Chebyshev Polynomials

The Chebyshev Polynomials are polynomials belonging to sine and cosine functions. They serve as solutions to the Chebyshev differential equations, which are differential equations with singular pointsGraph curve having a self-intersection. at $-1, 1, \infin$. They are used in approximating functions as well.

The Chebyshev Polynomials

A Chebyshev Polynomial of the first order is defined as follows:

From this definition, we can calculate:

Let's calculate $T_4(x)$ as follows:

A general recurrence relation for this can be written as follows:

Note: To find $T_{n+1}$ here, we need to first have the values of $T_n$ and $T_{n-1}$.

In case we want to find the $T_n(x)$ for this relation when needed without needing the previous values as above, we can use a generating function as follows:

Chebyshev Polynomials of the second order can be written as follows:

Properties

Some of the properties of the Chebyshev Polynomials are given below:

Symmetry

Chebyshev polynomials of odd order have odd symmetry, and for even order, an even symmetry:

Roots

Any degree Chebyshev polynomial has $n$ different simple roots called Chebyshev roots in the interval, $[-1,1]$.

Roots of the first-order polynomial can be written as follows:

Integral

Integration of the Polynomial can be defined as follows:

Product

The product of two Chebyshev Polynomials (first order) is given as:

Orthogonality

Polynomials of the first order are orthogonal. It means that there is a class of polynomials that obeys an orthogonality relationship where integrating the product of this class of polynomials would give Kronecker's deltaDiscrete version of the delta function.. This is shown with respect to the inner product.

On the interval $[1, -1]$: In a similar manner, we can define the orthogonality for polynomials of the second order.

Curve specification

Curves given by $T_n(x)$. They are special cases of Lissajous curvesGraph of a system of parametric equations which are responsible for parametric equations.

Applications

Some of the applications of Chebyshev polynomials are:

• Integral computation

• Solutions to differential equations

• Numerical analysis

• Waveform synthesis

• Trignometrical identities