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Chebyshev extremals and the exact disk Fekete polynomial

Statement

Assume the Axiom of Choice. Let $K = \overline{D (a, R)}$ be a closed disc with $R > 0$ , and let $[- 1, 1] \subseteq R \subseteq C$ be the real unit interval, with the extremal norms $t_{n}$ and Chebyshev constant $cheb$ of Chebyshev constant of a compact planar set and the capacity of Robin constant and logarithmic capacity of a compact set.

For the disc, $t_{n} (K) = R^{n}$ and $cheb (K) = R = cap (K)$ , attained by the monic polynomial $(z - a)^{n}$ .
For the interval, $t_{n} ([- 1, 1]) = 2^{1 - n}$ and $cheb ([- 1, 1]) = \frac{1}{2} = cap ([- 1, 1])$ , attained by the monic Chebyshev polynomial $2^{1 - n} T_{n}$ .
For each $n \geq 2$ the $n$ -th roots of unity form an $n$ -point Fekete tuple of the closed unit disc, with Fekete polynomial $F_{n} (z) = z^{n} - 1$ ; and $δ_{n} (\overline{D (0, 1)}) = n^{1 / (n - 1)}$ .
The empirical probability measures of these tuples converge weakly to normalized arclength on the unit circle, and $∥ z^{n} - 1 ∥_{\overline{D (0, 1)}} = 2$ , so the $n$ -th root of its norm tends to $1 = cheb (\overline{D (0, 1)})$ .

The Axiom of Choice is inherited from the equilibrium theory that supplies the two capacity values and the weak convergence; the Cauchy estimate, the minimax comparison, the Vandermonde determinant and the Gram–Schmidt bound are choice-free.

Facts & Assumptions

Given: a closed disc $K = \overline{D (a, R)}$ with $R > 0$ , the interval $[- 1, 1]$ , the closed unit disc $\overline{D (0, 1)}$ , and the conventions of Chebyshev constant of a compact planar set, Fekete points and the transfinite diameter of a compact set and Robin constant and logarithmic capacity of a compact set.

[F1]

For nonempty compact $K$ : $∥ p ∥_{K} = \sup_{z \in K} ∣ p (z) ∣$ is finite and attained, $t_{n} (K) = \inf {∥ p ∥_{K} : p monic of degree n}$ is a real number, $cheb (K) = \inf_{n \geq 1} t_{n} (K)^{1 / n}$ , and for the disc and the interval the capacities are $cap (\overline{D (a, R)}) = R$ and $cap ([- 1, 1]) = \frac{1}{2}$ (Chebyshev constant of a compact planar set, Capacity of a disc and its circular equilibrium measure, Arcsine equilibrium measure and capacity of a segment).

[F2]

For nonempty compact $K$ and $n \geq 2$ the $n$ -th Fekete diameter is $δ_{n} (K) = (\max_{z \in K^{n}} \prod_{i < j} ∣ z_{i} - z_{j} ∣)^{2 / [n (n - 1)]}$ , tuples attaining the maximum are Fekete tuples, and the associated polynomial $F_{n} (Z) = \prod_{j} (Z - z_{j})$ is monic of degree $n$ (Fekete points and the transfinite diameter of a compact set, Formal complex polynomials, evaluation, degree, leading coefficient, and monic polynomials).

[F3]

Assume the Axiom of Choice. For every compact $K$ one has $cap (K) = τ (K) = cheb (K)$ ; if $cap (K) > 0$ , then the empirical probability measures of any sequence of $n$ -point Fekete tuples of $K$ converge weakly to the unique equilibrium measure $μ_{K}$ (Fekete–Szegő equality of logarithmic capacity, transfinite diameter, and Chebyshev constant, Weak convergence of borel probability measures).

[F4]

Let $f$ be holomorphic on $D (a, R)$ and let $0 < r < R$ with $∣ f (ζ) ∣ \leq M$ on $∣ ζ - a ∣ = r$ ; then $∣ f^{(n)} (a) ∣ \leq n! M / r^{n}$ (Cauchy's inequalities bound every derivative by a boundary bound on a compactly contained circle). A complex polynomial $P (z) = \sum_{k = 0}^{n} a_{k} z^{k}$ is entire with $P^{'} (z) = \sum_{k = 1}^{n} k a_{k} z^{k - 1}$ , so by iteration the $n$ -th derivative of a monic polynomial of degree $n$ is the constant $n!$ (Complex polynomials are entire with the power-rule derivative, and rational functions are holomorphic wherever their denominator is nonzero).

[F5]

For $n \geq 1$ the polynomial $P_{n} = 2^{1 - n} T_{n}$ is monic of degree $n$ , and for every monic real polynomial $q$ of degree $n$ one has $\max_{x \in [- 1, 1]} ∣ q (x) ∣ \geq 2^{1 - n} = \max_{x \in [- 1, 1]} ∣ P_{n} (x) ∣$ (For $n \geq 1$ , $2^{1 - n} T_{n}$ is the minimax monic polynomial of degree $n$ on $[- 1, 1]$ , Chebyshev polynomials of the first and second kinds by their three-term recurrences, $T_{n} (\cos θ) = \cos (n θ)$ and $U_{n} (\cos θ) \sin θ = \sin ((n + 1) θ)$ for every $n \in N$ ).

[F6]

A complex polynomial $p$ of degree $n$ has real part $Re p$ , a real polynomial whose $x^{n}$ coefficient is the real part of the $x^{n}$ coefficient of $p$ ; hence $Re p$ is monic of degree $n$ when $p$ is monic, and $∣ Re p (x) ∣ \leq ∣ p (x) ∣$ for every real $x$ (Formal real polynomials, evaluation, degree, leading coefficient, and monic polynomials, $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ , Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[F7]

Determinant conventions and rules: the determinant is the Leibniz sum $\det (A) = \sum_{σ} sgn (σ) \prod_{i} a_{σ (i), i}$ (For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix); it is alternating and multilinear in the rows and columns (The Leibniz determinant is column-multilinear, alternating and normalized over every commutative ring); $\det (A B) = \det A \det B$ (For same-sized finite square matrices over a commutative ring, $\det (A B) = \det (A) \det (B)$ ); and the determinant of an upper triangular matrix is the product of its diagonal entries (The determinant of a triangular matrix is the product of its diagonal entries).

[F8]

Inner products, norms and orthogonalisation: $∥ v ∥^{2} = ⟨ v, v ⟩$ for the induced norm (Real and complex inner product spaces, with the inner product linear in the first argument, The norm $∥ v ∥ = \sqrt{⟨ v, v ⟩}$ induced by a real or complex inner product); a finite linearly independent list has an orthonormal list spanning the same successive spans (Gram–Schmidt turns every finite independent list into an orthonormal list with the same successive spans); for orthogonal vectors $∥ u + v ∥^{2} = ∥ u ∥^{2} + ∥ v ∥^{2}$ (Pythagoras, the parallelogram identity, and the real and complex polarisation identities); and a linear isometry carrying an orthonormal basis to an orthonormal basis satisfies $∣ \det T ∣ = 1$ (Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces, For an endomorphism in finite dimension, preserving lengths, preserving inner products, carrying orthonormal bases to orthonormal bases, and $T^{*} T = I$ are equivalent, Orthogonal and unitary operators form groups, and their determinants have modulus one).

[F9]

The $n$ -th roots of unity $ω_{j} = \exp (2 π i j / n)$ , $j = 0, \dots, n - 1$ , are $n$ distinct complex numbers of modulus one, and $z^{n} - 1$ has exactly these $n$ roots, each of multiplicity one (The $n$ -th roots of a complex number and the $n$ distinct roots of unity for every $n \geq 1$ , The complex exponential by its power series, $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , Integer powers in the complex field); a monic polynomial of degree $n$ with this root list is $\prod_{j} (z - ω_{j})$ , by uniqueness of the root factorisation (A complex polynomial of degree $n$ has exactly $n$ roots counted with multiplicity). The Vandermonde polynomial is $Δ_{n} (x_{1}, \dots, x_{n}) = \prod_{i < j} (x_{i} - x_{j})$ (The Vandermonde polynomial $Δ_{n} = \prod_{i < j} (x_{i} - x_{j})$ ).

[F10]

Limits and roots: $t^{1 / n} \to 1$ for every fixed $t > 0$ (For every $a > 0$ , $a^{1 / n} \to 1$ ); sums, products and quotients of convergent sequences converge to the corresponding combination (Algebra of limits: sums, scalar multiples, products and quotients); nonnegative $n$ -th roots exist and are monotone (Existence and uniqueness of $n$ -th roots: a unique $a^{1 / n} \geq 0$ with $(a^{1 / n})^{n} = a$ , Monotonicity of $x \mapsto x^{n}$ and of $n \mapsto a^{n}$ ).

Verification

technique · direct

1.1F1F4algebra

Disc: Cauchy lower bound. Let $p$ be a monic complex polynomial of degree $n$ and $K = \overline{D (a, R)}$ . By [F4] the polynomial $p$ is entire, so it is holomorphic on $D (a, R^{'})$ for every $R^{'} > R$ , and its $n$ -th derivative is the constant $p^{(n)} = n!$ ; applying Cauchy's inequality [F4] with $r = R$ and $M = ∥ p ∥_{K}$ , and noting that $∣ p ∣ \leq ∥ p ∥_{K}$ holds on the circle $∣ ζ - a ∣ = R$ , gives $n! = ∣ p^{(n)} (a) ∣ \leq n! ∥ p ∥_{K} / R^{n}$ , hence $∥ p ∥_{K} \geq R^{n}$ .

1.2F5F6algebra

Interval: complex-to-real reduction. Let $p$ be a monic complex polynomial of degree $n$ and put $q : = Re p$ . By [F6] the polynomial $q$ is a monic real polynomial of degree $n$ with $∣ q (x) ∣ \leq ∣ p (x) ∣$ for all real $x$ , so $\max_{x \in [- 1, 1]} ∣ q (x) ∣ \leq ∥ p ∥_{[- 1, 1]}$ ; the minimax theorem [F5] gives $2^{1 - n} \leq \max_{x \in [- 1, 1]} ∣ q (x) ∣$ , so $∥ p ∥_{[- 1, 1]} \geq 2^{1 - n}$ .

1.3F7algebra

Vandermonde determinant. For $m \geq 1$ and $w_{0}, \dots, w_{m - 1} \in C$ the matrix $V_{m} = (w_{j}^{k})_{j, k = 0}^{m - 1}$ satisfies $\det V_{m} = \prod_{i < j} (w_{j} - w_{i})$ . Indeed for $m = 1$ both sides are $1$ ; for $m \geq 2$ , replacing the $j$ -th row of $V_{m}$ by itself minus the first row does not change the determinant by [F7], because the determinant is multilinear and alternating and the subtracted term has two equal rows; the new $j$ -th row is $(w_{j} - w_{0}) (0, q_{1} (w_{j}), \dots, q_{m - 1} (w_{j}))$ with $q_{k} (z) = z^{k - 1} + z^{k - 2} w_{0} + \dots + w_{0}^{k - 1}$ , so multilinearity [F7] factors $\prod_{j \geq 1} (w_{j} - w_{0})$ out of the last $m - 1$ rows; the remaining matrix has first row $(1, w_{0}, \dots, w_{0}^{m - 1})$ and, below it, zeros in the first column, which Leibniz's formula [F7] evaluates as the determinant of the $(m - 1) \times (m - 1)$ matrix of the polynomials $q_{k}$ at $w_{1}, \dots, w_{m - 1}$ ; since $q_{k} (z) = z^{k - 1} + (lower order terms)$ , the basis change from $(1, z, \dots, z^{m - 2})$ to $(q_{1}, \dots, q_{m - 1})$ is unitriangular and hence a determinant-preserving sequence of column operations [F7], so that last determinant equals $\det V_{m - 1} (w_{1}, \dots, w_{m - 1})$ .

1.4F7F8algebra

Gram–Schmidt (Hadamard) bound. Let $c_{0}, \dots, c_{m - 1} \in C^{m}$ be the columns of a matrix $A$ . If they are linearly dependent then $\det A = 0$ by the alternating multilinearity in [F7]; otherwise [F8] supplies an orthonormal list $e_{0}, \dots, e_{m - 1}$ with $span (e_{0}, \dots, e_{k}) = span (c_{0}, \dots, c_{k})$ for every $k$ , and writing $u_{k} : = c_{k} - \sum_{j < k} ⟨ c_{k}, e_{j} ⟩ e_{j}$ one has $c_{k} = \sum_{j \leq k} ⟨ c_{k}, e_{j} ⟩ e_{j}$ , so $A = Q R$ with $Q$ the matrix of the $e_{j}$ and $R$ upper triangular with diagonal entries $R_{k k} = ⟨ c_{k}, e_{k} ⟩ = ∥ u_{k} ∥$ ; moreover $∥ u_{k} ∥ \leq ∥ c_{k} ∥$ , because $u_{k}$ is orthogonal to $\sum_{j < k} ⟨ c_{k}, e_{j} ⟩ e_{j}$ and Pythagoras [F8] gives $∥ c_{k} ∥^{2} = ∥ u_{k} ∥^{2} + ∥ \sum_{j < k} ⟨ c_{k}, e_{j} ⟩ e_{j} ∥^{2}$ . The matrix $Q$ has orthonormal columns, so it is a linear isometry of $C^{m}$ and $∣ \det Q ∣ = 1$ by [F8]; hence by [F7] $∣ \det A ∣ = ∣ \det Q ∣ ∣ \det R ∣ = \prod_{k} R_{k k} = \prod_{k} ∥ u_{k} ∥ \leq \prod_{k} ∥ c_{k} ∥$ .

2.1step 1.1F1algebra

Disc: extremal value. The monic polynomial $(z - a)^{n}$ has $∥ (z - a)^{n} ∥_{K} = R^{n}$ , so step 1.1 gives $t_{n} (K) = R^{n}$ for every $n \geq 1$ ; hence $cheb (K) = \inf_{n \geq 1} t_{n} (K)^{1 / n} = \inf_{n \geq 1} R = R$ .

2.2step 1.2F5F10algebra

Interval: extremal value and Chebyshev constant. By [F5] the monic polynomial $P_{n} = 2^{1 - n} T_{n}$ has $\max_{x \in [- 1, 1]} ∣ P_{n} (x) ∣ = 2^{1 - n}$ , so step 1.2 gives $t_{n} ([- 1, 1]) = 2^{1 - n}$ ; therefore $cheb ([- 1, 1]) = \inf_{n \geq 1} 2^{(1 - n) / n}$ , and since $2^{(1 - n) / n} = 2^{1 / n} / 2$ with $2^{1 / n} \to 1$ by [F10], that infimum is $\frac{1}{2}$ .

2.3step 1.3F9algebra

Vandermonde identity by induction. Induction on $m$ in the recursion of step 1.3 gives $\det V_{m} = \prod_{j \geq 1} (w_{j} - w_{0}) \cdot \prod_{1 \leq i < j \leq m - 1} (w_{j} - w_{i}) = \prod_{i < j} (w_{j} - w_{i})$ for every $m \geq 1$ ; in particular $∣ Δ_{n} (z_{0}, \dots, z_{n - 1}) ∣ = ∣ \det V_{n} ∣$ by [F9].

3.1step 2.1F1

Disc: capacity. By [F1], $cap (K) = R$ , so $cheb (K) = R = cap (K)$ .

3.2step 2.2F1

Interval: capacity. By [F1], $cap ([- 1, 1]) = \frac{1}{2}$ , so $cheb ([- 1, 1]) = \frac{1}{2} = cap ([- 1, 1])$ .

3.3step 2.3F2F8F9algebra

Unit disc: universal bound. Let $z_{0}, \dots, z_{n - 1} \in \overline{D (0, 1)}$ and $A = (z_{j}^{k})_{j, k = 0}^{n - 1}$ . Its columns are $c_{k} = (z_{j}^{k})_{j = 0}^{n - 1}$ with $∥ c_{k} ∥^{2} = \sum_{j} ∣ z_{j} ∣^{2 k} \leq n$ by [F8], [F9] and $∣ z_{j} ∣ \leq 1$ ; and $∣ \det A ∣ = ∣ Δ_{n} (z) ∣$ by step 2.3, so [F2] gives the Fekete bound $\prod_{i < j} ∣ z_{i} - z_{j} ∣ \leq n^{n / 2}$ .

4.1step 1.4step 3.3F2F9algebra

Unit disc: the roots of unity are Fekete. Let $ω_{j} = \exp (2 π i j / n)$ , $j = 0, \dots, n - 1$ , which are $n$ distinct points of the unit circle by [F9]. For $k \neq ℓ$ put $d : = k - ℓ$ and $r : = ω_{1}^{d}$ ; then $r^{n} = (ω_{1}^{n})^{d} = 1$ and $r \neq 1$ , since $r = 1$ would give $\exp (2 π i d / n) = 1$ , forcing $n ∣ d$ by the kernel statement of $\ker (\exp) = 2 π i Z$ , and $\exp z = \exp w$ exactly when $z - w \in 2 π i Z$ , contrary to $0 < ∣ d ∣ < n$ . The cyclic-shift computation of For $n \geq 2$ , the sum of all $n$ -th roots of unity is zero, applied to the list $1, r, \dots, r^{n - 1}$ in place of $1, ζ, \dots, ζ^{n - 1}$ (multiply $S = \sum_{j < n} r^{j}$ by $r$ and compare $r S$ with $S$ using $r^{n} = 1$ ), gives $(r - 1) S = 0$ , hence $S = 0$ ; since $ω_{j}^{d} = r^{j}$ by the integer power laws of [F9], the inner product $⟨ c_{k}, c_{ℓ} ⟩ = \sum_{j} ω_{j}^{k} \overline{ω_{j}^{ℓ}} = \sum_{j} ω_{j}^{k - ℓ}$ vanishes, the conjugate being the inverse because $∣ ω_{j} ∣ = 1$ . Each column has norm $\sqrt{n}$ , since $∣ ω_{j}^{2 k} ∣ = 1$ . The Gram–Schmidt residuals of an orthogonal list are the columns themselves, so step 1.4 gives $∣ \det A ∣ = n^{n / 2}$ ; by step 3.3 this is the maximum of $\prod_{i < j} ∣ z_{i} - z_{j} ∣$ over the closed unit disc, so the roots of unity form an $n$ -point Fekete tuple and $δ_{n} (\overline{D (0, 1)}) = n^{1 / (n - 1)}$ . The associated monic polynomial is $F_{n} (z) = \prod_{j} (z - ω_{j}) = z^{n} - 1$ by [F9].

5.1step 4.1F1F3

Weak convergence of the empirical measures. The closed unit disc is compact with $cap (\overline{D (0, 1)}) = 1 > 0$ by [F1], and its equilibrium measure is normalized arclength on the unit circle, which we denote $μ$ (Capacity of a disc and its circular equilibrium measure); since the tuple of step 4.1 is a Fekete tuple for every $n \geq 2$ , [F3] gives $\frac{1}{n} \sum_{j} δ_{ω_{j}} \Rightarrow μ$ .

5.2step 2.1step 3.1step 4.1F10algebra

The norm limit. On the closed unit disc, $∣ z^{n} - 1 ∣ \leq ∣ z ∣^{n} + 1 \leq 2$ by the modulus laws, with equality exactly at the points with $z^{n} = - 1$ , which exist on the unit circle; hence $∥ z^{n} - 1 ∥_{\overline{D (0, 1)}} = 2$ and $∥ F_{n} ∥^{1 / n} = 2^{1 / n} \to 1$ , and this limit is $cheb (\overline{D (0, 1)}) = cap (\overline{D (0, 1)}) = 1$ by steps 2.1 and 3.1 with $a = 0$ and $R = 1$ .

6.1step 2.1step 3.1step 2.2step 3.2step 4.1step 5.1step 5.2F1F3∎

Assembly. Steps 2.1, 3.1, 2.2, 3.2 prove the disc and interval identities $t_{n} = R^{n}$ , $t_{n} = 2^{1 - n}$ and the matching Chebyshev constants and capacities; step 4.1 proves the Fekete property of the roots of unity with $F_{n} (z) = z^{n} - 1$ and the value $δ_{n} = n^{1 / (n - 1)}$ ; step 5.1 proves weak convergence to normalized arclength; and step 5.2 proves the norm value $2$ and the limit of its $n$ -th roots. The Axiom of Choice is used only through [F3] and the capacity values of [F1].

Remarks

Where the two halves of the calculation meet. The first six steps compute the monic extremal norms and combine them with the known capacities of the disc and the interval; steps 1.3, 1.4, 2.3, 3.3 and 4.1 compute the Fekete diameters of the unit disc exactly, with the Vandermonde determinant reducing the Fekete problem to the classical Gram–Schmidt bound for column norms, and with the Fourier columns of the roots-of-unity matrix attaining equality. Step 5.1 then identifies the limit of the empirical measures with the equilibrium measure supplied by Capacity of a disc and its circular equilibrium measure, and step 5.2 confirms the consistency of the Fekete-polynomial norms with the general norm limit of Fekete–Szegő equality of logarithmic capacity, transfinite diameter, and Chebyshev constant.

Why the interval reduction is legitimate. For a monic complex polynomial $p$ on a real interval the real part $Re p$ is again monic — its leading coefficient is $Re (1) = 1$ — and it never exceeds $p$ in modulus on real arguments, so a minimax bound for monic real polynomials applies without loss. This is the only place where the real interval differs from the disc, where Cauchy's estimate handles complex coefficients directly.

Choice. The example states the Axiom of Choice because it quotes the two capacity values from the equilibrium examples and the weak convergence from Fekete–Szegő equality of logarithmic capacity, transfinite diameter, and Chebyshev constant; the determinant, Gram–Schmidt and minimax computations are choice-free.

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Sources

E. B. Saff, Logarithmic Potential Theory with Applications to Approximation Theory, §1 (standard reference, not scraped)