Chebyshev Polynomials

⚡ Key Takeaways

Chebyshev polynomials are a family of orthogonal polynomials $T_n(x)$ (first kind) and $U_n(x)$ (second kind) defined by $T_n(\cos\theta) = \cos(n\theta)$ . They are the unique monic polynomials of least supremum norm on $[-1,1]$ , making them central to approximation theory, numerical analysis, and quadrature. Introduced by Pafnuty Chebyshev in 1854.

💡 The Core Insight

Set $x = \cos\theta$ . The map $x \mapsto n\theta$ turns multiplication of angles into a polynomial in $x$ , because $\cos(n\theta)$ is always a polynomial in $\cos\theta$ (by De Moivre's theorem). Every hard question about $T_n$ — its roots, extrema, orthogonality, minimax property — becomes an easy question about the cosine of a multiple angle. Chebyshev polynomials are "cosines in disguise."

DefinitionChebyshev Polynomials of the First Kind

The Chebyshev polynomials of the first kind $T_n : [-1,1] \to \mathbb{R}$ are defined for integer $n \ge 0$ by

1

\quad T_n(\cos\theta) = \cos(n\theta), \qquad \theta \in [0,\pi].

Derivation hint: Substituting $x=\cos\theta$ bijects $[0,\pi]\to[-1,1]$ ; the right side is a polynomial in $\cos\theta$ by induction on De Moivre's formula. Equivalently, for $x \in [-1,1]$ , $T_n(x) = \cos(n \arccos x)$ . Outside $[-1,1]$ one uses the hyperbolic form

2

\quad T_n(x) = \cosh(n\,\mathrm{arccosh}\, x), \qquad x \ge 1.

Derivation hint: Replace $\theta$ by $i t$ so $\cos\to\cosh$ ; both forms agree as analytic continuations of the same polynomial. Here $n$ is the degree, $x$ the real argument, and $\theta$ the auxiliary angle.

DefinitionChebyshev Polynomials of the Second Kind

The Chebyshev polynomials of the second kind $U_n(x)$ are defined by

3

\quad U_n(\cos\theta) = \frac{\sin\big((n+1)\theta\big)}{\sin\theta}.

Derivation hint: The ratio is a polynomial of degree $n$ in $\cos\theta$ because $\sin((n+1)\theta)/\sin\theta$ is the Dirichlet-type kernel expandable via product-to-sum identities. They satisfy the link $T_n'(x) = n\,U_{n-1}(x)$ for $n \ge 1$ .

TheoremThree-Term Recurrence

Both families obey the same recurrence:

4

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

5

\quad U_{n+1}(x) = 2x\,U_n(x) - U_{n-1}(x), \qquad U_0 = 1,\; U_1 = 2x.

Derivation hint for [4]: Add the identities $\cos((n+1)\theta)+\cos((n-1)\theta)=2\cos\theta\cos(n\theta)$ and substitute $x=\cos\theta$ .

TheoremOrthogonality

On $[-1,1]$ , the first-kind polynomials are orthogonal with weight $w(x)=(1-x^2)^{-1/2}$ :

6

\quad \int_{-1}^{1} \frac{T_m(x)\,T_n(x)}{\sqrt{1-x^2}}\,dx = \begin{cases} 0, & m \ne n,\\ \pi, & m=n=0,\\ \tfrac{\pi}{2}, & m=n \ge 1. \end{cases}

Derivation hint: Substitute $x=\cos\theta$ , $dx=-\sin\theta\,d\theta$ ; the integral becomes $\int_0^\pi \cos(m\theta)\cos(n\theta)\,d\theta$ , the classical Fourier-cosine orthogonality. The second-kind polynomials are orthogonal with weight $w(x)=\sqrt{1-x^2}$ :

7

\quad \int_{-1}^{1} U_m(x)\,U_n(x)\,\sqrt{1-x^2}\,dx = \frac{\pi}{2}\,\delta_{mn}.

Derivation hint: Substitution $x=\cos\theta$ turns this into $\int_0^\pi \sin((m+1)\theta)\sin((n+1)\theta)\,d\theta$ .

TheoremMinimax / Extremal Property

Among all monic polynomials of degree $n \ge 1$ , the scaled Chebyshev polynomial $2^{1-n}T_n(x)$ has the smallest possible maximum absolute value on $[-1,1]$ :

8

\quad \min_{p \text{ monic}, \deg p = n}\;\max_{x\in[-1,1]} |p(x)| = \frac{1}{2^{\,n-1}}, \quad \text{attained by } p = 2^{1-n}T_n.

Derivation hint: $T_n$ equioscillates between $\pm1$ at $n+1$ points; any competing monic polynomial with smaller norm would force the difference to change sign $n$ times yet vanish too often — a contradiction with degree.

ProofEquioscillation forces minimality

We prove the extremal property [8].

Step 1. The leading coefficient of $T_n$ is $2^{n-1}$ (read off from the recurrence [4]: each step multiplies the leading coefficient by 2 starting from $T_1=x$ ). Hence $\tilde T_n := 2^{1-n}T_n$ is monic.

Step 2. Since $T_n(\cos\theta)=\cos(n\theta)$ , the values $\pm 1$ are attained at the $n+1$ points $x_k=\cos(k\pi/n)$ , $k=0,\dots,n$ , with alternating signs: $T_n(x_k)=(-1)^k$ . Thus $\tilde T_n$ attains $\pm 2^{1-n}$ alternately at these $n+1$ nodes.

Step 3. Suppose for contradiction some monic $q$ of degree $n$ has $\max_{[-1,1]}|q| < 2^{1-n}$ . Consider $d(x)=\tilde T_n(x)-q(x)$ . Both are monic of degree $n$ , so $d$ has degree $\le n-1$ .

Step 4. At each node $x_k$ , $\tilde T_n(x_k)=(-1)^k 2^{1-n}$ dominates $|q(x_k)|<2^{1-n}$ in magnitude, so $\mathrm{sign}\,d(x_k)=(-1)^k$ . The sign of $d$ alternates across $n+1$ nodes, forcing at least $n$ sign changes, hence $\ge n$ roots.

Step 5. A polynomial of degree $\le n-1$ with $\ge n$ roots must be identically zero, so $q=\tilde T_n$ — contradicting the strict inequality. $\blacksquare$

∎

ProofClosed-form power expansion

From De Moivre, $\cos(n\theta)+i\sin(n\theta)=(\cos\theta+i\sin\theta)^n$ . Taking real parts and using $\sin^2\theta=1-\cos^2\theta$ :

9

\quad T_n(x) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k}(x^2-1)^k\,x^{\,n-2k}.

Derivation hint: Expand $(\cos\theta + i\sin\theta)^n$ by the binomial theorem, keep even powers of $i\sin\theta$ , replace $\sin^2\theta = -(x^2-1)$ . This confirms $T_n$ is a genuine polynomial of degree $n$ with integer coefficients.

∎

PL

Pafnuty Lvovich Chebyshev

1821–1894

A founder of the St. Petersburg mathematical school, Chebyshev worked across number theory, probability, and mechanics. His 1854 memoir Théorie des mécanismes connus sous le nom de parallélogrammes sought the linkage geometry that best approximates straight-line motion — a practical engineering question about steam-engine mechanisms. Minimizing the deviation of a polynomial from zero led him directly to the polynomials now bearing his name. His student lineage (Markov, Lyapunov, Voronoy) shaped 20th-century analysis and probability.

🧠 Intuition

Picture wrapping the interval $[-1,1]$ onto the upper half of a unit circle via $x=\cos\theta$ . Equally spaced angles $\theta$ correspond to points $x$ that cluster near the endpoints $\pm 1$ . The function $T_n(x)=\cos(n\theta)$ is just a cosine running $n$ times faster in $\theta$ . So all of $T_n$ 's oscillations — exactly $n$ of them between $-1$ and $1$ — come from a uniform cosine seen through the "stretchy" lens of $\arccos$ . That lens-stretching at the endpoints is why Chebyshev interpolation nodes cluster there and avoid the Runge oscillation that ruins equally spaced interpolation.

ExampleComputing

T_5

via recurrence

Given: $T_0=1$ , $T_1=x$ . Find $T_5(x)$ and evaluate at $x=0.5$ .

Steps using [4]:

$T_2 = 2x\cdot x - 1 = 2x^2 - 1$
$T_3 = 2x(2x^2-1) - x = 4x^3 - 3x$
$T_4 = 2x(4x^3-3x) - (2x^2-1) = 8x^4 - 8x^2 + 1$
$T_5 = 2x(8x^4-8x^2+1) - (4x^3-3x) = 16x^5 - 20x^3 + 5x$

Check via trig: with $x=0.5=\cos(60^\circ)$ , $T_5(0.5)=\cos(300^\circ)=0.5$ . Direct substitution: $16(0.03125)-20(0.125)+5(0.5)=0.5-2.5+2.5=0.5$ . ✓

\boxed{T_5(x)=16x^5-20x^3+5x,\qquad T_5(0.5)=0.5}

ExampleMinimax error of monic interpolation

Given: We approximate a function on $[-1,1]$ by a monic cubic ( $n=3$ ). What is the smallest achievable peak value of the cubic, and where does it occur?

Steps: By [8], the minimizing monic cubic is $2^{1-3}T_3 = \tfrac14(4x^3-3x)=x^3-\tfrac34 x$ . Its max norm is $2^{1-3}=1/4=0.25$ . The extrema occur at $x_k=\cos(k\pi/3)$ , $k=0,1,2,3$ , i.e. $x=1,\;0.5,\;-0.5,\;-1$ , where the value equals $\pm 0.25$ alternately.

\boxed{\min\max|p| = \tfrac14 = 0.25,\quad p(x)=x^3-\tfrac34 x}

No monic cubic does better on $[-1,1]$ .

⚠️ Common mistake

$T_n(x)=\cos(n\arccos x)$ is only valid for $|x|\le 1$ . For $|x|>1$ , $\arccos x$ is not real; you must switch to the $\cosh$ form [2] or, more safely, just use the polynomial expansion or recurrence [4], which are valid for all real (and complex) $x$ . Students who plug $x=2$ into the $\arccos$ formula get nonsense; the polynomial $T_n(2)$ grows like $\cosh(n\,\mathrm{arccosh}\,2)$ , i.e. exponentially.

Counterexample

One might assume that equally spaced interpolation nodes are "fairest." Take $f(x)=1/(1+25x^2)$ on $[-1,1]$ (the Runge function) with 11 equally spaced nodes: the interpolating polynomial oscillates wildly near $\pm1$ , with error exceeding $7$ in magnitude. Replacing those with the 11 Chebyshev nodes $x_k=\cos\big((2k+1)\pi/22\big)$ tames the error to under $0.1$ . The equal-spacing intuition is exactly wrong — node clustering at the endpoints is essential.

connection

Chebyshev polynomials sit at a crossroads. As a special case of Jacobi Polynomials with parameters $\alpha=\beta=-\tfrac12$ , they inherit the general orthogonal-polynomial machinery (Rodrigues formula, second-order ODE). Via $x=\cos\theta$ they are a thin disguise of the Fourier Series cosine basis, so the Discrete Cosine Transform is essentially a Chebyshev transform. Their extremal property connects to Approximation Theory and the Weierstrass Approximation Theorem. The recurrence [4] is a model instance of Three-Term Recurrence Relations for orthogonal polynomials, and the second-kind family $U_n$ arises as characters of $SU(2)$ , linking to Representation Theory.

🔬 Where this appears

⏳ History

- 1854 — Chebyshev publishes his memoir on linkage mechanisms, defining the polynomials of least deviation from zero.

- 1859 — Chebyshev extends the theory to weighted approximation and continued fractions, foundational for orthogonal polynomials.

- 1885 — Weierstrass proves the approximation theorem; Chebyshev's minimax framework becomes the quantitative complement.

- 1901 — Runge publishes his counterexample on equally spaced interpolation, retroactively highlighting Chebyshev nodes' superiority.

- 1911 — D. Jackson's theorems quantify best-approximation rates, building on the Chebyshev extremal property.

- 1934 — A. N. Kolmogorov and the Moscow/St. Petersburg schools formalize Chebyshev systems and best-approximation theory.

- 1955 — Clenshaw publishes his recurrence-based summation algorithm for Chebyshev series.

- 1969 — Clenshaw and Curtis present their practical, FFT-friendly quadrature scheme.

- 2000s–present — The `chebfun` project (Trefethen et al., Oxford) revives Chebyshev technology as a complete numerical computing paradigm.

open-problem

While the univariate theory is essentially complete, the multivariate analogue is far less settled. There is no canonical "Chebyshev polynomial" on general multidimensional domains: minimizing the sup-norm over, say, a triangle or disk leads to hard, often unsolved extremal problems. The optimal point configurations (the multivariate analogue of Chebyshev nodes) for interpolation on simplices and balls remain active research, tied to the Fekete point and Lebesgue constant minimization problems. Even on $[-1,1]^2$ , characterizing the polynomial of least deviation among monic-in-each-variable polynomials is not fully resolved.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also