Chebyshev Polynomials

1. Definition via Recurrence

Chebyshev polynomials \(T_n(x)\) are a family of polynomials that play a central role in approximation theory, numerical analysis, and interpolation. The simplest way to define them is through the recurrence relation:

\[ T_0(x) = 1, \quad T_1(x) = x, \quad T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x), \quad n \geq 1 \]

By induction, \(T_n(x)\) is a polynomial of degree \(n\), and the coefficient of the highest power term \(x^n\) is \(2^{n-1}\) for \(n \ge 1\). This recurrence formula allows for efficient computation of Chebyshev polynomials for any degree.

2. Trigonometric Form

Chebyshev polynomials can also be expressed using trigonometric functions. Let \( x = \cos(\theta) \). Using the identity

\[ \cos(a + b) + \cos(a - b) = 2\cos(a)\cos(b) \]

we can show by induction that

\[ T_n(x) = \cos(n \cdot \arccos(x)), \quad x \in [-1, 1] \]

This form emphasizes that Chebyshev polynomials oscillate between \(-1\) and \(1\) on the interval \([-1,1]\) and provides intuition about their roots and extrema.

Proof Sketch (Induction)

• For \( n=0 \): \( T_0(x) = \cos(0) = 1 \)
• For \( n=1 \): \( T_1(x) = \cos(\arccos(x)) = x \)
• Assume true for \( n \) and \( n-1 \). Then, using the recurrence:

\[ T_{n+1}(x) = 2\cos(\theta) \cos(n\theta) - \cos((n-1)\theta) = \cos((n+1)\theta), \quad \theta = \arccos(x) \]

3. Key Properties of \(T_n\)

• \(|T_n(x)| \leq 1\) for all \( x \in [-1, 1] \)
• \(T_n(\cos(\frac{i \pi}{n})) = (-1)^i\), for \( i = 0, 1, \dots, n \)
• \(\sum_{i=1}^{n} T_n\left(\cos\left(\frac{2i - 1}{2n}\pi\right)\right) = 0\)

4. Monic Chebyshev Polynomials

Let \(\mathcal{P}_n([a,b])\) denote the set of degree-\(n\) polynomials with \(n\) distinct roots in \([a,b]\) and leading coefficient 1. Among all such polynomials in \([-1,1]\), the Chebyshev polynomials minimize the maximum absolute value on the interval:

If \( p \in \mathcal{P}_n([-1,1]) \), then \(\max_{x \in [-1,1]} |p(x)| \geq \frac{1}{2^{n-1}} \).

By defining the monic Chebyshev polynomial

\[ Q_n(x) = \frac{1}{2^{n-1}} T_n(x) \]

we achieve equality: the maximum of \(Q_n(x)\) on \([-1,1]\) is exactly \( 1/2^{n-1} \), making them optimal for minimizing the uniform norm.

5. Interpolation and Chebyshev Points

When interpolating a function \( f \in \mathcal{C}^{n+1}([-1,1]) \), the error is minimized if the interpolation points are chosen as the roots of \( T_{n+1}(x) \):

\[ x_i = \cos\left(\frac{2i + 1}{2n + 2} \pi\right), \quad i = 0, 1, \dots, n \]

The corresponding interpolation error satisfies:

\[ |E_n(x)| = |f(x) - p_n(x)| \leq \frac{1}{2^n (n+1)!} \max_{x \in [-1,1]} |f^{(n+1)}(x)| \]

This is the smallest possible maximum error among all choices of \( n+1 \) points in \([-1,1]\). It shows why Chebyshev points are fundamental in numerical interpolation: they provide an optimal distribution of nodes that minimizes oscillations and ensures uniform accuracy.

Example 2 — Degree 3 Polynomial Interpolation

Suppose we interpolate \( f(x) = \cos(x) \) on \([-1,1]\) with a cubic polynomial. Using Chebyshev points:

\[ x_0 = \cos( \frac{\pi}{8} ) \approx 0.9239, \quad x_1 = \cos( \frac{3\pi}{8} ) \approx 0.383, \quad x_2 = \cos( \frac{5\pi}{8} ) \approx -0.383, \quad x_3 = \cos( \frac{7\pi}{8} ) \approx -0.924 \]

Interpolating \( f(x) \) at these points gives a polynomial that approximates \( \cos(x) \) very closely across the entire interval, with the error uniformly small. If we used equally spaced points instead, the polynomial would oscillate near \(-1\) and \(1\), demonstrating the practical advantage of Chebyshev points.