Chebyshev Polynomials

Chebyshev polynomials are a family of orthogonal polynomials

T_{n} (x)

(first kind) and

U_{n} (x)

(second kind) defined by

T_{n} (cos θ) = cos (n θ)

. 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 Chebyshev polynomials of the first kind, $T_{n} (x)$ , are degree- $n$ polynomials satisfying the deceptively simple identity $T_{n} (cos θ) = cos (n θ)$ . From this trigonometric disguise emerge nearly all their remarkable properties: orthogonality on $[- 1, 1]$ with weight $(1 - x^{2})^{- 1/2}$ , a three-term recurrence, and an extremal "equioscillation" property that minimizes the maximum error. They matter because, among all monic polynomials of fixed degree, the (scaled) Chebyshev polynomial deviates least from zero on $[- 1, 1]$ — a fact that underlies optimal polynomial interpolation, spectral methods, and economical function approximation. The surprising fact: a problem about wiggly polynomials becomes trivial once you change variables to an angle and recognize $cos (n θ)$ .

The Core Insight

Set $x = cos θ$ . The map $x \mapsto n θ$ turns multiplication of angles into a polynomial in $x$ , because $cos (n θ)$ is always a polynomial in $cos θ$ (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."

Formal Statement

Chebyshev Polynomials of the First Kind

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

T_{n} (cos θ) = cos (n θ), θ \in [0, π] .

⟳Substituting

x = cos θ

bijects

[0, π] \to [- 1, 1]

; the right side is a polynomial in

cos θ

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

T_{n} (x) = cosh (n arccosh x), x \geq 1.

⟳Replace

θ

i t

cos \to cosh

; both forms agree as analytic continuations of the same polynomial.

Here $n$ is the degree, $x$ the real argument, and $θ$ the auxiliary angle.

Chebyshev Polynomials of the Second Kind

The Chebyshev polynomials of the second kind $U_{n} (x)$ are defined by

U_{n} (cos θ) = \frac{sin ( ( n + 1 ) θ )}{sin θ} .

⟳The ratio is a polynomial of degree

n

cos θ

because

sin ((n + 1) θ) / sin θ

is the Dirichlet-type kernel expandable via product-to-sum identities.

They satisfy the link $T_{n}^{'} (x) = n U_{n - 1} (x)$ for $n \geq 1$ .

Three-Term Recurrence

Both families obey the same recurrence:

T_{n + 1} (x) = 2 x T_{n} (x) - T_{n - 1} (x), T_{0} = 1, T_{1} = x,

U_{n + 1} (x) = 2 x U_{n} (x) - U_{n - 1} (x), U_{0} = 1, U_{1} = 2 x .

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

Orthogonality

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

\int_{- 1}^{1} \frac{T _{m} ( x ) T _{n} ( x )}{1 - x ^{2}} d x = ⎩ ⎨ ⎧ 0, π, \frac{π}{2}, m \neq = n, m = n = 0, m = n \geq 1.

⟳Substitute

x = cos θ

d x = - sin θ d θ

; the integral becomes

\int_{0}^{π} cos (m θ) cos (n θ) d θ

, the classical Fourier-cosine orthogonality.

The second-kind polynomials are orthogonal with weight $w (x) = 1 - x^{2}$ :

\int_{- 1}^{1} U_{m} (x) U_{n} (x) 1 - x^{2} d x = \frac{π}{2} δ_{mn} .

⟳Substitution

x = cos θ

turns this into

\int_{0}^{π} sin ((m + 1) θ) sin ((n + 1) θ) d θ

Minimax / Extremal Property

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

p monic, d e g p = n min x \in [- 1, 1] max ∣ p (x) ∣ = \frac{1}{2 ^{n - 1}}, attained by p = 2^{1 - n} T_{n} .

⟳

T_{n}

equioscillates between

\pm 1

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.

Why It Holds / Derivation

Equioscillation forces minimality

We prove the extremal property [8].

Step 1. The leading coefficient of

T_{n}

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 θ) = cos (n θ)

, the values

\pm 1

are attained at the

n + 1

points

x_{k} = cos (k π / 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

\leq 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

sign d (x_{k}) = (- 1)^{k}

. The sign of

d

alternates across

n + 1

nodes, forcing at least

n

sign changes, hence

\geq n

roots. Step 5. A polynomial of degree

\leq n - 1

with

\geq n

roots must be identically zero, so

q = \tilde{T}_{n}

— contradicting the strict inequality.

■

Closed-form power expansion

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

T_{n} (x) = k = 0 \sum ⌊ n /2 ⌋ (2 k n) (x^{2} - 1)^{k} x^{n - 2 k} .

⟳Expand

(cos θ + i sin θ)^{n}

by the binomial theorem, keep even powers of

i sin θ

, replace

sin^{2} θ = - (x^{2} - 1)

This confirms $T_{n}$ is a genuine polynomial of degree $n$ with integer coefficients.

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.

Building Intuition

Picture wrapping the interval $[- 1, 1]$ onto the upper half of a unit circle via $x = cos θ$ . Equally spaced angles $θ$ correspond to points $x$ that cluster near the endpoints $\pm 1$ . The function $T_{n} (x) = cos (n θ)$ is just a cosine running $n$ times faster in $θ$ . 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.

:::svg[The first four Chebyshev polynomials $T_{n} (x)$ on $[- 1, 1]$ , oscillating between $\pm 1$ .]

Worked Examples

Computing $T_5$ via recurrence

Given:

T_{0} = 1

T_{1} = x

. Find

T_{5} (x)

and evaluate at

x = 0.5

. Steps using [4]:

$T_{2} = 2 x \cdot x - 1 = 2 x^{2} - 1$
$T_{3} = 2 x (2 x^{2} - 1) - x = 4 x^{3} - 3 x$
$T_{4} = 2 x (4 x^{3} - 3 x) - (2 x^{2} - 1) = 8 x^{4} - 8 x^{2} + 1$
$T_{5} = 2 x (8 x^{4} - 8 x^{2} + 1) - (4 x^{3} - 3 x) = 16 x^{5} - 20 x^{3} + 5 x$

Check via trig: with

x = 0.5 = cos (6 0^{\circ})

T_{5} (0.5) = cos (30 0^{\circ}) = 0.5

. Direct substitution:

16 (0.03125) - 20 (0.125) + 5 (0.5) = 0.5 - 2.5 + 2.5 = 0.5

. ✓

T_{5} (x) = 16 x^{5} - 20 x^{3} + 5 x, T_{5} (0.5) = 0.5

Minimax 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} = \frac{1}{4} (4 x^{3} - 3 x) = x^{3} - \frac{3}{4} x

. Its max norm is

2^{1 - 3} = 1/4 = 0.25

The extrema occur at $x_{k} = cos (k π /3)$ , $k = 0, 1, 2, 3$ , i.e. $x = 1, 0.5, - 0.5, - 1$ , where the value equals $\pm 0.25$ alternately.

min max ∣ p ∣ = \frac{1}{4} = 0.25, p (x) = x^{3} - \frac{3}{4} x

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

Common Misconceptions

$T_{n} (x) = cos (n arccos x)$ is only valid for $∣ x ∣ \leq 1$ . 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 arccosh 2)

, i.e. exponentially.

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

Connections

Chebyshev polynomials sit at a crossroads. As a special case of Jacobi Polynomials with parameters $α = β = - \frac{1}{2}$ , they inherit the general orthogonal-polynomial machinery (Rodrigues formula, second-order ODE). Via $x = cos θ$ 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 $S U (2)$ , linking to Representation Theory.

Real-World Applications

Chebyshev (Clenshaw–Curtis) quadrature — numerical integration using Chebyshev nodes converges rapidly for smooth integrands; built into MATLAB's chebfun.
Spectral methods for PDEs — Chebyshev collocation discretizes derivatives in fluid dynamics and weather modeling, achieving exponential accuracy on bounded domains.
Filter design (electronics) — Chebyshev filters trade equiripple passband behavior for sharper roll-off than Butterworth filters, derived directly from $T_{n}$ .
Function approximation in libraries — cos, exp, and special functions in math libraries are often evaluated via economized Chebyshev series ("Chebyshev economization").
Clenshaw's algorithm — stable summation of Chebyshev series, used in spacecraft ephemeris evaluation (JPL planetary tables store positions as Chebyshev coefficients).
Image/Signal compression — the Discrete Cosine Transform underlying JPEG is the Chebyshev transform in disguise.
Solving polynomial equations — Chebyshev's minimal-deviation property informs conditioning and root-finding in numerical linear algebra (companion-matrix colleague matrices).

Historical Development

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.

Subtleties & Limits

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.