Faber Polynomials in Analysis

• Faber Polynomials are families of polynomials generated from conformal maps that capture the polynomial part of exterior functions and underpin approximation theory.
• They bridge conformal geometry with polynomial approximation, playing a pivotal role in operator theory, Krylov methods, and non-Hermitian time propagation.
• Recent generalizations extend their use to modular forms, weighted analytic spaces, and bi-univalent function theory, highlighting their broad utility in modern research.

Faber polynomials are polynomial families generated from conformal maps of planar domains. In the classical setting, they are attached to a compact set $E \subset \mathbb{C}$ with simply connected complement and encode the polynomial part of the exterior conformal map; in later literatures, closely related generalized forms appear in weighted analytic spaces, inverse function theory, and modular forms. Across these settings, they serve as a bridge between conformal geometry and polynomial approximation, and they now occupy a technical role in operator theory, Krylov methods, non-Hermitian time propagation, zero-distribution problems, and inverse boundary-value problems (Beckermann et al., 2013, Kane, 2010).

1. Classical construction and generalized variants

For a compact set $E \subset \mathbb{C}$ with rectifiable Jordan boundary, let $\Phi$ be the Riemann conformal map from the exterior of $E$ to the exterior of the unit disk, normalized by $\Phi(\infty)=\infty$ and $\Phi'(\infty)>0$. The classical Faber polynomial $F_n$ is the polynomial part of the Laurent expansion of $\Phi(z)^n$ at infinity, equivalently characterized by

$F_n(z)=\Phi(z)^n+O(1/z)\qquad (z\to\infty).$

An equivalent generating description uses the inverse map $\Psi=\Phi^{-1}$: $E \subset \mathbb{C}$0 This formulation underlies much of the approximation-theoretic and zero-distribution literature (Beckermann et al., 2013, Levenberg et al., 2018).

Several generalized constructions preserve the same conformal-mapping logic while changing the ambient function space. For a regular curve $E \subset \mathbb{C}$1 with bounded and unbounded complementary components $E \subset \mathbb{C}$2 and $E \subset \mathbb{C}$3, generalized Faber polynomials $E \subset \mathbb{C}$4 and $E \subset \mathbb{C}$5 are defined as principal parts of

$E \subset \mathbb{C}$6

where $E \subset \mathbb{C}$7 and $E \subset \mathbb{C}$8 are conformal maps associated with $E \subset \mathbb{C}$9 and $\Phi$0, respectively. These versions are adapted to weighted Smirnov and Lebesgue spaces rather than uniform polynomial approximation (Bilalov et al., 2019).

A distinct but standard modular-form usage employs polynomials in the Hauptmodul $\Phi$1. In that setting, basis elements of spaces of weakly holomorphic modular forms are written in the form

$\Phi$2

and the corresponding $\Phi$3 are explicitly described in the source as not being the classical Faber polynomials of function theory, but rather special polynomials in $\Phi$4 or $\Phi$5 determined recursively to cancel unwanted negative powers of $\Phi$6 (Kane, 2010). This terminological split is essential: the geometric and modular theories share a name and some structural analogies, but they are not the same construction.

2. Geometry on planar domains: norms, corners, cusps, and zeros

The asymptotics of Faber polynomials are highly sensitive to boundary geometry. For Joukowski airfoils, whose boundary has an outward cusp, the normalized counting measures of the zeros of the Faber polynomials converge weak-* to an explicit limit that is never equal to the potential-theoretic equilibrium measure of the set. Depending on the regime, the limiting support is either a simple arc or a combination of an arc and a loop, and this behavior explains a class of examples related to electrostatic skeletons and Ullman’s Chebyshev quadrature (Levenberg et al., 2018).

Recent work on piecewise Dini-smooth Jordan curves with corners and cusps sharpened the norm asymptotics. If $\Phi$7 has corners with exterior angles $\Phi$8 and $\Phi$9, then

$E$0

and pointwise

$E$1

In parallel, the same paper proves that the $E$2th Chebyshev polynomial $E$3 of a piecewise Dini-smooth Jordan curve satisfies

$E$4

using weighted Faber polynomials and a Fourier analytic representation due to Pommerenke (Miña-Díaz et al., 26 Sep 2025).

These results correct a common overgeneralization from smooth boundaries. Classical asymptotic minimality of Faber polynomials is stable on Dini-smooth arcs, but outward corners and cusps can create norm spikes and non-equilibrium zero distributions. The modern picture is therefore geometric rather than purely potential-theoretic.

3. Matrix and operator inequalities

In operator theory, Faber polynomials are evaluated at a bounded linear operator $E$5 on a Hilbert space. When $E$6 is convex, compact, and contains the numerical range $E$7, a previously known result gives the uniform estimate

$E$8

This estimate is useful in numerical linear algebra because it yields explicit polynomial bounds relevant to Krylov subspace error analysis (Beckermann et al., 2013).

The non-convex extension is the central contribution of Beckermann and Crouzeix. Let

$E$9

where $\Phi(\infty)=\infty$0 is a convex compact set containing $\Phi(\infty)=\infty$1, $\Phi(\infty)=\infty$2, and $\Phi(\infty)=\infty$3 is simply connected. If $\Phi(\infty)=\infty$4, then:

• If $\Phi(\infty)=\infty$5, one has

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

• If $\Phi(\infty)=\infty$7, one has

$\Phi(\infty)=\infty$8

Here $\Phi(\infty)=\infty$9 is a geometric constant satisfying $\Phi'(\infty)>0$0 and $\Phi'(\infty)>0$1 when $\Phi'(\infty)>0$2 is convex, so the convex estimate is recovered as a special case (Beckermann et al., 2013).

The technical mechanism is an integral representation for $\Phi'(\infty)>0$3 over $\Phi'(\infty)>0$4, leading to the estimate

$\Phi'(\infty)>0$5

where $\Phi'(\infty)>0$6 is the negative part of the minimum eigenvalue of a self-adjoint operator $\Phi'(\infty)>0$7. In application, the quantity

$\Phi'(\infty)>0$8

controls ideal GMRES approximation, and bounds on $\Phi'(\infty)>0$9 provide explicit residual estimates. The non-convex theory thus enlarges the admissible spectral sets beyond the classical convex-numerical-range regime (Beckermann et al., 2013).

4. Non-Hermitian propagation, exponential integrators, and accelerated power methods

For non-unitary quantum many-body dynamics, Faber polynomials provide a polynomial basis for the expansion of analytic functions of a non-Hermitian generator. With an ellipse chosen to contain the spectrum, the evolution operator is expanded as

$F_n$0

and for the elliptic contour the coefficients $F_n$1 are expressed באמצעות Bessel functions. The associated recurrence

$F_n$2

permits a memory-efficient implementation, since only two state vectors are required at each step. The method directly simulates non-Hermitian dynamics and quantum-jump trajectories without hermitization or doubling of the Hilbert space, and it is presented as a generalization of Chebyshev propagation to non-Hermitian scenarios (Soares et al., 2024).

A closely related numerical analysis appears in seismic wave modelling. There, Faber polynomial exponential integrators are used for non-symmetric discrete operators arising from absorbing boundaries. The method generalizes Chebyshev-based exponential integrators from spectra on real intervals to spectra enclosed by ellipses in $F_n$3. The paper emphasizes two facts: the practical importance of determining an optimal ellipse encompassing the full spectrum of the discrete operator, and a sharp bound for the approximation error of the exponential of a normal matrix. It also reports numerical investigations of stability, dispersion, convergence, and computational efficiency for the Faber exponential scheme (Ravelo et al., 2022).

Recent work on deltoid and random-walk constructions pushes the approximation viewpoint further. In a deltoid region, a polynomial family $F_n$4 satisfying the same recurrence relation as the Faber polynomials obeys

$F_n$5

with $F_n$6 in the deltoid region and

$F_n$7

The same paper gives a constructive proof that $F_n$8 is approximately a polynomial of degree $F_n$9 within the deltoid region and applies this to a higher-order momentum-based acceleration of power iteration for matrices with complex eigenvalues (Cowal et al., 2 Jul 2025).

The random-walk generalization replaces the specific deltoid recurrence by polynomial families defined from mean-zero random walks. The associated conformal map

$\Phi(z)^n$0

links the recurrence to Faber polynomials for radially convex domains, and the resulting families both approximate $\Phi(z)^n$1 by degree $\Phi(z)^n$2 polynomials and exhibit a rapid growth property outside the stability region. These properties are used to build arbitrary-order dynamic momentum power iteration methods for classes of non-symmetric matrices (Cowal et al., 28 Oct 2025).

5. Modular, weakly holomorphic, and quasimodular settings

In modular-form theory, Faber polynomials encode basis elements and zero sets through the modular invariant $\Phi(z)^n$3. For weakly holomorphic modular forms of weight $\Phi(z)^n$4 on $\Phi(z)^n$5, one has

$\Phi(z)^n$6

and the main analytic result of the cited paper is an asymptotic description of the coefficients of these $\Phi(z)^n$7 in terms of derivatives of Maass–Poincaré series. The paper explicitly stresses that these are not the classical Faber polynomials of function theory, but special polynomials in $\Phi(z)^n$8 arising from modular basis construction (Kane, 2010).

For cusp forms of large weight with very large order of vanishing at infinity and a fixed number $\Phi(z)^n$9 of finite zeros, the associated Faber polynomial is defined by

$F_n(z)=\Phi(z)^n+O(1/z)\qquad (z\to\infty).$0

with $F_n(z)=\Phi(z)^n+O(1/z)\qquad (z\to\infty).$1. The renormalized polynomials satisfy

$F_n(z)=\Phi(z)^n+O(1/z)\qquad (z\to\infty).$2

so they converge to the truncated exponential polynomial of degree $F_n(z)=\Phi(z)^n+O(1/z)\qquad (z\to\infty).$3. Consequently, the zeros of the modular forms cluster near $F_n(z)=\Phi(z)^n+O(1/z)\qquad (z\to\infty).$4 vertical lines, at height approximately $F_n(z)=\Phi(z)^n+O(1/z)\qquad (z\to\infty).$5, rather than following the boundary-circle or uniform-distribution patterns known from Eisenstein series and Hecke cusp forms (Rudnick, 2023).

Generalized Faber polynomials also enter orthogonal polynomial expansions. For even $F_n(z)=\Phi(z)^n+O(1/z)\qquad (z\to\infty).$6, the basis elements

$F_n(z)=\Phi(z)^n+O(1/z)\qquad (z\to\infty).$7

define monic generalized Faber polynomials in $F_n(z)=\Phi(z)^n+O(1/z)\qquad (z\to\infty).$8, and these admit expansions in an Atkin-like orthogonal basis. The expansion coefficients are identified with Fourier coefficients of normalized extremal quasimodular forms multiplied by explicit modular factors (Nakaya, 2023).

A more recent development concerns the Miller basis. For $F_n(z)=\Phi(z)^n+O(1/z)\qquad (z\to\infty).$9, if $\Psi=\Phi^{-1}$0 are the zeros of $\Psi=\Phi^{-1}$1, then for every $\Psi=\Phi^{-1}$2,

$\Psi=\Phi^{-1}$3

This linearity of moments extends the range in which one can prove that at least one zero leaves the arc on the boundary of the modular fundamental domain, and it yields a limit distribution depending on the asymptotic ratio of the Miller index to the weight (Zilka, 7 Oct 2025).

6. Geometric function theory, weighted bases, and coefficient problems

In the theory of bi-univalent functions, Faber polynomials are used to encode inverse coefficients. For a normalized analytic function

$\Psi=\Phi^{-1}$4

the inverse $\Psi=\Phi^{-1}$5 is expanded in terms of Faber polynomials $\Psi=\Phi^{-1}$6. This permits systematic coefficient extraction under subordination conditions and yields explicit bounds for Taylor–Maclaurin coefficients. Representative results include

$\Psi=\Phi^{-1}$7

under vanishing assumptions on the intermediate coefficients, and

$\Psi=\Phi^{-1}$8

for a comprehensive subclass $\Psi=\Phi^{-1}$9. The same framework produces Fekete–Szegő type inequalities such as bounds for $E \subset \mathbb{C}$00 (Amourah, 2018, Saleh et al., 2019).

This coefficient technology has been combined with other analytic structures. One paper defines a new class of bi-univalent functions using the Tremblay fractional derivative operator and a Fibonacci-based subordination function

$E \subset \mathbb{C}$01

then uses Faber polynomial expansions to derive bounds for the general coefficient $E \subset \mathbb{C}$02 (Altinkaya et al., 2019).

On regular curves, generalized Faber polynomials also form bases in weighted analytic spaces. If the weight satisfies the Muckenhoupt $E \subset \mathbb{C}$03 condition on the curve, then $E \subset \mathbb{C}$04 and $E \subset \mathbb{C}$05 are bases in the corresponding weighted Smirnov spaces. The same paper proves that a double system with complex-valued coefficients,

$E \subset \mathbb{C}$06

forms a basis in weighted Lebesgue spaces over regular curves (Bilalov et al., 2019).

A further structural result concerns common zeros. If the first $E \subset \mathbb{C}$07 Faber polynomials of a meromorphic univalent function vanish at $E \subset \mathbb{C}$08, then

$E \subset \mathbb{C}$09

and

$E \subset \mathbb{C}$10

If instead $E \subset \mathbb{C}$11 and $E \subset \mathbb{C}$12 for all $E \subset \mathbb{C}$13, then necessarily

$E \subset \mathbb{C}$14

with univalence equivalent to $E \subset \mathbb{C}$15 (Savchuk, 2015).

7. Inverse problems and geometric flows

Faber polynomials have also been adapted to inverse conductivity problems through Faber Polynomial Polarization Tensors (FPTs). For a simply connected inclusion $E \subset \mathbb{C}$16, the exterior conformal map is written

$E \subset \mathbb{C}$17

and the corresponding Faber polynomials are defined by a generating function involving $E \subset \mathbb{C}$18. In the extreme-conductivity regime, the shape-recovery formula is explicit: $E \subset \mathbb{C}$19 This yields an exact recovery of the conformal mapping coefficients from FPT data up to the order of measurement (Choi et al., 2019).

A later paper extends the same strategy to two non-iterative analytical methods based on GPTs and FPTs. One reconstructs inclusions with extreme or near-extreme conductivity through explicit conformal-map coefficients, and the other treats arbitrary conductivity by approximating the inclusion as a perturbation of its equivalent ellipse. The paper states that this second method can non-iteratively approximate an inclusion of general shape with arbitrary conductivity, including a straight or asymmetric shape (Choi et al., 2020).

A very different geometric application appears in the study of plane loops and integrable systems. For a loop $E \subset \mathbb{C}$20, the generating function

$E \subset \mathbb{C}$21

has coefficients that are identified with curvature and higher local invariants, including $E \subset \mathbb{C}$22, $E \subset \mathbb{C}$23, $E \subset \mathbb{C}$24, and the Schwarzian derivative. Imposing isometry and isoenergy conditions leads to a recursion operator

$E \subset \mathbb{C}$25

and the resulting hierarchy of curvature flows is the mKdV hierarchy (Matsutani et al., 2015).

Taken together, these developments show that Faber polynomials are not a single-purpose approximation tool but a conformally organized algebraic mechanism. Their classical role as polynomial parts of exterior maps persists, yet the same mechanism now supports operator-norm bounds on non-convex spectral sets, non-Hermitian propagation schemes, modular zero asymptotics, weighted basis theory, explicit inverse reconstructions, and integrable curve dynamics.