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Orthogonal polynomials

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Set of polynomials where any two are orthogonal to each other

Inmathematics, anorthogonal polynomial sequence is a family ofpolynomials such that any two different polynomials in the sequence areorthogonal to each other under someinner product.

The most widely used orthogonal polynomials are theclassical orthogonal polynomials, consisting of theHermite polynomials, theLaguerre polynomials and theJacobi polynomials. TheGegenbauer polynomials form the most important class of Jacobi polynomials; they include theChebyshev polynomials, and theLegendre polynomials as special cases. These are frequently given by theRodrigues' formula.

The field of orthogonal polynomials developed in the late 19th century from a study ofcontinued fractions byP. L. Chebyshev and was pursued byA. A. Markov andT. J. Stieltjes. They appear in a wide variety of fields:numerical analysis (quadrature rules),probability theory,representation theory (ofLie groups,quantum groups, and related objects),enumerative combinatorics,algebraic combinatorics,mathematical physics (the theory ofrandom matrices,integrable systems, etc.), andnumber theory. Some of the mathematicians who have worked on orthogonal polynomials includeGábor Szegő,Sergei Bernstein,Naum Akhiezer,Arthur Erdélyi,Yakov Geronimus,Wolfgang Hahn,Theodore Seio Chihara,Mourad Ismail,Waleed Al-Salam,Richard Askey, andRehuel Lobatto.

Definition for 1-variable case for a real measure

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Given any non-decreasing functionα on the real numbers, we can define theLebesgue–Stieltjes integral $\int f(x)\,d\alpha (x)$ of a functionf. If this integral is finite for all polynomialsf, we can define an inner product on pairs of polynomialsf andg by $\langle f,g\rangle =\int f(x)g(x)\,d\alpha (x).$

This operation is a positive semidefiniteinner product on thevector space of all polynomials, and is positive definite if the function α has an infinite number of points of growth. It induces a notion oforthogonality 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 $\deg P_{n}=n~,\quad \langle P_{m},\,P_{n}\rangle =0\quad {\text{for}}\quad m\neq n~.$

In other words, the sequence is obtained from the sequence of monomials 1,x,x², … by theGram–Schmidt process with respect to this inner product.

Usually the sequence is required to beorthonormal, namely, $\langle P_{n},P_{n}\rangle =1,$ however, other normalisations are sometimes used.

Absolutely continuous case

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Sometimes we have $d\alpha (x)=W(x)\,dx$ where $W:[x_{1},x_{2}]\to \mathbb {R}$ is a non-negative function with support on some interval[x₁,x₂] in the real line (wherex₁ = −∞ andx₂ = ∞ are allowed). Such aW is called aweight function.^[1] Then the inner product is given by $\langle f,g\rangle =\int _{x_{1}}^{x_{2}}f(x)g(x)W(x)\,dx.$ However, there are many examples of orthogonal polynomials where the measuredα(x) has points with non-zero measure where the functionα is discontinuous, so cannot be given by a weight functionW as above.

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:

The classical orthogonal polynomials (Jacobi polynomials,Laguerre polynomials,Hermite polynomials, and their special casesGegenbauer polynomials,Chebyshev polynomials andLegendre polynomials).
TheWilson polynomials, which generalize the Jacobi polynomials. They include many orthogonal polynomials as special cases, such as theMeixner–Pollaczek polynomials, thecontinuous Hahn polynomials, thecontinuous dual Hahn polynomials, and the classical polynomials, described by theAskey scheme
TheAskey–Wilson polynomials introduce an extra parameterq into the Wilson polynomials.

Discrete orthogonal polynomials are orthogonal with respect to somediscrete measure. Sometimes the measure has finite support, in which case the family of orthogonal polynomials is finite, rather than an infinite sequence. TheRacah polynomials are examples of discrete orthogonal polynomials, and include as special cases theHahn polynomials anddual Hahn polynomials, which in turn include as special cases theMeixner polynomials,Krawtchouk polynomials, andCharlier polynomials.

Meixner classified all the orthogonalSheffer 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 theNEF-QVFs and aremartingale polynomials for certainLévy processes.

Sieved orthogonal polynomials, such as thesieved ultraspherical polynomials,sieved Jacobi polynomials, andsieved Pollaczek polynomials, have modifiedrecurrence relations.

One can also consider orthogonal polynomials for some curve in thecomplex plane. The most important case (other than real intervals) is when the curve is the unit circle, givingorthogonal polynomials on the unit circle, such as theRogers–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 theunit disk.

The advantage of orthogonality between different orders ofHermite polynomials is applied to Generalized frequency division multiplexing (GFDM) structure. More than one symbol can be carried in each grid of time-frequency lattice.^[2]

Properties

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

Relation to moments

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The orthogonal polynomialsP_n can be expressed in terms of themoments

m_{n}=\int x^{n}\,d\alpha (x)

as follows:

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}}~,

where the constantsc_n are arbitrary (depend on the normalization ofP_n).

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 $P_{0}$ prescribes that $P_{1}$ must have the form $P_{1}(x)=c_{1}\left(x-{\frac {\langle P_{0},x\rangle P_{0}}{\langle P_{0},P_{0}\rangle }}\right)=c_{1}(x-m_{1}),$ which can be seen to be consistent with the previously given expression with the determinant.

Recurrence relation

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The polynomialsP_n satisfy a three-termrecurrence relation of the form

P_{n}(x)=(A_{n}x+B_{n})P_{n-1}(x)+C_{n}P_{n-2}(x)

whereA_n is not 0. The converse is also true; seeFavard's theorem. These recurrence relations are key for deriving properties of orthogonal polynomials.

Christoffel–Darboux formula

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Main article:Christoffel–Darboux formula

Zeros

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If the measure dα is supported on an interval [a, b], all the zeros ofP_n lie in [a, b]. Moreover, the zeros have the following interlacing property: ifm < n, there is a zero ofP_n between any two zeros of P_m.

Combinatorial interpretation

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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.^[3]

Other types of orthogonal polynomials

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Multivariate orthogonal polynomials

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TheMacdonald polynomials are orthogonal polynomials in several variables, depending on the choice of anaffine root system. They include many other families of multivariable orthogonal polynomials as special cases, including theJack polynomials, theHall–Littlewood polynomials, theHeckman–Opdam polynomials, and theKoornwinder polynomials. TheAskey–Wilson polynomials are the special case of Macdonald polynomials for a certain non-reduced root system of rank 1.

Multiple orthogonal polynomials

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Main article: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

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Main article:Sobolev orthogonal polynomials

These are orthogonal polynomials with respect to aSobolev 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

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Orthogonal polynomials with matrices have either coefficients that are matrices or the indeterminate is a matrix.

There are two popular examples: either the coefficients $\{a_{i}\}$ are matrices or $x {\displaystyle x}$ :

Variante 1: $P(x)=A_{n}x^{n}+A_{n-1}x^{n-1}+\cdots +A_{1}x+A_{0}$ , where $\{A_{i}\}$ are $p\times p$ matrices.
Variante 2: $P(X)=a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}I_{p}$ where $X {\displaystyle X}$ is a $p\times p$ -matrix and $I_{p}$ is theidentity matrix.

Quantum polynomials

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Quantum polynomials or q-polynomials are theq-analogs of orthogonal polynomials.

Skew-orthogonal polynomials

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Main article:Skew-orthogonal polynomials

Orthogonal polynomials can be defined as a vector basis set $P_{0},P_{1},P_{2},\dots$ of asymmetric bilinear form $B {\displaystyle B}$ on polynomials. In the basis of the orthogonal polynomials, the bilinear form diagonalizes as $B(P_{i},P_{j})=B(P_{i},P_{i})\delta _{ij}$ . Similarly, given anondegenerate skew-symmetric bilinear form on polynomials, we can find a pair of vector basis sets $P_{0},P_{1},P_{2},\dots$ and $Q_{0},Q_{1},Q_{2},\dots$ , such that the bilinear form skew-diagonalizes as $B(P_{i},Q_{j})=B(P_{i},Q_{i})\delta _{ij}$ .

References

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^Demo of orthonormal polynomials obtained for different weight functions
^Catak, E.; Durak-Ata, L. (2017). "An efficient transceiver design for superimposed waveforms with orthogonal polynomials".2017 IEEE International Black Sea Conference on Communications and Networking (BlackSeaCom). pp. 1–5.doi:10.1109/BlackSeaCom.2017.8277657.ISBN 978-1-5090-5049-9.S2CID 22592277.
^Viennot, Xavier (2017)."The Art of Bijective Combinatorics, Part IV, Combinatorial theory of orthogonal polynomials and continued fractions". Chennai: IMSc.

Abramowitz, Milton;Stegun, Irene Ann, eds. (1983) [June 1964]."Chapter 22".Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 773.ISBN 978-0-486-61272-0.LCCN 64-60036.MR 0167642.LCCN 65-12253.
Chihara, Theodore Seio (1978).An Introduction to Orthogonal Polynomials. Gordon and Breach, New York.ISBN 0-677-04150-0.
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.Bibcode:2001JCoAM.133...13C.doi:10.1016/S0377-0427(00)00632-4.ISSN 0377-0427.MR 1858267.
Foncannon, J. J.; Foncannon, J. J.; Pekonen, Osmo (2008). "Review ofClassical and quantum orthogonal polynomials in one variable by Mourad Ismail".The Mathematical Intelligencer.30. Springer New York:54–60.doi:10.1007/BF02985757.ISSN 0343-6993.S2CID 118133026.
Ismail, Mourad E. H. (2005).Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge: Cambridge Univ. Press.ISBN 0-521-78201-5.
Jackson, Dunham (2004) [1941].Fourier Series and Orthogonal Polynomials. New York: Dover.ISBN 0-486-43808-2.
Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010),"Orthogonal Polynomials", inOlver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.),NIST Handbook of Mathematical Functions, Cambridge University Press,ISBN 978-0-521-19225-5,MR 2723248.
"Orthogonal polynomials",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
Szegő, Gábor (1939).Orthogonal Polynomials. Colloquium Publications. Vol. XXIII. American Mathematical Society.ISBN 978-0-8218-1023-1.MR 0372517.{{cite book}}:ISBN / Date incompatibility (help)
Totik, Vilmos (2005). "Orthogonal Polynomials".Surveys in Approximation Theory.1:70–125.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).