Chebyshev Polynomials

Introduction

Chebyshev polynomials, named after the Russian mathematician Pafnuty Chebyshev, are a sequence of orthogonal polynomials that arise in various areas of mathematics, including approximation theory, numerical analysis, and algebra. They are particularly significant in the context of approximation theory and Fourier analysis.

Definition and Properties

Chebyshev Polynomials of the First Kind

The Chebyshev polynomials of the first kind, denoted as \( T_n(x) \), are defined by the recurrence relation:

\[ T_0(x) = 1 \] \[ T_1(x) = x \] \[ T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x) \]

Alternatively, they can be defined using the trigonometric identity:

\[ T_n(\cos \theta) = \cos(n \theta) \]

These polynomials are orthogonal with respect to the weight function \( \frac{1}{\sqrt{1-x^2}} \) on the interval \([-1, 1]\).

Chebyshev Polynomials of the Second Kind

The Chebyshev polynomials of the second kind, denoted as \( U_n(x) \), are defined by the recurrence relation:

\[ U_0(x) = 1 \] \[ U_1(x) = 2x \] \[ U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x) \]

Alternatively, they can be defined using the trigonometric identity:

\[ U_n(\cos \theta) = \frac{\sin((n+1) \theta)}{\sin \theta} \]

These polynomials are orthogonal with respect to the weight function \( \sqrt{1-x^2} \) on the interval \([-1, 1]\).

Applications

Approximation Theory

Chebyshev polynomials play a crucial role in approximation theory. The Chebyshev approximation minimizes the maximum error, which is known as the minimax property. This makes them particularly useful in polynomial interpolation and the design of filters.

Numerical Analysis

In numerical analysis, Chebyshev polynomials are used to solve differential equations and integral equations. They are also employed in the Chebyshev iterative methods for solving linear systems.

Spectral Methods

Chebyshev polynomials are used in spectral methods for solving partial differential equations. These methods involve expanding the solution in terms of Chebyshev polynomials and using their orthogonality properties to simplify the problem.

Control Theory

In control theory, Chebyshev polynomials are used to design optimal controllers. The polynomials help in approximating the desired control law and in analyzing the stability of the system.

Properties and Theorems

Orthogonality

The orthogonality of Chebyshev polynomials is a fundamental property that underlies many of their applications. For the first kind, the orthogonality condition is:

\[ \int_{-1}^{1} \frac{T_m(x) T_n(x)}{\sqrt{1-x^2}} \, dx = \begin{cases} 0 & \text{if } m \neq n \\ \frac{\pi}{2} & \text{if } m = n \neq 0 \\ \pi & \text{if } m = n = 0 \end{cases} \]

For the second kind, the orthogonality condition is:

\[ \int_{-1}^{1} U_m(x) U_n(x) \sqrt{1-x^2} \, dx = \begin{cases} 0 & \text{if } m \neq n \\ \frac{\pi}{2} & \text{if } m = n \end{cases} \]

Zeros and Extrema

The zeros of the Chebyshev polynomials of the first kind are given by:

\[ x_k = \cos \left( \frac{(2k-1)\pi}{2n} \right), \quad k = 1, 2, \ldots, n \]

The extrema occur at:

\[ x_k = \cos \left( \frac{k\pi}{n} \right), \quad k = 0, 1, \ldots, n \]

For the second kind, the zeros are given by:

\[ x_k = \cos \left( \frac{k\pi}{n+1} \right), \quad k = 1, 2, \ldots, n \]

Symmetry

Chebyshev polynomials exhibit symmetry properties. For the first kind:

\[ T_n(-x) = (-1)^n T_n(x) \]

For the second kind:

\[ U_n(-x) = (-1)^n U_n(x) \]

Recurrence Relations

In addition to the primary recurrence relations, Chebyshev polynomials satisfy several other recurrence relations and differential equations. For instance, the differential equation for \( T_n(x) \) is:

\[ (1-x^2) T_n(x) - x T_n'(x) + n^2 T_n(x) = 0 \]

Generalizations and Extensions

Generalized Chebyshev Polynomials

Generalized Chebyshev polynomials extend the concept to other weight functions and intervals. These polynomials retain many properties of the classical Chebyshev polynomials but are adapted to different contexts.

Multivariate Chebyshev Polynomials

Multivariate Chebyshev polynomials are used in higher-dimensional approximation problems. They are defined on multi-dimensional domains and have applications in multi-dimensional integration and approximation.

Chebyshev Rational Functions

Chebyshev rational functions are another extension, defined as rational functions that minimize the maximum error in approximation problems. They are particularly useful in problems where the function to be approximated has poles.

References