Chebyshev Polynomial Approximation

• Chebyshev polynomial approximation is defined by expressing continuous functions on [-1,1] as sums of orthogonal Chebyshev polynomials, using a three-term recursion for efficient computation.
• The method leverages a generating function and spectral convergence properties for rapid decay of coefficients in analytic cases, ensuring near-minimax accuracy.
• It provides numerical stability in interpolation and avoids the Runge phenomenon through clustered Chebyshev nodes, making it ideal for solving differential equations and boundary-value problems.

Chebyshev polynomial approximation is a central technique in numerical analysis and computational mathematics for representing continuous functions as sums of orthogonal polynomials over the interval $[-1,1]$. Compared to Fourier series, the Chebyshev polynomial interpolant delivers higher accuracy in the uniform norm when approximating polynomial data, and exhibits superior numerical stability when applied to problems such as ordinary and partial differential equations, boundary-value problems, and spectral methods.

1. Definition and Three-Term Recursion

Chebyshev polynomials of the first kind $T_n(x)$ for %%%%2%%%% are defined on $[-1,1]$ by

$T_n(x) = \cos(n \arccos x) .$

Equivalently, when $x = \cos\theta$, $T_n(x) = \cos(n\theta)$. The standard three-term recurrence satisfied by these polynomials is

$$T_{n+1}(x) = 2x\,T_n(x) - T_{n-1}(x), \quad T_0(x) = 1,\ T_1(x) = x.$$

This recursion, obtained by trigonometric identities on the cosine function, is computationally efficient and numerically stable for evaluating high-degree polynomials.

2. Generating Function

The ordinary generating function for Chebyshev polynomials is

$$G(x,z) = \sum_{n=0}^\infty T_n(x) z^n = \frac{1 - x z}{1 - 2 x z + z^2}, \quad |z|<1.$$

This compact formula enables formal manipulations and provides insight into the spectral structure of Chebyshev expansions.

3. Orthogonality and Parseval Identity

Chebyshev polynomials are orthogonal with respect to the Chebyshev weight $(1-x^2)^{-1/2}$ on $[-1,1]$: $$\int_{-1}^1 T_m(x) T_n(x) \frac{dx}{\sqrt{1-x^2}} = \begin{cases} 0, & m \neq n, \ \pi, & m = n = 0, \ \pi/2, & m = n \neq 0. \end{cases}$$ This orthogonality implies that Chebyshev polynomials form a Hilbert-space basis in $L^2([-1,1], (1-x^2)^{-1/2}dx)$. Parseval's identity holds as

$$\int_{-1}^1 \frac{[f(x)]^2}{\sqrt{1 - x^2}}\, dx = \pi C_0^2 + \frac{\pi}{2} \sum_{n=1}^\infty C_n^2,$$

where $C_0, C_n$ are the Chebyshev series coefficients of $f$.

4. Expansion, Truncation, and Best Approximation

The Chebyshev series for any $f \in L^2([-1,1], (1-x^2)^{-1/2})$ reads

$f(x) = \sum_{n=0}^\infty C_n T_n(x),$

with coefficients

$$C_0 = \frac{1}{\pi} \int_{-1}^1 \frac{f(x)}{\sqrt{1-x^2}}\,dx, \quad C_n = \frac{2}{\pi} \int_{-1}^1 \frac{f(x) T_n(x)}{\sqrt{1-x^2}}\,dx, \quad n \geq 1.$$

The truncated sum $f_N(x) = \sum_{n=0}^N C_n T_n(x)$ provides the $L^2$-optimal approximation to $f$ in the Chebyshev basis.

For analytic functions, the coefficients decay geometrically: if $f$ is analytic in the Bernstein ellipse $E_\rho$ (with $|f(z)| \le M$ in $E_\rho$, $\rho > 1$),

$$|C_n| \le \frac{2M}{\rho^n}, \qquad \|f - f_N\|_\infty \le \frac{2M}{\rho^N (\rho-1)},$$

resulting in spectral convergence of the partial sums.

5. Interpolation Nodes and the Near-Minimax Property

Chebyshev polynomial interpolation is usually performed at the nodes

$$x_k = \cos\left( \frac{2k+1}{2(N+1)} \pi \right),\quad k=0,1,\ldots,N,$$

which are the roots of $T_{N+1}(x)$. The Lagrange interpolant at these nodes,

$p_N(x) = \sum_{k=0}^N f(x_k) \ell_k(x),$

where $\ell_k$ are the Lagrange cardinal polynomials, possesses a Lebesgue constant that grows only logarithmically in $N$, ensuring good stability in the uniform norm. The error oscillates ("equiripple") exactly $N+2$ times, and for analytic $f$, exponential convergence occurs.

For polynomial data, the Chebyshev interpolant recovers the original polynomial exactly if the degree does not exceed $N$. For general continuous functions, Chebyshev interpolation avoids the Runge phenomenon associated with equispaced interpolation.

6. Practical Comparison: Chebyshev versus Fourier

For functions such as polynomials (e.g., $f(x) = x^7 - 14x^5 + 49x^3 - 36x$), Chebyshev interpolation at degree $7$ yields exact recovery, whereas Fourier interpolation requires considerably more terms. For discontinuous functions (e.g., the Heaviside function), Chebyshev and Fourier series exhibit the Gibbs phenomenon, but Fourier series may converge more rapidly in the $L^2$ norm for jump discontinuities.

7. Computational and Theoretical Impact

Chebyshev polynomial approximation is fundamentally important for numerical schemes addressing ordinary differential equations, boundary value problems, spectral methods for PDEs, and digital signal processing. The Chebyshev polynomial, as a solution to the Sturm-Liouville problem,

$(1-x^2) y'' - x y' + n^2 y = 0,$

embodies optimality properties that extend to minimax approximation, interpolation stability, and best uniform polynomial approximation.

Chebyshev expansions offer "near-minimax" accuracy in the uniform norm for sufficiently smooth functions and exhibit uniformly bounded Lebesgue constants at clustered interpolation nodes. These features, combined with explicit recursive evaluations, robust orthogonality, and Parseval-type identities, give Chebyshev methods their recognized superiority for polynomial approximation tasks on $[-1,1]$.