Orthogonal polynomials
Introduction
Orthogonal polynomials are a class of polynomials that arise in the context of various mathematical and physical problems. They are defined by the property that they are orthogonal to each other with respect to a given inner product. This orthogonality condition leads to many useful properties and applications in areas such as numerical analysis, approximation theory, and quantum mechanics.
Definition and Properties
Orthogonal polynomials \( P_n(x) \) are a sequence of polynomials where each polynomial \( P_n(x) \) is of degree \( n \), and they satisfy the orthogonality condition: \[ \int_a^b P_m(x) P_n(x) w(x) \, dx = 0 \quad \text{for} \quad m \neq n, \] where \( w(x) \) is a weight function defined on the interval \([a, b]\).
The weight function \( w(x) \) is a non-negative function that ensures the orthogonality of the polynomials over the interval \([a, b]\). The choice of \( w(x) \) and the interval \([a, b]\) determines the specific family of orthogonal polynomials.
Classical Orthogonal Polynomials
Several families of orthogonal polynomials are well-studied and have significant applications. These include:
Legendre Polynomials
Legendre polynomials \( P_n(x) \) are orthogonal with respect to the weight function \( w(x) = 1 \) on the interval \([-1, 1]\). They satisfy the orthogonality condition: \[ \int_{-1}^1 P_m(x) P_n(x) \, dx = 0 \quad \text{for} \quad m \neq n. \]
Chebyshev Polynomials
Chebyshev polynomials come in two kinds: the first kind \( T_n(x) \) and the second kind \( U_n(x) \). They are orthogonal with respect to the weight functions \( w(x) = \frac{1}{\sqrt{1-x^2}} \) and \( w(x) = \sqrt{1-x^2} \), respectively, on the interval \([-1, 1]\).
Hermite Polynomials
Hermite polynomials \( H_n(x) \) are orthogonal with respect to the weight function \( w(x) = e^{-x^2} \) on the entire real line \((-\infty, \infty)\). They satisfy the orthogonality condition: \[ \int_{-\infty}^\infty H_m(x) H_n(x) e^{-x^2} \, dx = 0 \quad \text{for} \quad m \neq n. \]
Laguerre Polynomials
Laguerre polynomials \( L_n(x) \) are orthogonal with respect to the weight function \( w(x) = e^{-x} \) on the interval \([0, \infty)\). They satisfy the orthogonality condition: \[ \int_0^\infty L_m(x) L_n(x) e^{-x} \, dx = 0 \quad \text{for} \quad m \neq n. \]
Generating Functions
Orthogonal polynomials often have associated generating functions that provide a compact representation of the entire sequence. For example, the generating function for the Hermite polynomials is given by: \[ e^{2xt - t^2} = \sum_{n=0}^\infty H_n(x) \frac{t^n}{n!}. \]
Recurrence Relations
Orthogonal polynomials satisfy recurrence relations that allow the computation of higher-degree polynomials from lower-degree ones. For instance, the recurrence relation for Legendre polynomials is: \[ (n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x). \]
Differential Equations
Many orthogonal polynomials are solutions to specific differential equations. For example, the Legendre polynomials satisfy the Legendre differential equation: \[ (1-x^2) \frac{d^2 P_n(x)}{dx^2} - 2x \frac{d P_n(x)}{dx} + n(n+1) P_n(x) = 0. \]
Applications
Orthogonal polynomials have numerous applications across various fields:
Numerical Analysis
In numerical analysis, orthogonal polynomials are used in techniques such as Gaussian quadrature, where they help in approximating the integral of a function.
Approximation Theory
Orthogonal polynomials are used in approximation theory to construct polynomial approximations to functions. The Chebyshev polynomials are particularly important in minimizing the maximum error in polynomial approximation.
Quantum Mechanics
In quantum mechanics, orthogonal polynomials appear in the solutions to the Schrödinger equation for certain potentials. For example, the Hermite polynomials are used in the solution of the quantum harmonic oscillator.