bernstein|Sergei Bernstein]], [[Naum Akhiezer]], [[Arthur Erdélyi]], [[Yakov Geronimus]], [[Wolfgang Hahn]], [[Theodore Seio Chihara]], [[Mourad Ismail]], [[Waleed Al-Salam]], and [[Richard Askey]].
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| == Definition for 1-variable case for a real measure ==
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| Given any non-decreasing function ''α'' on the real numbers, we can define the [[Lebesgue–Stieltjes integral]]
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| :<math>\int f(x)d\alpha(x)</math>
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| 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
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| :<math>\langle f, g \rangle = \int f(x) g(x) \; d\alpha(x).</math>
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| This operation is a positive semidefinite [[inner product space|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.
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| Then the sequence (''P''<sub>''n''</sub>)<sub>''n''=0</sub><sup>∞</sup> of orthogonal polynomials is defined by the relations
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| :<math> \deg P_n = n~, \quad \langle P_m, \, P_n \rangle = 0 \quad \text{for} \quad m \neq n~.</math> | |
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| In other words, the sequence is obtained from the sequence of monomials 1, ''x'', ''x''<sup>2</sup>, ... by the [[Gram–Schmidt process]] with respect to this inner product.
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| Usually the sequence is required to be [[orthonormal]], namely,
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| :<math> \langle P_n, P_n \rangle = 1~, </math>
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| however, other normalisations are sometimes used.
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| ===Absolutely continuous case===
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| Sometimes we have
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| :<math>\displaystyle d\alpha(x) = W(x)dx</math>
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| where
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| :<math>W : [x_1, x_2] \to \mathbb{R}</math>
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| is a non-negative function with support on some interval [''x''<sub>1</sub>, ''x''<sub>2</sub>] in the real line (where ''x''<sub>1</sub> = −∞ and ''x''<sub>2</sub> = ∞ are allowed). Such a ''W'' is called a '''weight function'''. | |
| Then the inner product is given by
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| :<math>\langle f, g \rangle = \int_{x_1}^{x_2} f(x) g(x) W(x) \; dx.</math>
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| However there are many examples of orthogonal polynomials where the measure dα(''x'') has points with non-zero measure where the function α is discontinuous, so cannot be given by a weight function ''W'' as above.
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| ==Examples of orthogonal polynomials==
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| The most commonly used orthogonal polynomials are orthogonal for a measure with support in a real interval. This includes:
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| *The classical orthogonal polynomials ([[Jacobi polynomials]], [[Laguerre polynomials]], [[Hermite polynomials]], and their special cases [[Gegenbauer polynomials]], [[Chebyshev polynomials]] and [[Legendre polynomials]]).
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| *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]]
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| *The [[Askey–Wilson polynomials]] introduce an extra parameter ''q'' into the Wilson polynomials.
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| [[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]].
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| [[Sieved orthogonal polynomials]], such as the [[sieved ultraspherical polynomials]], [[sieved Jacobi polynomials]], and [[sieved Pollaczek polynomials]], have modified recurrence relations.
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| 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]].
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| 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.
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| ==Properties==
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| Orthogonal polynomials of one variable defined by a non-negative measure on the real line have the following properties.
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| ===Relation to moments===
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| The orthogonal polynomials ''P''<sub>''n''</sub> can be expressed in terms of the [[moment (mathematics)|moments]]
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| :<math> m_n = \int x^n d\alpha(x) </math>
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| as follows:
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| :<math> P_n(x) = c_n \, \det \begin{bmatrix}
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| m_0 & m_1 & m_2 &\cdots & m_n \\
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| m_1 & m_2 & m_3 &\cdots & m_{n+1} \\
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| &&\cdots&& \\
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| m_{n-1} &m_n& m_{n+1} &\cdots &m_{2n-1}\\
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| 1 & x & x^2 & \cdots & x^{n}
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| \end{bmatrix}~,</math>
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| where the constants ''c''<sub>''n''</sub> are arbitrary (depend on the normalisation of ''P''<sub>''n''</sub>).
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| ===Recurrence relation===
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| The polynomials ''P''<sub>''n''</sub> satisfy a recurrence relation of the form
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| :<math> P_n(x) = (A_n x + B_n) P_{n-1}(x) + C_n P_{n-2}(x)~.</math>
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| See [[Favard's theorem]] for a converse result.
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| ===Christoffel–Darboux formula===
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| {{main|Christoffel–Darboux formula}}
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| ===Zeros===
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| If the measure d''α'' is supported on an interval [''a'', ''b''], all the zeros of ''P''<sub>''n''</sub> lie in [''a'', ''b'']. Moreover, the zeros have the following interlacing property: if ''m''>''n'', there is a zero of ''P''<sub>''m''</sub> between any two zeros of ''P''<sub>''n''</sub>.
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| ==Multivariate orthogonal polynomials==
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| 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 non-reduced root system of rank 1.
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| == See also ==
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| * [[Appell sequence]]
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| * [[Askey scheme]] of hypergeometric orthogonal polynomials
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| * [[Binomial type|Polynomial sequences of binomial type]]
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| * [[Biorthogonal polynomials]]
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| * [[Generalized Fourier series]]
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| * [[Secondary measure]]
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| * [[Sheffer sequence]]
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| * [[Umbral calculus]]
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| == References ==
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| * {{Abramowitz_Stegun_ref|22|773}}
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| * {{cite book | first=Theodore Seio|last= Chihara | title= An Introduction to Orthogonal Polynomials | publisher= Gordon and Breach, New York | year=1978 | isbn = 0-677-04150-0}}
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| *{{Cite journal | last1=Chihara | first1=Theodore Seio | title=Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999) | doi=10.1016/S0377-0427(00)00632-4 | mr=1858267 | year=2001 | journal=Journal of Computational and Applied Mathematics | issn=0377-0427 | volume=133 | issue=1 | chapter=45 years of orthogonal polynomials: a view from the wings | pages=13–21 | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}
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| *{{Cite journal | last1=Foncannon | first1=J. J. | title=Review of ''Classical and quantum orthogonal polynomials in one variable'' by Mourad Ismail | publisher=Springer New York | doi=10.1007/BF02985757 | year=2008 | journal=[[The Mathematical Intelligencer]] | issn=0343-6993 | volume=30 | pages=54–60 | last2=Foncannon | first2=J. J. | last3=Pekonen | first3=Osmo | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}
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| *{{cite book | last=Ismail|first=Mourad E. H. | title=Classical and Quantum Orthogonal Polynomials in One Variable | year=2005 | isbn=0-521-78201-5 | url = http://www.cambridge.org/us/catalogue/catalogue.asp?isbn=9780521782012 | publisher=Cambridge Univ. Press | location=Cambridge}}
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| * {{cite book | first=Dunham |last=Jackson | title= Fourier Series and Orthogonal Polynomials | location= New York | publisher=Dover | origyear=1941|year= 2004 | isbn = 0-486-43808-2}}
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| *{{dlmf|id=18|title=Orthogonal Polynomials|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
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| *{{springer|title=Orthogonal polynomials|id=p/o070340}}
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| *{{Cite book | last1=Szegő | first1=Gábor | title=Orthogonal Polynomials | url=http://books.google.com/books?id=3hcW8HBh7gsC | publisher= American Mathematical Society | series=Colloquium Publications | isbn=978-0-8218-1023-1 | mr=0372517 | year=1939 | volume=XXIII | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}
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| *{{cite journal | first= Vilmos|last= Totik | authorlink = Vilmos Totik | year = 2005 | title = Orthogonal Polynomials | journal = Surveys in Approximation Theory | volume = 1 | pages = 70–125 | arxiv = math.CA/0512424}}
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| {{DEFAULTSORT:Orthogonal Polynomials}}
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| [[Category:Articles containing proofs]]
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| [[Category:Orthogonal polynomials| ]]
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