Chebyshev Polynomial of the First Kind — Definition, Formula & Examples

Chebyshev polynomials of the first kind are a sequence of orthogonal polynomials

T_n(x)

defined on

[-1,1]

that satisfy

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

. They arise naturally in approximation theory, where they minimize the maximum error when approximating functions by polynomials.

For each non-negative integer $n$ , the Chebyshev polynomial of the first kind $T_n(x)$ is the unique polynomial of degree $n$ satisfying $T_n(\cos\theta) = \cos(n\theta)$ for all $\theta \in \mathbb{R}$ . Equivalently, $T_n$ can be defined by the recurrence $T_0(x)=1$ , $T_1(x)=x$ , and $T_{n+1}(x)=2xT_n(x)-T_{n-1}(x)$ . These polynomials are orthogonal on $[-1,1]$ with respect to the weight function $w(x)=(1-x^2)^{-1/2}$ .

Key Formula

T_{n+1}(x) = 2x\,T_n(x) - T_{n-1}(x)

Where:

$T_n(x)$ = Chebyshev polynomial of the first kind of degree $n$
$x$ = Variable, typically restricted to $[-1,1]$
$n$ = Non-negative integer degree

How It Works

You can generate any Chebyshev polynomial using the three-term recurrence relation: start with

T_0(x)=1

and

T_1(x)=x

, then repeatedly apply

T_{n+1}(x)=2xT_n(x)-T_{n-1}(x)

. For small arguments, the trigonometric definition

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

lets you evaluate the polynomial directly. The roots of

T_n(x)

are

x_k = \cos\!\left(\frac{2k-1}{2n}\pi\right)

for

k=1,\dots,n

, and these roots serve as optimal interpolation nodes that minimize Runge's phenomenon.

Worked Example

Problem: Use the recurrence relation to find

T_3(x)

Base cases: Start with the first two Chebyshev polynomials.

T_0(x) = 1, \quad T_1(x) = x

Compute T₂: Apply the recurrence with

n=1

T_2(x) = 2x \cdot T_1(x) - T_0(x) = 2x^2 - 1

Compute T₃: Apply the recurrence again with

n=2

T_3(x) = 2x \cdot T_2(x) - T_1(x) = 2x(2x^2 - 1) - x = 4x^3 - 3x

Answer:

T_3(x) = 4x^3 - 3x

. You can verify:

T_3(\cos\theta) = 4\cos^3\theta - 3\cos\theta = \cos(3\theta)

, which matches the triple-angle formula.

Why It Matters

Chebyshev polynomials are the backbone of spectral methods in numerical analysis and scientific computing. Their roots provide the optimal nodes for polynomial interpolation, directly reducing approximation error in finite element analysis, signal processing filters, and computational physics simulations.

Common Mistakes

Mistake: Forgetting the factor of 2 in the recurrence and writing

T_{n+1}(x) = xT_n(x) - T_{n-1}(x)

Correction: The correct recurrence is

T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)

. Missing the 2 produces incorrect polynomials for

n \geq 2