Orthogonal polynomials

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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 Al-Salam, and Richard Askey.

Definition for 1-variable case for a real measure

Given any non-decreasing function α on the real numbers, we can define the Lebesgue–Stieltjes integral

$\int f(x)d\alpha(x)$

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

$\langle f, g \rangle = \int f(x) g(x) \; d\alpha(x).$

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 (Pn)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, obtained from the sequence of monomials 1, x, x2, ... by the Gram–Schmidt process.

Usually the sequence is required to be orthonormal, namely,

$\langle P_n, P_n \rangle = 1~,$

however, other normalisations are sometimes used.

Absolutely continuous case

Sometimes we have

$\displaystyle d\alpha(x) = W(x)dx$

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 (where $x_1 = -\infty$ and $x_2 = \infty$ are allowed). Such a W is called a weight function. 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 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.

Examples of orthogonal polynomials

The most commonly used orthogonal polynomials are orthogonal for a measure with support in a real interval. This includes:

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 non-negative measure on the real line have the following properties.

Relation to moments

The orthogonal polynomials Pn can be expressed in terms of the moments

$m_n = \int x^n d\alpha(x)$

as follows:

$P_n(x) = c_n \, \det \begin{bmatrix} m_0 & m_1 & m_2 &\cdots & m_n \\ m_1 & m_2 & m_3 &\cdots & m_{n+1} \\ &&\cdots&& \\ m_{n-1} &m_n& m_{n+1} &\cdots &m_{2n-1}\\ 1 & x & x^2 & \cdots & x^{n} \end{bmatrix}~,$

where the constants cn are arbitrary (depend on the normalisation of Pn).

Recurrent relation

The polynomials Pn satisfy a recurrent relation of the form

$P_n(x) = (A_n x + B_n) P_{n-1}(x) + C_n P_{n-2}(x)~.$

See Favard's theorem for a converse result.

Zeros

If the measure dα is supported on an interval [a, b], all the zeros of Pn lie in [a, b]. Moreover, the zeros have the following interlacing property: if m>n, there is a zero of Pm between any two zeros of Pn.

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.

References

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