Weighted Chebyshev Polynomials

• Weighted Chebyshev polynomials are monic polynomials that minimize a weighted supremum norm over compact sets, extending classical Chebyshev concepts with weight-modified norms.
• They employ a weighted minimax framework involving Widom and Szegő factors, providing sharp non-asymptotic and asymptotic bounds in extremal problems.
• Applications include orthogonal polynomial systems on intervals, circular arcs, and simplices, influencing numerical integration, approximation theory, and potential theory.

Weighted Chebyshev polynomials are not a single universally standardized family. In contemporary usage, the phrase most commonly denotes monic degree-$n$ polynomials minimizing a weighted supremum norm on a compact set, but it also refers to orthogonal systems associated with Chebyshev-type weights, endpoint-modified and simplicial generalizations, and weighted inequalities or trigonometric extremal problems built from Chebyshev kernels. The unifying theme is that the classical Chebyshev structure is altered by a weight, either in the norm, in the orthogonality measure, or in the underlying geometry (Christiansen et al., 17 Feb 2025, AlQudah, 2015).

1. Extremal definition and the weighted minimax problem

In the extremal-theoretic formulation, let $e\subset \mathbb C$ be an infinite compact set and let $w:e\to[0,\infty)$ be an upper semicontinuous weight that is positive at infinitely many points of $e$. The weighted norm of a polynomial $P_n$ is

$\|wP_n\|_e:=\sup_{z\in e}|w(z)P_n(z)|.$

The $n$-th weighted Chebyshev polynomial is the unique minimizer among monic degree-$n$ polynomials,

$$T_{n,w}=\arg\min\{\|wP_n\|_e:\ P_n \text{ monic of degree }n\},$$

and its minimal deviation is

$t_n(e,w):=\|wT_{n,w}\|_e.$

If $e\subset \mathbb C$0 is the logarithmic capacity, the associated Widom factor is

$e\subset \mathbb C$1

This is the basic framework for current weighted Chebyshev theory on subsets of the real line and, with suitable modifications, on arcs and other compact sets (Christiansen et al., 17 Feb 2025).

A closely related object is the weighted residual polynomial. For $e\subset \mathbb C$2, one considers

$e\subset \mathbb C$3

where $e\subset \mathbb C$4 if $e\subset \mathbb C$5, while $e\subset \mathbb C$6 imposes monicity. The generalized Widom factor is then

$e\subset \mathbb C$7

This unifies weighted Chebyshev and weighted residual problems in a single extremal scheme (Christiansen et al., 17 Feb 2025).

For compact $e\subset \mathbb C$8, the alternation theorem remains central. A real polynomial $e\subset \mathbb C$9 is extremal if and only if there are $w:e\to[0,\infty)$0 points $w:e\to[0,\infty)$1 in $w:e\to[0,\infty)$2 such that

$w:e\to[0,\infty)$3

with the convention $w:e\to[0,\infty)$4. This yields the real and simple zero structure of the extremal polynomials (Christiansen et al., 17 Feb 2025).

2. Widom factors, Szegő factors, and sharp bounds

The potential-theoretic normalization is governed by the Szegő factor

$w:e\to[0,\infty)$5

or, in the residual setting,

$w:e\to[0,\infty)$6

A universal lower bound is

$w:e\to[0,\infty)$7

On the real line, much of the recent theory concerns when the stronger doubled bound $w:e\to[0,\infty)$8 is valid (Christiansen et al., 17 Feb 2025, Alpan et al., 2024).

For the special class

$w:e\to[0,\infty)$9

where $e$0 is a real polynomial of degree $e$1 with all zeros outside $e$2, the real-line theory gives non-asymptotic lower and upper bounds. If $e$3 is compact of positive capacity and $e$4, then

$e$5

and if $e$6 is compact, regular, and a Parreau–Widom set, then

$e$7

These are weighted analogues of the classical doubled lower bound and Totik–Widom upper bound (Christiansen et al., 17 Feb 2025).

The asymptotic theory extends beyond reciprocal-polynomial weights. For compact $e$8, if $e$9 is upper semicontinuous, continuous at almost every point of $P_n$0, and satisfies $P_n$1 on $P_n$2 for some nonzero polynomial $P_n$3, then

$P_n$4

If $P_n$5 is compact, regular, and Parreau–Widom, then

$P_n$6

Moreover, for such $P_n$7, the conditions

$P_n$8

are equivalent, and either implies $P_n$9 (Christiansen et al., 17 Feb 2025).

A parallel lower-bound theory on regular compact subsets of $\|wP_n\|_e:=\sup_{z\in e}|w(z)P_n(z)|.$0 establishes optimal non-asymptotic and asymptotic estimates for large classes of weights, including rational product weights and weights with finitely many zeros, infinitely many zeros, and strong zeros. In this setting the doubled lower bound

$\|wP_n\|_e:=\sup_{z\in e}|w(z)P_n(z)|.$1

is obtained non-asymptotically for broad classes of regular weights, and asymptotically on intervals this yields an extension of Bernstein’s 1931 result to arbitrary Riemann integrable weights with finitely many zeros and to some continuous weights with infinitely many zeros (Alpan et al., 2024).

3. Jacobi, circular, and arc models

A particularly tractable family is given by Jacobi weights on $\|wP_n\|_e:=\sup_{z\in e}|w(z)P_n(z)|.$2,

$\|wP_n\|_e:=\sup_{z\in e}|w(z)P_n(z)|.$3

The corresponding Jacobi-weighted Chebyshev polynomial is the unique monic minimizer of

$\|wP_n\|_e:=\sup_{z\in e}|w(z)P_n(z)|.$4

and the normalized extremal quantity is

$\|wP_n\|_e:=\sup_{z\in e}|w(z)P_n(z)|.$5

For all nonnegative parameters,

$\|wP_n\|_e:=\sup_{z\in e}|w(z)P_n(z)|.$6

If $\|wP_n\|_e:=\sup_{z\in e}|w(z)P_n(z)|.$7, the sequence is constant in $\|wP_n\|_e:=\sup_{z\in e}|w(z)P_n(z)|.$8; if $\|wP_n\|_e:=\sup_{z\in e}|w(z)P_n(z)|.$9, then $n$0 for every $n$1; and if $n$2, then $n$3 (Christiansen et al., 2024).

The interval problem is linked to a unit-circle problem. For $n$4, if $n$5 denotes the Jacobi interval extremal value and $n$6 the corresponding unit-circle extremal value, then

$n$7

This circle correspondence generalizes earlier special-parameter results and is derived through the Erdős–Lax inequality and a Pólya–Szegő lemma (Christiansen et al., 2024).

Another classical model is the vanishing boundary weight on the unit circle,

$n$8

The minimizer $n$9 satisfies the derivative identity

$n$0

together with

$n$1

For every $n$2, the norms $n$3 are monotonically decreasing in $n$4 and converge to $n$5. In the case $n$6, the normalized zero counting measures converge weakly to $n$7 on the unit circle (Bergman et al., 2023).

On the circular arc

$n$8

weighted Chebyshev polynomials arise as the special case $n$9 of weighted residual polynomials. For reciprocal-polynomial weights and then for broader classes of weights, the weighted Widom factors satisfy

$$T_{n,w}=\arg\min\{\|wP_n\|_e:\ P_n \text{ monic of degree }n\},$$0

This generalizes the unweighted circular-arc constant $$T_{n,w}=\arg\min\{\|wP_n\|_e:\ P_n \text{ monic of degree }n\},$$1 and extends the theory to algebraic zeros and singular factors in the weight (Christiansen et al., 5 Feb 2026).

4. Orthogonality-based weighted and generalized families

Weighted Chebyshev theory also includes orthogonal-polynomial families whose measure is a modification of the classical Chebyshev weight. One such family is the generalized Chebyshev-type polynomial of the first kind,

$$T_{n,w}=\arg\min\{\|wP_n\|_e:\ P_n \text{ monic of degree }n\},$$2

where $$T_{n,w}=\arg\min\{\|wP_n\|_e:\ P_n \text{ monic of degree }n\},$$3, $$T_{n,w}=\arg\min\{\|wP_n\|_e:\ P_n \text{ monic of degree }n\},$$4, and $$T_{n,w}=\arg\min\{\|wP_n\|_e:\ P_n \text{ monic of degree }n\},$$5 are explicit correction terms built from $$T_{n,w}=\arg\min\{\|wP_n\|_e:\ P_n \text{ monic of degree }n\},$$6 and its derivatives. These polynomials are orthogonal on $$T_{n,w}=\arg\min\{\|wP_n\|_e:\ P_n \text{ monic of degree }n\},$$7 with respect to

$$T_{n,w}=\arg\min\{\|wP_n\|_e:\ P_n \text{ monic of degree }n\},$$8

so the parameters $$T_{n,w}=\arg\min\{\|wP_n\|_e:\ P_n \text{ monic of degree }n\},$$9 and $t_n(e,w):=\|wT_{n,w}\|_e.$0 act as endpoint-mass modifiers. The family admits an explicit Bernstein-basis representation and closed-form weighted integrals against Bernstein polynomials in terms of beta functions (AlQudah, 2015).

A different weighted orthogonality problem uses

$t_n(e,w):=\|wT_{n,w}\|_e.$1

The corresponding orthogonal polynomials can be described through the polynomial mapping $t_n(e,w):=\|wT_{n,w}\|_e.$2, with the higher-degree members written in terms of monic Jacobi polynomials and Chebyshev polynomials of the second kind. This setting unifies several earlier special cases and makes the role of the Chebyshev factor $t_n(e,w):=\|wT_{n,w}\|_e.$3 completely explicit (Castillo et al., 2019).

Weighted Chebyshev systems also appear on simplices. On the triangular domain

$t_n(e,w):=\|wT_{n,w}\|_e.$4

the bivariate Chebyshev-I weighted orthogonal polynomials $t_n(e,w):=\|wT_{n,w}\|_e.$5 are constructed with respect to

$t_n(e,w):=\|wT_{n,w}\|_e.$6

with the main separation argument using $t_n(e,w):=\|wT_{n,w}\|_e.$7. They admit the closed form

$t_n(e,w):=\|wT_{n,w}\|_e.$8

and form a degree-ordered orthogonal system on the simplex (AlQudah, 2015).

The $t_n(e,w):=\|wT_{n,w}\|_e.$9-Chebyshev functions provide a further generalization: $e\subset \mathbb C$00 These are generally not polynomials, but they are orthogonal on

$e\subset \mathbb C$01

with respect to

$e\subset \mathbb C$02

For special parameter choices they reduce to classical Chebyshev polynomials or sign-modified variants, and the corresponding node sets become subsets of ordinary Chebyshev and Chebyshev–Lobatto points (Marchi et al., 2021).

5. Weighted inequalities and trigonometric extremal analogues

A distinct line of work studies weighted inequalities involving Chebyshev polynomials as functions of their index. Extending $e\subset \mathbb C$03 to real $e\subset \mathbb C$04 by

$e\subset \mathbb C$05

one obtains a Jensen-type inequality for weighted geometric means. If $e\subset \mathbb C$06, $e\subset \mathbb C$07, and

$e\subset \mathbb C$08

then

$e\subset \mathbb C$09

Equality holds if and only if $e\subset \mathbb C$10, or $e\subset \mathbb C$11 for all indices $e\subset \mathbb C$12 with $e\subset \mathbb C$13; conversely, if the inequality holds for all $e\subset \mathbb C$14, then the parameter condition is necessary. The proof proceeds through the two-parameter inequality

$e\subset \mathbb C$15

and the concavity of $e\subset \mathbb C$16 (Javaheri et al., 2022).

Weighted Chebyshev extremality also appears in periodic trigonometric form. On the torus $e\subset \mathbb C$17, one studies products

$e\subset \mathbb C$18

with weighted norm

$e\subset \mathbb C$19

Writing

$e\subset \mathbb C$20

reduces the problem to a sum-of-translates minimax problem. Existence of minimax and maximin node systems and equioscillation follow from Fenton’s method, but the weighted torus case is substantially less rigid than the unweighted or interval cases: equioscillation need not characterize the minimax point uniquely, and minimax and maximin values can differ (Nagy et al., 2023).

6. Quadrature, adjacent theories, and terminological boundaries

The phrase “Chebyshev-type” often appears in quadrature theory, where it has a different meaning. A Chebyshev-type quadrature is an equal-weight quadrature formula,

$e\subset \mathbb C$21

or its trigonometric analogue on the circle. For doubling weights, the minimum number of nodes required for exactness up to degree $e\subset \mathbb C$22 is controlled, up to constants depending only on the doubling constant, by the smallest local mass of the weight at scale $e\subset \mathbb C$23 (Gilboa et al., 2015). This theory concerns equal-weight integration rather than weighted Chebyshev polynomials as extremal or orthogonal objects.

A constructive dyadic variant introduces classes $e\subset \mathbb C$24 of weights satisfying symmetry relations induced by maps $e\subset \mathbb C$25. For $e\subset \mathbb C$26, the resulting equal-weight quadratures are exact for all polynomials in $e\subset \mathbb C$27, and for $e\subset \mathbb C$28 nodes they achieve the maximal possible degree $e\subset \mathbb C$29 (Vagharshakyan, 2011). Although closely related in spirit, these are quadrature formulas rather than a family of weighted Chebyshev polynomials.

A recurrent misconception is that every parameterized use of Chebyshev polynomials is part of weighted Chebyshev theory. This is not the case. In Lucas–Lehmer-type primality testing, Chebyshev polynomials serve as the arithmetic engine behind generalized recurrences through the compositional law

$e\subset \mathbb C$30

but the work is explicitly not about “weighted Chebyshev polynomials” in the standard orthogonal-polynomial sense (Chua, 2020). Likewise, in the study of Demazure multiplicities for $e\subset \mathbb C$31-modules, numerical multiplicities are expressed as coefficients of quotients of normalized second-kind Chebyshev polynomials $e\subset \mathbb C$32, with the partition data entering through products and quotients; the paper explicitly states that this is not a new formal weighted Chebyshev theory (Biswal, 19 Apr 2026).

Across these settings, “weighted Chebyshev polynomials” therefore denotes a family of related but non-identical constructions. The most stable core is the weighted minimax problem for monic polynomials under a weighted sup norm, together with the accompanying Widom-factor and Szegő-factor theory. Around that core lie orthogonality-based weighted families, weighted trigonometric extremal problems, and several adjacent theories in which Chebyshev polynomials interact with weights without themselves constituting a standard weighted Chebyshev family (Christiansen et al., 17 Feb 2025, Alpan et al., 2024).