Orthogonal polynomials

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In mathematics, two polynomials f and g are orthogonal to each other with respect to a nonnegative "weight function" w precisely if

<math>\int_{x_1}^{x_2} f(x)g(x)w(x)\,dx=0.<math>

In other words, if polynomials are treated as vectors and the inner product of two polynomials f(x) and g(x) is defined as

<math>\langle f,g \rangle=\int_{x_1}^{x_2} f(x)g(x)\,w(x)\,dx<math>

then the orthogonal polynomials are simply orthogonal vectors in this inner product space.

A polynomial sequence pn(x) for n = 0, 1, 2, ... , where pn(x) has degree n, is said to be a sequence of orthogonal polynomials with respect to a "weight function" w when any two of them are orthogonal with respect to that weight function, i.e.,

<math>\langle p_n, p_m \rangle=\int_{x_1}^{x_2} p_n(x) p_m(x)\,w(x)\,dx=0\ \mbox{whenever}\ n\neq m.<math>

The sequence of orthogonal polynomials can be successively constructed by carrying out the Gram-Schmidt process with the sequence of powers <math> x^k, \; k \ge 0<math>, where the positive-definite inner product <math>\langle p,q \rangle <math> on the space of polynomials is given by the integral above.

Contents

General properties of orthogonal polynomials

  • Three-term recurrence relations

For each non-negative weight function the corresponding orthogonal polynomials obey a recurrence relation

<math>f_{n+1}=(a_n+xb_n)f_n - c_nf_{n-1}<math>

where the constants <math>a_n<math>, <math>b_n<math> and <math>c_n<math> are given by

<math>b_n=\frac{k_{n+1}}{k_n}<math>
<math>a_n=b_n \left(\frac{k_{n+1}'}{k_{n+1}} - \frac{k_n'}{k_n} \right)<math>
<math>c_n=\frac{k_{n+1}k_{n-1}h_n} {k_n^2 h_{n-1}}<math>

and <math>k_n<math> and <math>k_n'<math> are the leading terms in the expansion of the polynomial:

<math>f_n(x)=k_nx^n+k_n'x^{n-1}+\cdots<math>

and <math>h_n<math> is the normalization, defined below.

The classical orthogonal polynomials

The collective name "classical orthogonal polynomials" refers to a class of orthogonal polynomials which are distinguished by several characteristic properties. They occur in many applications including mathematical physics, interpolation theory, the theory of random matrices and many others and have been therefore studied in mathematics since a long time. Some of their characteristic properties will be outlined in the following subsections:

Differential equation

The classical orthogonal polynomials satisfy a second-order differential equation

<math> g_2(x)f_n''(x) + g_1(x)f_n'(x) + d_n f_n(x) = 0<math>

where <math>g_1(x)<math> and <math>g_2(x)<math> are independent of n and <math>d_n<math> is a constant that depends only on n. The coefficient function <math> g_2(x) <math> is a polynomial of degree <math> \le 2<math>, the coefficient <math> g_1(x) <math> is a polynomial of degree <math> \le 1<math>.

Existence of a Rodrigues formula

Every classical orthogonal polynomial can be obtained via a so-called Rodrigues formula:

<math>f_n(x)=\frac{1}{e_n w(x)}\, \frac{d^n}{dx^n} w(x)[g(x)]^n<math>

where <math>w(x)<math> is the defining weight function of the series of orthogonal polynomials (defined in the list below) and <math>e_n<math> is a constant depending only on n, and <math>g(x)<math> is a polynomial independent of n.

The orthogonality relationship is

<math>\int_{x_1}^{x_2}p_n(x)p_m(x)w(x)\,dx=\delta_{mn}h_n<math>

where δmn is the Kronecker delta and hn is defined in the table below for each weight function.

Table Of Classical Orthogonal Polynomials
Name x1  x2 w(x)hn
Chebyshev polynomials (first kind) <math>-1<math> <math>1<math> <math>(1-x^2)^{-1/2}<math> <math>\left\{

\begin{matrix} \pi &:~n=0 \\ \pi/2 &:~n\ne 0 \end{matrix}\right. <math>

Chebyshev polynomials (second kind) <math>-1<math> <math>1<math> <math>(1-x^2)^{1/2}<math> <math>\pi/2<math>
Legendre polynomials <math>-1<math> <math>1<math> <math>1<math> <math>\frac{2}{2n+1}<math>
Laguerre polynomials <math>0<math> <math>\infty<math> <math>e^{-x}<math> <math>1<math>
Hermite polynomials <math>-\infty<math> <math>\infty<math> <math>e^{-x^2}<math> <math>n!\,\sqrt{2\pi}<math>

See also

References

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