Chebyshev Approximation

Chebyshev polynomials provide optimal approximations by minimizing maximum error, and Chebyshev nodes mitigate Runge oscillations in polynomial interpolation through strategic point placement.

DefinitionChebyshev Polynomials

The Chebyshev polynomials of the first kind are defined by: $T_n(x) = \cos(n \arccos x), \quad x \in [-1,1]$

They satisfy the recurrence relation: $T_0(x) = 1, \quad T_1(x) = x, \quad T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x)$

The Chebyshev polynomials are orthogonal on $[-1,1]$ with weight $(1-x^2)^{-1/2}$ and satisfy $|T_n(x)| \leq 1$ for $|x| \leq 1$ .

The minimax property of Chebyshev polynomials makes them fundamental in approximation theory. Among all monic polynomials of degree $n$ , $2^{1-n}T_n(x)$ has the smallest maximum absolute value on $[-1,1]$ , equal to $2^{1-n}$ .

DefinitionChebyshev Nodes

The Chebyshev nodes of the first kind on $[-1,1]$ are: $x_k = \cos\left(\frac{2k+1}{2(n+1)}\pi\right), \quad k = 0, 1, \ldots, n$

These are the zeros of $T_{n+1}(x)$ . For an arbitrary interval $[a,b]$ , transform via: $x_k = \frac{a+b}{2} + \frac{b-a}{2}\cos\left(\frac{2k+1}{2(n+1)}\pi\right)$

ExampleInterpolation Error Comparison

Interpolating $f(x) = \frac{1}{1+25x^2}$ (Runge's example) on $[-1,1]$ with $n=10$ points:

Equally-spaced nodes: Maximum error $\approx 2.5$ near $x = \pm 1$ (wild oscillations)

Chebyshev nodes: Maximum error $\approx 0.04$ (well-behaved approximation)

The error improvement is dramatic: Chebyshev nodes cluster near endpoints where interpolation error is typically largest, balancing error across the interval.

Remark

Interpolation Error Bound: For polynomial interpolation at Chebyshev nodes, the error satisfies: $|f(x) - p_n(x)| \leq \frac{1}{2^n n!} \max_{\xi \in [a,b]} |f^{(n+1)}(\xi)|$

This is nearly optimal (within a factor of 2) among all possible node choices. The product $\prod_{i=0}^n |x - x_i|$ achieves its minimum maximum value when the $x_i$ are Chebyshev nodes, a consequence of the minimax property of Chebyshev polynomials.

Chebyshev approximations extend to best approximation theory. The Chebyshev expansion of $f$ on $[-1,1]$ is: $f(x) \approx \sum_{k=0}^n a_k T_k(x)$ where coefficients $a_k$ are computed via orthogonality. This expansion converges rapidly for smooth functions and forms the basis of spectral methods in numerical PDE solving.

The Remez exchange algorithm constructs the polynomial of degree $n$ that minimizes $\max_{x \in [a,b]} |f(x) - p_n(x)|$ (best uniform approximation). This polynomial oscillates between $\pm$ maximum error at $n+2$ points, characterized by the equioscillation theorem. Chebyshev interpolation provides a near-optimal approximation that is computationally simpler than the exact minimax polynomial.