View source for Orthogonal polynomials

{{Short description|Set of polynomials where any two are orthogonal to each other}}
{{Use American English|date = March 2019}}
In [[mathematics]], an '''orthogonal polynomial sequence''' is a family of  [[polynomial]]s such that any two different polynomials in the sequence are [[orthogonality|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]] and the [[Jacobi polynomials]]. The [[Gegenbauer polynomials]] form the most important class of Jacobi polynomials; they include the [[Chebyshev polynomials]], and the [[Legendre polynomials]] as special cases.  These are frequently given by the [[Rodrigues' formula]].

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]]. They appear in a wide variety of fields: [[numerical analysis]] ([[Gaussian quadrature|quadrature rules]]), [[probability theory]], [[representation theory]] (of [[Lie group|Lie groups]], [[quantum group|quantum groups]], and related objects), [[enumerative combinatorics]], [[algebraic combinatorics]], [[mathematical physics]] (the theory of [[random matrix|random matrices]], [[integrable system|integrable systems]], etc.), and [[number theory]]. 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]], [[Richard Askey]], and [[Rehuel Lobatto]].

== Definition for 1-variable case for a real measure ==

Given any non-decreasing function {{math|''α''}} on the real numbers, we can define the [[Lebesgue–Stieltjes integral]] 
<math display="block">\int f(x) \, d\alpha(x)</math>
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
<math display="block">\langle f, g \rangle = \int f(x) g(x) \, d\alpha(x).</math>

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.

Then the sequence {{math|(''P''<sub>''n''</sub>){{su|b=''n''=0|p=∞}}}} of orthogonal polynomials is defined by the relations
<math display="block"> \deg P_n = n~, \quad \langle P_m, \, P_n \rangle = 0 \quad \text{for} \quad m \neq n~.</math>

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.

Usually the sequence is required to be [[orthonormal]], namely,
<math display="block"> \langle P_n, P_n \rangle = 1 , </math>
however, other normalisations are sometimes used.

===Absolutely continuous case===

Sometimes we have 
<math display="block"> d\alpha(x) = W(x) \, dx</math>
where 
<math display="block">W : [x_1, x_2] \to \R</math>
is a non-negative function with support on some interval {{math|[''x''<sub>1</sub>, ''x''<sub>2</sub>]}} in the real line (where {{math|1=''x''<sub>1</sub>&nbsp;=&nbsp;−∞}} and {{math|1=''x''<sub>2</sub> = ∞}} are allowed). Such a {{math|''W''}} is called a '''weight function'''.<ref>[https://demonstrations.wolfram.com/OrthonormalPolynomialsUnderDifferentInnerProductMeasures/ Demo of orthonormal polynomials obtained for different weight functions]</ref> Then the inner product is given by
<math display="block">\langle f, g \rangle = \int_{x_1}^{x_2} f(x) g(x) W(x) \, dx.</math>
However, there are many examples of orthogonal polynomials where the measure {{math|''dα''(''x'')}} has points with non-zero measure where the function {{math|''α''}} is discontinuous, so cannot be given by a weight function {{math|''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 [[Gegenbauer 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]].

Meixner classified all the orthogonal [[Sheffer sequence|Sheffer sequences]]: there are only Hermite, Laguerre, Charlier, Meixner, and Meixner–Pollaczek. In some sense Krawtchouk should be on this list too, but they are a finite sequence. These six families correspond to the [[Natural_exponential_family#The_six_NEF-QVFs|NEF-QVFs]] and are [[Martingale_(probability_theory)|martingale]] polynomials for certain [[Lévy process|Lévy processes]].

[[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.

The advantage of orthogonality between different orders of [[Hermite polynomials]] is applied to Generalized frequency division multiplexing (GFDM) structure. More than one symbol can be carried in each grid of time-frequency lattice.<ref>{{cite book |last1=Catak |first1=E. |last2=Durak-Ata |first2=L. |title=2017 IEEE International Black Sea Conference on Communications and Networking (BlackSeaCom) |chapter=An efficient transceiver design for superimposed waveforms with orthogonal polynomials |date=2017 |pages=1–5 |doi=10.1109/BlackSeaCom.2017.8277657 |isbn=978-1-5090-5049-9 |s2cid=22592277 }}</ref>

==Properties==

Orthogonal polynomials of one variable defined by a non-negative measure on the real line have the following properties.

===Relation to moments===

The orthogonal polynomials ''P''<sub>''n''</sub> can be expressed in terms of the [[moment (mathematics)|moments]]

:<math> m_n = \int x^n \, d\alpha(x) </math>

as follows:

:<math> P_n(x) = c_n \, \det \begin{bmatrix}
m_0 & m_1 &  m_2 &\cdots & m_n \\
m_1 & m_2 &  m_3 &\cdots & m_{n+1} \\
\vdots&\vdots&\vdots&\ddots& \vdots \\
m_{n-1} &m_n& m_{n+1} &\cdots &m_{2n-1}\\
1 & x & x^2 & \cdots & x^n
\end{bmatrix}~,</math>

where the constants ''c''<sub>''n''</sub> are arbitrary (depend on the normalization of ''P''<sub>''n''</sub>).

This comes directly from applying the Gram–Schmidt process to the monomials, imposing each polynomial to be orthogonal with respect to the previous ones. For example, orthogonality with <math>P_0</math> prescribes that <math>P_1</math> must have the form<math display="block">P_1(x) = c_1 \left(x- \frac{\langle P_0, x\rangle P_0}{\langle P_0,P_0\rangle} \right)
= c_1 ( x - m_1),</math>which can be seen to be consistent with the previously given expression with the determinant.

===Recurrence relation===

The polynomials ''P''<sub>''n''</sub> satisfy a recurrence relation of the form
:<math> P_n(x) = (A_n x + B_n) P_{n-1}(x) + C_n P_{n-2}(x)</math>
where ''A<sub>n</sub>'' is not 0.  The converse is also true; see [[Favard's theorem]].

===Christoffel–Darboux formula===

{{main|Christoffel–Darboux formula}}

===Zeros===

If the measure d''&alpha;'' is supported on an interval [''a'',&nbsp;''b''], all the zeros of ''P''<sub>''n''</sub> lie in [''a'',&nbsp;''b'']. Moreover, the zeros have the following interlacing property: if ''m''&nbsp;&lt;&nbsp;''n'', there is a zero of ''P''<sub>''n''</sub> between any two zeros of&nbsp;''P''<sub>''m''</sub>. [[Electrostatic]] interpretations of the zeros can be given.{{Citation needed|date=December 2021}}

===Combinatorial interpretation===

From the 1980s, with the work of X. G. Viennot, J. Labelle, Y.-N. Yeh, D. Foata, and others, combinatorial interpretations were found for all the classical orthogonal polynomials. <ref>{{cite web |last1=Viennot |first1=Xavier |url=https://viennot.org/abjc4-ch5.html |title=The Art of Bijective Combinatorics, Part IV, Combinatorial theory of orthogonal polynomials and continued fractions. |date=2017 |publisher=IMSc |location=Chennai  }}</ref>

==Other types of orthogonal polynomials==
===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 non-reduced root system of rank 1.
===Multiple orthogonal polynomials===
{{main|Multiple orthogonal polynomials}}
Multiple orthogonal polynomials are polynomials in one variable that are orthogonal with respect to a finite family of measures.
===Sobolev orthogonal polynomials===
{{main|Sobolev orthogonal polynomials}}
These are orthogonal polynomials with respect to a [[Sobolev space|Sobolev]] inner product, i.e. an inner product with derivatives. Including derivatives has big consequences for the polynomials, in general they no longer share some of the nice features of the classical orthogonal polynomials.
===Orthogonal polynomials with matrices===
Orthogonal polynomials with matrices have either coefficients that are matrices or the indeterminate is a matrix.

There are two popular examples: either the coefficients <math>\{a_i\}</math> are matrices or <math>x</math>:
* Variante 1: <math>P(x)=A_nx^n+A_{n-1}x^{n-1}+\cdots + A_1x +A_0</math>, where <math>\{A_{i}\}</math> are <math>p\times p</math> matrices.
* Variante 2: <math>P(X)=a_nX^n+a_{n-1}X^{n-1}+\cdots + a_1X +a_0I_p</math> where <math>X</math> is a <math>p\times p</math>-matrix and <math>I_p</math> is the identity matrix.

===Quantum polynomials===
Quantum polynomials or q-polynomials are the [[q-analog]]s of orthogonal polynomials.

== See also ==

* [[Appell sequence]]
* [[Askey scheme]] of hypergeometric orthogonal polynomials
* [[Favard's theorem]]
* [[Binomial type|Polynomial sequences of binomial type]]
* [[Biorthogonal polynomials]]
* [[Generalized Fourier series]]
* [[Secondary measure]]
* [[Sheffer sequence]]
* [[Sturm–Liouville theory]]
* [[Umbral calculus]]
* [[Plancherel–Rotach asymptotics]]

== References ==
{{reflist}}

* {{Abramowitz_Stegun_ref|22|773}}
* {{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}}
*{{Cite journal | last1=Chihara | first1=Theodore Seio | department=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 | title=45 years of orthogonal polynomials: a view from the wings | pages=13–21| bibcode=2001JCoAM.133...13C | doi-access=free }}
*{{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| s2cid=118133026 }}
*{{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}}
* {{cite book | first=Dunham |last=Jackson | title= Fourier Series and Orthogonal Polynomials | location= New York | publisher=Dover | orig-year=1941|year= 2004 | isbn = 0-486-43808-2}}
*{{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}}
*{{springer|title=Orthogonal polynomials|id=p/o070340}}
*{{Cite book | last1=Szegő | first1=Gábor | title=Orthogonal Polynomials | url=https://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 }}
*{{cite journal | first=  Vilmos|last= Totik | author-link = Vilmos Totik | year = 2005 | title = Orthogonal Polynomials | journal = Surveys in Approximation Theory | volume = 1 | pages = 70–125 | arxiv = math.CA/0512424}}
* C. Chan, A. Mironov, A. Morozov, A. Sleptsov, {{arXiv|1712.03155}}.
* Herbert Stahl and Vilmos Totik: General Orthogonal Polynomials, Cambridge Univ. Press, ISBN 978-0-521-41534-7 (1992).
* G. Sansone: Orthogonal Functions, (Revised English Edition), Dover, ISBN 978-0-486-77730-0 (1991).
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