Weighted Faber Polynomials

• Weighted Faber Polynomials are extensions of classical Faber polynomials that incorporate auxiliary weights to modify conformal-power expansions for domains with corners, cusps, or geometric singularities.
• They enable the construction of asymptotically minimal trial functions in Chebyshev polynomial approximations by damping boundary peaks while preserving leading coefficients.
• These frameworks also bridge to modular forms, quadrature domains, and basis theory in weighted function spaces, revealing deep ties with random-walk recurrences and arithmetic weights.

Searching arXiv for recent and relevant papers on weighted Faber polynomials and closely related Faber-polynomial frameworks. Weighted Faber polynomials are extensions of classical Faber polynomials in which the polynomial part of a conformal-power expansion is modified by an auxiliary weight, or, in modular and weighted-function-space settings, by weight-dependent prefactors, recurrences, or norms. In current usage the term does not denote a single universal construction. Rather, it covers several related frameworks: analytic weights inserted into $G_m(z)\Phi(z)^n$ on Jordan curves, generalized modular polynomials attached to $j(\tau)$ and modular weight data, Faber-transform reconstructions for power-weighted and log-weighted quadrature domains, random-walk-induced recurrences, and generalized $p$-Faber systems in weighted Smirnov spaces (Miña-Díaz et al., 26 Sep 2025, Nakaya, 2023, Graven et al., 4 Sep 2025, Cowal et al., 28 Oct 2025, Bilalov et al., 2019).

1. Classical background and the range of the term

For a Jordan curve $\Gamma$ with unbounded complement $\Omega$, the classical exterior conformal map

$\Phi:\Omega\to\{w\in\mathbb{C}:|w|>1\}$

is normalized by $\Phi(\infty)=\infty$ and $\Phi(z)/z\to \mathrm{cap}(\Gamma)^{-1}$ as $z\to\infty$. The $n$th Faber polynomial $j(\tau)$0 is the polynomial part of the Laurent expansion of $j(\tau)$1 at infinity; equivalently,

$j(\tau)$2

where $j(\tau)$3 and $j(\tau)$4 (Miña-Díaz et al., 26 Sep 2025).

Within this classical framework, a weighted Faber polynomial is obtained by replacing $j(\tau)$5 by $j(\tau)$6, where $j(\tau)$7 is analytic in $j(\tau)$8 and satisfies $j(\tau)$9. If $p$0, with $p$1 analytic in $p$2 and $p$3, then the weighted polynomial $p$4 is the polynomial part of $p$5. This is the construction developed for Jordan curves with corners and cusps, where unweighted Faber polynomials cease to be asymptotically minimal (Miña-Díaz et al., 26 Sep 2025).

A distinct but related use occurs in the Faber transform literature. For simply connected domains, the exterior Faber transform $p$6 sends analytic functions on the disk or its complement to analytic functions on the complementary domain, and the corresponding Faber polynomials are $p$7. In that setting, weighted constructions are often realized not by introducing a new polynomial family, but by applying the standard Faber transform and inverse Faber polynomials to weighted geometric objects such as $p$8 or $p$9 (Graven et al., 4 Sep 2025).

This suggests that the unifying feature is not a single formula but a recurrent mechanism: classical Faber expansions are modified so that geometric singularities, modular weights, weighted area forms, or operator-theoretic constraints are absorbed into the polynomial model.

2. Weighted construction on Jordan curves with corners and cusps

The construction used for piecewise Dini-smooth Jordan curves begins from the observation that corners and cusps amplify the boundary values of $\Gamma$0. To suppress these peaks, one fixes $\Gamma$1, chooses $\Gamma$2 analytic in $\Gamma$3 with $\Gamma$4, and defines $\Gamma$5 as the polynomial part of $\Gamma$6. It has degree $\Gamma$7, leading coefficient $\Gamma$8, and integral representation

$\Gamma$9

with $\Omega$0 as above (Miña-Díaz et al., 26 Sep 2025).

The explicit weight is built from the images $\Omega$1 of the corner points $\Omega$2. For $\Omega$3,

$\Omega$4

with branches chosen so that $\Omega$5. Truncating the Laurent series

$\Omega$6

yields

$\Omega$7

where $\Omega$8 is chosen so that $\Omega$9 on $\Phi:\Omega\to\{w\in\mathbb{C}:|w|>1\}$0, and near each corner point $\Phi:\Omega\to\{w\in\mathbb{C}:|w|>1\}$1 one has $\Phi:\Omega\to\{w\in\mathbb{C}:|w|>1\}$2 (Miña-Díaz et al., 26 Sep 2025).

Algebraically,

$\Phi:\Omega\to\{w\in\mathbb{C}:|w|>1\}$3

The weighting is therefore a finite linear combination of nearby Faber polynomials. This finite-band structure is crucial: it permits local damping near singular boundary points without changing the leading coefficient or degree (Miña-Díaz et al., 26 Sep 2025).

The analytic control comes from Pommerenke’s Fourier representation. If $\Phi:\Omega\to\{w\in\mathbb{C}:|w|>1\}$4 has bounded secant variation, then for $\Phi:\Omega\to\{w\in\mathbb{C}:|w|>1\}$5,

$\Phi:\Omega\to\{w\in\mathbb{C}:|w|>1\}$6

where

$\Phi:\Omega\to\{w\in\mathbb{C}:|w|>1\}$7

The weighted polynomials satisfy

$\Phi:\Omega\to\{w\in\mathbb{C}:|w|>1\}$8

The factor $\Phi:\Omega\to\{w\in\mathbb{C}:|w|>1\}$9 is small near the corner images $\Phi(\infty)=\infty$0, exactly where the singular point masses in $\Phi(\infty)=\infty$1 would otherwise force large boundary values (Miña-Díaz et al., 26 Sep 2025).

3. Asymptotic minimality, Chebyshev polynomials, and corner geometry

The main application of the weighted construction is to the $\Phi(\infty)=\infty$2th Chebyshev polynomial $\Phi(\infty)=\infty$3 of a compact set, defined as the unique monic polynomial of degree $\Phi(\infty)=\infty$4 minimizing the supremum norm. For a piecewise Dini-smooth Jordan curve $\Phi(\infty)=\infty$5,

$\Phi(\infty)=\infty$6

This extends the smooth-curve result to curves with corners and cusps (Miña-Díaz et al., 26 Sep 2025).

The proof uses the weighted polynomials $\Phi(\infty)=\infty$7 as trial polynomials. Since $\Phi(\infty)=\infty$8 has degree $\Phi(\infty)=\infty$9 and leading coefficient $\Phi(z)/z\to \mathrm{cap}(\Gamma)^{-1}$0, extremality gives

$\Phi(z)/z\to \mathrm{cap}(\Gamma)^{-1}$1

The weighted Fourier analysis yields

$\Phi(z)/z\to \mathrm{cap}(\Gamma)^{-1}$2

Letting $\Phi(z)/z\to \mathrm{cap}(\Gamma)^{-1}$3 and then $\Phi(z)/z\to \mathrm{cap}(\Gamma)^{-1}$4, and combining this with Szegő’s lower bound

$\Phi(z)/z\to \mathrm{cap}(\Gamma)^{-1}$5

produces the limit $\Phi(z)/z\to \mathrm{cap}(\Gamma)^{-1}$6 (Miña-Díaz et al., 26 Sep 2025).

A central misconception addressed by this result is that ordinary Faber polynomials already provide asymptotically minimal trial functions on all sufficiently regular curves. They do so for smooth Dini-smooth curves, where $\Phi(z)/z\to \mathrm{cap}(\Gamma)^{-1}$7, but not in general when corners or cusps are present. At a corner $\Phi(z)/z\to \mathrm{cap}(\Gamma)^{-1}$8 with exterior angle $\Phi(z)/z\to \mathrm{cap}(\Gamma)^{-1}$9, the singular part of $z\to\infty$0 contributes a persistent term, and

$z\to\infty$1

whereas away from the corners,

$z\to\infty$2

Thus corner singularities obstruct uniform convergence to $z\to\infty$3 on $z\to\infty$4 (Miña-Díaz et al., 26 Sep 2025).

The same analysis gives new asymptotic norm bounds for unweighted Faber polynomials. If $z\to\infty$5 are the corners and

$z\to\infty$6

then

$z\to\infty$7

This bound is sharp in several regimes, including convex cases and cases where the maximal $z\to\infty$8 is realized by an exterior angle $z\to\infty$9. When the maximum is instead governed by $n$0, as for certain inward corners, the exact value of $n$1 remains unknown (Miña-Díaz et al., 26 Sep 2025).

4. Modular and arithmetic formulations

In the modular-forms literature, weighted Faber polynomials usually mean weight-dependent polynomials in the Hauptmodul $n$2, together with explicit modular prefactors. One standard construction writes the canonical basis elements of $n$3 as

$n$4

where $n$5, $n$6 is monic of degree $n$7, and

$n$8

Here the adjective “weighted” refers to the dependence on the modular weight $n$9 through the prefactor $j(\tau)$00 (Nakaya, 2023).

Nakaya proves that these generalized Faber polynomials admit orthogonal expansions in the Atkin-like polynomial families $j(\tau)$01, with coefficients recovered from the Atkin inner product

$j(\tau)$02

and encoded by normalized extremal quasimodular forms $j(\tau)$03. The resulting generating series identify the expansion coefficients with Fourier coefficients of $j(\tau)$04 multiplied by explicit powers of $j(\tau)$05, $j(\tau)$06, and $j(\tau)$07 (Nakaya, 2023).

A related but different classification of weakly holomorphic modular forms writes

$j(\tau)$08

for $j(\tau)$09, where $j(\tau)$10 and $j(\tau)$11 is determined by the weight class modulo $j(\tau)$12. In this setting, Kane uses harmonic weak Maass forms and Maass–Poincaré series to derive asymptotics for the coefficients of the shifted Faber polynomials $j(\tau)$13, including explicit coefficient asymptotics in terms of the constants $j(\tau)$14 and $j(\tau)$15 (Kane, 2010).

Further modular reinterpretations emphasize weighting by parameters rather than prefactors. For the Miller basis $j(\tau)$16, the zeros of the associated Faber polynomials $j(\tau)$17 satisfy linear moment identities

$j(\tau)$18

so that the modular weight $j(\tau)$19 and the Miller index $j(\tau)$20 act as linear weights on the zero distribution. After normalization, the moments depend asymptotically only on the ratio $j(\tau)$21 (Zilka, 7 Oct 2025).

Another arithmetic use of weighting is renormalization. For cusp forms of large weight $j(\tau)$22 with $j(\tau)$23, the associated Faber polynomial $j(\tau)$24 is rescaled as

$j(\tau)$25

This renormalized polynomial satisfies

$j(\tau)$26

coefficientwise, where $j(\tau)$27. Consequently the zeros of $j(\tau)$28 satisfy $j(\tau)$29, which forces the zeros of the underlying modular form to cluster near $j(\tau)$30 vertical lines at height approximately $j(\tau)$31 (Rudnick, 2023).

5. Faber transforms, weighted quadrature domains, and random-walk recurrences

In the theory of quadrature domains, the weighted setting is formulated through weighted area measures rather than through a separately defined polynomial family. For power weights

$j(\tau)$32

and the log-weighted case $j(\tau)$33, reconstruction proceeds by the classical Faber transform and inverse Faber polynomials. The decisive objects are $j(\tau)$34 in the power-weighted case and $j(\tau)$35 in the log-weighted case. The paper explicitly states that it does not introduce a separate family of weighted Faber polynomials; instead, the standard Faber transform is applied to these weighted geometric quantities (Graven et al., 4 Sep 2025).

For simply connected power-weighted quadrature domains, one obtains characterizations such as

$j(\tau)$36

in the unbounded case, where $j(\tau)$37 is the inverse Faber polynomial and $j(\tau)$38 is recovered by an inverse Faber transform. More generally, $j(\tau)$39 if and only if $j(\tau)$40 extends to a rational function. For log-weighted quadrature domains, the corresponding criterion is rationality of $j(\tau)$41, leading to explicit representations

$j(\tau)$42

according to whether the domain is bounded or unbounded (Graven et al., 4 Sep 2025).

A different notion of weighting appears in the random-walk construction of polynomial recurrences. Here one fixes a probability vector $j(\tau)$43 with $j(\tau)$44, $j(\tau)$45, and mean-zero condition

$j(\tau)$46

The associated polynomial family $j(\tau)$47 satisfies

$j(\tau)$48

equivalently

$j(\tau)$49

The same coefficients define the rational exterior map

$j(\tau)$50

and the paper shows that the corresponding Faber polynomials satisfy the same recurrence. In this sense the weights $j(\tau)$51 are built into both the geometry of the domain and the Faber family itself (Cowal et al., 28 Oct 2025).

This random-walk weighting has sharp analytic consequences. If $j(\tau)$52 is the enclosed radially convex domain and

$j(\tau)$53

then

$j(\tau)$54

Moreover $j(\tau)$55 can be approximated on $j(\tau)$56 by a polynomial of degree $j(\tau)$57, and these filters lead to arbitrary-order dynamic momentum power iteration methods for certain non-symmetric matrices (Cowal et al., 28 Oct 2025).

6. Weighted Smirnov spaces and basis theory

A further development treats generalized Faber polynomials as bases in weighted analytic function spaces on regular curves. Let $j(\tau)$58 be a rectifiable Jordan curve, $j(\tau)$59 its bounded interior, and $j(\tau)$60 its exterior. If $j(\tau)$61 is the exterior conformal map, the generalized $j(\tau)$62-Faber polynomials $j(\tau)$63 are defined as the principal polynomial part of

$j(\tau)$64

If $j(\tau)$65 is the interior conformal map normalized by $j(\tau)$66, the interior generalized $j(\tau)$67-Faber polynomials $j(\tau)$68 arise from the principal part of

$j(\tau)$69

These constructions recover the usual monomials on the unit circle (Bilalov et al., 2019).

The function-space setting is the weighted Smirnov spaces $j(\tau)$70 and $j(\tau)$71, where the boundary norm is taken in $j(\tau)$72. The decisive hypothesis is the Muckenhoupt condition $j(\tau)$73, together with corresponding pullback conditions on the unit circle. Under these assumptions, conformal transplantation operators $j(\tau)$74 map weighted Hardy bases on the circle to the generalized Faber systems on $j(\tau)$75 (Bilalov et al., 2019).

The main basis theorem states that if $j(\tau)$76 is regular, $j(\tau)$77, $j(\tau)$78, and

$j(\tau)$79

then $j(\tau)$80 is a basis of $j(\tau)$81 and $j(\tau)$82 is a basis of $j(\tau)$83. The same paper establishes basis properties for a double system

$j(\tau)$84

with complex-valued coefficients $j(\tau)$85, $j(\tau)$86, in weighted Lebesgue spaces $j(\tau)$87, using a Riemann–Hilbert analysis and boundedness of the Cauchy singular integral operator on $j(\tau)$88 (Bilalov et al., 2019).

This branch of the theory makes the role of weighting especially explicit. The weights are neither auxiliary damping factors nor modular prefactors, but structural hypotheses on the ambient Banach space. The generalized Faber systems are shown to be Schauder bases, not orthogonal or Riesz bases, and the hypotheses $j(\tau)$89, regularity of $j(\tau)$90, and Muckenhoupt-type conditions are essential (Bilalov et al., 2019).