1 Introduction

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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} T_N(\cos \theta ) = \cos (N\theta ), \end{equation}

or via the three-term recurrence relation

\begin{equation} T_0(x) = 1, \quad T_1(x) = x, \quad T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x). \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} \omega _k = \mathrm{e}^{2\pi i k/N}, \quad k = 0, 1, \ldots , N-1. \end{equation}

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

Definition 1.0.1 Rotated roots

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When the \(N\)-th roots of unity are rotated by an angle \(\theta \) (equivalently, multiplied by \(\mathrm{e}^{i\theta }\)) and projected onto the real axis, we obtain the \(N\) real numbers

\begin{equation} r_k(\theta ) = \cos \left(\theta + \frac{2\pi k}{N}\right), \quad k = 0, 1, \ldots , N-1. \end{equation}

The rotation angle \(\theta \) controls the orientation of the \(N\)-gon, while the projection yields a multiset of real values in \([-1, 1]\).

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 1.0.2 Unscaled polynomial

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Let \(P_N(x; \theta )\) denote the monic polynomial having the rotated roots as roots:

\begin{equation} P_N(x; \theta ) = \prod _{k=0}^{N-1} \left(x - r_k(\theta )\right). \end{equation}

Since the roots are real and lie in \([-1, 1]\), this polynomial has real coefficients and degree \(N\), and each \(r_k(\theta )\) is a root.

Definition 1.0.3 Scaled polynomial

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To match the normalization of Chebyshev polynomials, which have leading coefficient \(2^{N-1}\) for \(N \geq 1\), we define the scaled polynomial:

\begin{equation} S_N(x; \theta ) = 2^{N-1} \cdot P_N(x; \theta ). \end{equation}

The scaled polynomial inherits the degree and roots from \(P_N\).

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

Theorem 1.0.4 Main Theorem

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For any positive integer \(N \geq 1\) and any angle \(\theta \in \mathbb {R}\),

\begin{equation} \label{eq:main_result} S_N(x; \theta ) = T_N(x) + c(\theta ), \end{equation}

where the constant \(c(\theta )\) is given explicitly by

\begin{equation} \label{eq:explicit_constant} c(\theta ) = 2^{N-1} \cdot (-1)^N \cdot \prod _{k=0}^{N-1} \cos \left(\theta + \frac{2\pi k}{N}\right) - T_N(0). \end{equation}

Moreover, all coefficients of \(S_N(x; \theta )\) of degree \(k \geq 1\) are independent of \(\theta \) and equal the corresponding coefficients of \(T_N(x)\).

Remark 1.0.5

(See ../docs/ChebyshevCircles/ProofOfMainTheorem.html#mainTheoremAPI Documentation | ../paper/chebyshev_circles.pdfPaper Theorem 1.1)