Chapter 1: Introduction

A compiled PDF version of this blueprint is available at print/blueprint.pdf.

The Chebyshev polynomials of the first kind, denoted $T_N(x)$, form one of the most important families of orthogonal polynomials, with applications spanning approximation theory, numerical analysis, and harmonic analysis. These polynomials are typically defined either through the trigonometric identity

\begin{equation}...\end{equation}

or via the three-term recurrence relation

\begin{equation}...\end{equation}

In this project, we present a novel geometric construction of Chebyshev polynomials using rotated roots of unity. Consider the $N$-th roots of unity on the complex unit circle:

\begin{equation}...\end{equation}

These form a regular $N$-gon inscribed in the unit circle, with vertices equally spaced at angular intervals of $2\pi/N$.

The Chebyshev polynomial construction begins with the $N$-th roots of unity on the complex unit circle. These form a regular $N$-gon inscribed in the unit circle, with vertices equally spaced at angular intervals of $2\pi/N$.

Definition 1.1

[blueprint]

When the $N$-th roots of unity are rotated by an angle $\theta$ (equivalently, multiplied by $e^{i\theta}$), we obtain the $N$ complex numbers $e^{i(\theta + 2\pi k/N)}$ for $k = 0, 1, \ldots, N-1$. The rotation angle $\theta$ controls the orientation of the $N$-gon.

def rotatedRootsOfUnityList (N : ℕ) (θ : ℝ) : List ℂ :=
  List.range N |>.map (fun k => exp (I * (θ + 2 * π * k / N)))

As $\theta$ varies from $0$ to $2\pi$, the rotated roots sweep through all possible configurations of $N$ equally-spaced points on the unit circle. Projecting these onto the real axis yields the inputs to the polynomial construction.

As $\theta$ varies from $0$ to $2\pi$, the projected roots sweep through all possible configurations of $N$ points on $[-1, 1]$ constrained by the symmetry of the roots of unity.

Definition 1.2

[blueprint]

Given a list of real numbers $[r_1, r_2, \ldots, r_n]$, form the monic polynomial $\prod_{i=1}^{n}(X - r_i)$. This is the unique monic polynomial of degree $n$ having exactly those roots (counted with multiplicity).

def polynomialFromRealRoots (roots : List ℝ) : Polynomial ℝ :=
  roots.foldr (fun r p => (X - C r) * p) 1

To match the normalization of Chebyshev polynomials, which have leading coefficient $2^{N-1}$ for $N \geq 1$, we scale the monic polynomial $P_N$ by this factor.

Definition 1.3

[blueprint]

The scaled polynomial is defined as $$S_N(x;\theta) = 2^{N-1} \cdot P_N(x;\theta).$$ It inherits the degree and roots from $P_N$, with leading coefficient $2^{N-1}$.

def scaledPolynomial (N : ℕ) (θ : ℝ) : Polynomial ℝ :=
  C (2 ^ (N - 1)) * unscaledPolynomial N θ

The main theorem establishes that $S_N(x;\theta) = T_N(x) + c(\theta)$, so only the constant term varies with $\theta$. All higher-degree coefficients are $\theta$-independent and match those of $T_N$.

Our main result establishes that this construction yields the Chebyshev polynomial up to an additive constant.

We now assemble the pieces to prove the main theorem. By the power sum equality theorem, the rotated root projections $r_k(\theta)$ and the Chebyshev roots $\xi_k$ have matching power sums for $1 \leq j < N$. Newton's identities then imply that the elementary symmetric polynomials match, so the polynomial coefficients of degree $k \geq 1$ are identical.

Theorem 1.4

[blueprint]

For any $N \geq 1$ and $\theta \in \mathbb{R}$, $S_N(x;\theta) = T_N(x) + c(N,\theta)$.

theorem mainTheorem : StatementOfTheorem := by

Proof ▼

The proof proceeds by coefficient comparison via \texttt{ext n}. For $n \geq 1$: the $n$-th coefficient of $S_N$ is $\theta$-invariant (\texttt{constant\_term\_only\_varies}), so it equals the coefficient at $\theta = 0$, which matches $T_N$ by \texttt{scaledPolynomial\_matches\_chebyshev\_at\_zero}. For $n = 0$: the explicit constant $c(N,\theta)$ is chosen precisely so that the constant terms agree.


  intro N θ hN
  -- Prove equality coefficient by coefficient
  ext n
  simp only [coeff_add, coeff_C]
  by_cases hn : n = 0
  · -- Constant term: explicitConstant is chosen exactly to make this work
    simp [hn]
    rw [explicitConstant_correct N θ hN]
    ring
  · -- Non-constant terms: θ-invariance + match at θ=0
    simp [hn]
    have h_pos : 0 < n := Nat.pos_of_ne_zero hn
    calc (scaledPolynomial N θ).coeff n
        = (scaledPolynomial N 0).coeff n := constant_term_only_varies N θ 0 n hN h_pos
      _ = (Chebyshev.T ℝ N).coeff n := scaledPolynomial_matches_chebyshev_at_zero N n hN h_pos

This completes the proof that Chebyshev polynomials emerge from projections of rotated roots of unity. The only degree of freedom under rotation is the constant term, reflecting that the $N$-th power sum is the only one that can depend on $\theta$.

Remark

(See \href{../docs/ChebyshevCircles/ProofOfMainTheorem.html mainTheorem}{API Documentation} | \href{../paper/chebyshev_circles.pdf}{Paper Theorem 1.1})

Significance and context

This theorem reveals several deep connections:

Discrete geometry meets polynomial theory. The result shows that Chebyshev polynomials emerge naturally from the simplest discrete geometric object---equally-spaced points on a circle---through elementary operations (rotation and projection).
Power sum invariance. The proof hinges on showing that certain discrete sums of powers of cosines are invariant under the phase shift $\theta$. This invariance reflects discrete orthogonality properties of trigonometric functions, a finite analog of continuous Fourier analysis.
Symmetric function theory. Newton's identities provide the bridge from power sums to elementary symmetric polynomials, and hence to polynomial coefficients. The theorem demonstrates this classical algebraic machinery in action.
Roots and quadrature. The projected roots $r_k(0) = \cos(2\pi k/N)$ are closely related to the Chebyshev-Gauss quadrature nodes $\cos((2k+1)\pi/(2N))$, both arising from equidistributed points on the circle.

Computational illustration: N = 5

To make the construction concrete, consider $N = 5$ with $\theta = 0$. The five roots of unity are

\begin{equation}...\end{equation}

positioned at angles $0°, 72°, 144°, 216°, 288°$ on the unit circle. Their real projections are

\begin{equation}...\end{equation}

The monic polynomial with these roots is

\begin{equation}...\end{equation}

which simplifies to a polynomial of degree 5. Scaling by $2^4 = 16$ gives

\begin{equation}...\end{equation}

where the coefficients of $x^5, x^3, x$ match those of $T_5(x) = 16x^5 - 20x^3 + 5x$, differing only in the constant term.

Rotating by $\theta = \pi/10$ shifts the roots but preserves the polynomial coefficients (except the constant), demonstrating the invariance principle.

Proof strategy

The proof of Theorem~thm:main proceeds through several key steps:

Discrete orthogonality. We establish that sums of complex exponentials and cosines over the $N$-th roots of unity satisfy discrete orthogonality relations analogous to Fourier orthogonality.
Power sum invariance. Using binomial expansion of $\cos^j(\theta + 2\pi k/N)$ and the orthogonality relations, we prove that the power sums \begin

equation

\sum_

k=0

^

N-1

\cos^j\left(\theta + \frac

2\pi k

N

\right)

are independent of $\theta$ for all $1 \leq j < N$.

We apply Newton's identities to show that if two sets of roots have identical power sums (except for the 0-th power sum, which counts the number of roots), their elementary symmetric polynomials match. Consequently, the monic polynomials with these roots have identical coefficients except possibly the constant term.

We identify the roots of $T_N(x)$ as $\cos((2k + 1)\pi/(2N))$ for $k = 0, \ldots, N-1$, and verify that these roots have the same power sums as the rotated roots $r_k(\theta)$. This establishes coefficient-wise equality for degrees $k \geq 1$.

For $\theta = 0$, explicit calculation shows $S_N(x; 0) = T_N(x)$, confirming the identification up to the constant term.

1.1 Significance and context

This theorem reveals several deep connections:

Discrete geometry meets polynomial theory. The result shows that Chebyshev polynomials emerge naturally from the simplest discrete geometric object---equally-spaced points on a circle---through elementary operations (rotation and projection).
Power sum invariance. The proof hinges on showing that certain discrete sums of powers of cosines are invariant under the phase shift $\theta$. This invariance reflects discrete orthogonality properties of trigonometric functions, a finite analog of continuous Fourier analysis.
Symmetric function theory. Newton's identities provide the bridge from power sums to elementary symmetric polynomials, and hence to polynomial coefficients. The theorem demonstrates this classical algebraic machinery in action.
Roots and quadrature. The projected roots $r_k(0) = \cos(2\pi k/N)$ are closely related to the Chebyshev-Gauss quadrature nodes $\cos((2k+1)\pi/(2N))$, both arising from equidistributed points on the circle.

1.2 Computational illustration: N = 5

To make the construction concrete, consider $N = 5$ with $\theta = 0$. The five roots of unity are

\begin{equation}...\end{equation}

positioned at angles $0°, 72°, 144°, 216°, 288°$ on the unit circle. Their real projections are

\begin{equation}...\end{equation}

The monic polynomial with these roots is

\begin{equation}...\end{equation}

which simplifies to a polynomial of degree 5. Scaling by $2^4 = 16$ gives

\begin{equation}...\end{equation}

where the coefficients of $x^5, x^3, x$ match those of $T_5(x) = 16x^5 - 20x^3 + 5x$, differing only in the constant term.

Rotating by $\theta = \pi/10$ shifts the roots but preserves the polynomial coefficients (except the constant), demonstrating the invariance principle.

1.3 Proof strategy

The proof of Theorem~thm:main proceeds through several key steps:

Discrete orthogonality. We establish that sums of complex exponentials and cosines over the $N$-th roots of unity satisfy discrete orthogonality relations analogous to Fourier orthogonality.
Power sum invariance. Using binomial expansion of $\cos^j(\theta + 2\pi k/N)$ and the orthogonality relations, we prove that the power sums \begin

equation

\sum_

k=0

^

N-1

\cos^j\left(\theta + \frac

2\pi k

N

\right)

are independent of $\theta$ for all $1 \leq j < N$.

For $\theta = 0$, explicit calculation shows $S_N(x; 0) = T_N(x)$, confirming the identification up to the constant term.