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Chebyshev polynomials on a Jordan arc
Benedikt Buchecker, Benjamin Eichinger, Olof Rubin, Aron Wennman
TL;DR
The paper addresses the open Christiansen–Simon–Zinchenko conjecture on Chebyshev-polynomial asymptotics for analytic Jordan arcs, where the real-line alternation theorem fails. Using weighted Faber polynomials, extremal signatures, discrete orthogonal polynomials, and sampling inequalities, it proves the conjecture for every analytic Jordan arc and obtains Szegő–Widom asymptotics for the polynomials themselves.
Problem
The revised Christiansen–Simon–Zinchenko conjecture on Chebyshev-polynomial asymptotics for Jordan arcs remained open after the real-line alternation theorem failed beyond the real line.
Method
The proof uses Faber polynomials constructed from Laurent expansions of a canonical exterior conformal map, together with extremal-point polynomials.
Results
The Christiansen–Simon–Zinchenko conjecture holds for any analytic Jordan arc, with the Chebyshev polynomials themselves also satisfying locally uniform Szegő–Widom asymptotics.
Takeaways & Limitations
The paper establishes the conjectured asymptotic description for Chebyshev polynomials on analytic Jordan arcs and extends it from norm asymptotics to the polynomials themselves.
Takeaways & Limitations
The analysis does not resolve the corresponding questions for several disjoint arcs and curves, for which the authors state that new ideas appear necessary.
Abstract
from arXiv · showhide
We describe the asymptotics of Chebyshev polynomials on an analytic Jordan arc in the plane. This gives an affirmative answer to a conjecture of Christiansen-Simon-Zinchenko, based on predictions of Widom from 1969. The proof combines weighted Faber polynomials with extremal signatures, discrete orthogonal polynomials and a Marcinkiewicz-Zygmund sampling inequality, and yields Szegő-Widom asymptotics for the Chebyshev polynomials themselves.
1. Introduction
The paper proves the revised Christiansen–Simon–Zinchenko conjecture for every analytic Jordan arc, determining the asymptotic Widom factor and strengthening norm asymptotics to Szegő–Widom asymptotics for the polynomials themselves. The proof combines weighted Faber polynomials for the upper bound with extremal signatures and discrete orthogonal polynomials for the lower bound.
- Main result: For circular arcs, the Widom factors converge to 2 cos^2(α/4), with values ranging from 1 to 2.This behavior motivated the revised conjecture after showing that the original universal limit 2 was incorrect.
- Main result: The Christiansen–Simon–Zinchenko conjecture holds for any analytic Jordan arc γ.This resolves the revised conjecture prompted by the failure of Widom’s original prediction for circular arcs.
- Further asymptotics: The norm asymptotics are strengthened to locally uniform Szegő–Widom asymptotics for the Chebyshev polynomials on the complement of γ.The formulation uses the outer function g(z) = ρ(z)/ρ(∞).
- Proof strategy: The proof obtains the upper bound from weighted Faber polynomials and the matching lower bound from extremal signatures and discrete orthogonal polynomials.The lower-bound construction uses discrete measures supported on n + 1 suitably chosen points.
- Further constructions: The paper introduces dual Faber polynomials whose zeros are the extremal points of the weighted Faber polynomial used in the upper-bound construction.These degree-n + 1 polynomials are presented as a potentially independent object and are compared with classical Chebyshev polynomials of the second kind.
- Open questions: Open questions include limit points for several disjoint arcs and curves and whether the original Widom conjecture holds when zeros are constrained to the arc.The constrained-zero question is known for intervals and circular arcs, while the general lower bound appears to require new ideas.
2. Faber polynomials on curves and arcs
This section outlines Faber polynomials through exterior conformal maps and develops their asymptotics for analytic Jordan curves and arcs. For arcs, the two boundary sides are handled by opening the arc into a Jordan-curve domain, while analytic regularity yields holomorphic non-vanishing weights to which the asymptotics apply.
- Jordan curves: For analytic Jordan curves, conformal-map and weight extensions to neighborhoods provide the contour framework underlying weighted Chebyshev asymptotics.The construction uses nested positively oriented Jordan curves around the original curve.
- Jordan arcs: For analytic Jordan arcs, Faber polynomials receive contributions from both boundary sides because ϕ and g have two sets of boundary values.The map z + √(z^2 − 1) opens the arc into the exterior of an analytic Jordan curve, reducing the analysis to two Faber-polynomial evaluations.
- Jordan arcs: Near the arc, the resulting asymptotic formula holds uniformly on the arc and in a neighborhood when interpreted by analytic extension.The two-sided approach is reflected through boundary values denoted g+ and g−.
- Regularity: If γ is analytic, ρ ◦ φ extends holomorphically and non-vanishingly near D, so g = ρ/ρ(∞) satisfies the asymptotic formula for Fn(g, z).For γ of class C1+α, ρ ◦ φ is instead Hӧlder-α continuous on D.
3. The upper bound
The upper bound is established by constructing weighted Faber-polynomial trial functions and optimizing their asymptotic boundary norm. For C2+α Jordan arcs, the unique extremizer is g = ρ/ρ(∞), yielding the limit ∥Fn(g, ·)∥γ → 1/ρ(∞) and the claimed upper bound.
- Trial-polynomial construction: Trial polynomials Fn(g, z) have degree n and leading coefficient ϕ′(∞)^n, reducing the upper-bound problem to controlling their boundary norm.The function g is selected from a normalized H∞ class to minimize ∥Fn(g, ·)∥γ asymptotically.
- Asymptotic norm: For a C2+α Jordan arc with 0 < α < 1, Theorem 3.1 gives lim n→∞ ∥Fn(g, ·)∥γ = 1/ρ(∞).The admissible family consists of g ∈ H∞(Ω) with g(∞) = 1 and g ◦ φ ∈ Cη(D) for some η > 0.
- Extremal problem: The extremal problem has a unique solution g(z) = ρ(z)/ρ(∞), and its corresponding minimal value equals the value required for the upper bound.The proof uses harmonic measure and the sub-mean value inequality for the subharmonic modulus of g/ρ; the extremizer has |g+(z)| + |g−(z)| = 1/ρ(∞).
- Conclusion: Combining Lemma 3.2 with Theorem 3.3 proves Theorem 3.1 and establishes the section’s upper bound.The argument also proves uniqueness through the equality case of the mean-value inequality.
4. The lower bound
The lower bound is obtained by using extremal signatures and an explicit optimal prediction measure on carefully chosen extremal points. A dual Faber-polynomial construction places these points appropriately and yields the estimates needed to match the upper bound.
- Discrete optimal prediction measures: For any set of n + 1 distinct points, an explicit measure realizes the optimal prediction measure and identifies the associated Chebyshev polynomial.The construction bypasses an abstract maximizing-measure argument by giving a formula attributed to Vidensky.
- Extremal points: For sufficiently large n, nθ(z) + β(z) increases strictly from 0 to 2πn, producing precisely n + 1 extremal points z_j on the arc.They satisfy nθ(z_j) + β(z_j) = 2πj, and their spacing is sufficiently separated for the lower-bound argument.
- Dual Faber polynomial: The natural dual Faber polynomial has a square-root singularity that switches the interaction between the two branches, changing cosine oscillations into sine oscillations.Its asymptotic expansion holds uniformly on a fixed neighborhood of γ, with an exponentially small error for some 0 < r < 1.
- Zero placement: A zero-free holomorphic function f is constructed so that the zeros of the resulting polynomial coincide exactly with the n + 1 extremal points z_j.The construction matches the phase difference of f to that of g and controls the zeros uniformly near the arc.
- Lower-bound estimate: There exists C > 0 such that the constructed polynomial satisfies the required lower-bound estimate at every extremal point, uniformly for 0 ≤ j ≤ n.The estimate is expressed through the normalization involving ρ(∞) and an additive constant C depending only on γ.
5. Szeg˝o-Widom asymptotics
The section proves Szegő–Widom asymptotics for Chebyshev polynomials on analytic Jordan arcs. It establishes this by showing that weighted Faber polynomials satisfy the hypotheses of a stability lemma and then transferring discrete L2 control to standard L2 and exterior asymptotics.
- Stability lemma: Approximate extremality and orthogonality force a monic polynomial P_n to be close to the monic Chebyshev polynomial T_n.The stability lemma applies when P_n is nearly extremal on the support points and almost orthogonal with respect to the discrete measure μ_n.
- Sampling inequality: A Marcinkiewicz–Zygmund sampling inequality transfers the resulting discrete L2 stability to the standard L2 norm on γ.The sampling estimate is obtained using separated points and control of the Muckenhoupt A2-characteristic.
- Weighted Faber polynomial: The weighted Faber polynomial F_n(g, z) satisfies both the near-extremality and approximate orthogonality conditions required by the stability lemma.Its behavior at the extreme points is linked to orthogonal-polynomial values, including at the endpoints.
- Proof of Theorem 1.2: Combining Proposition 5.3, the stability lemma, and the sampling inequality proves strong L2 asymptotics for the Chebyshev polynomials.The proof normalizes Cap(γ)=1 before applying these ingredients.
- Proof of Theorem 1.2: Bernstein–Walsh control converts the L2 asymptotics into Szegő–Widom asymptotics at points bounded away from γ.The resulting exterior estimate includes a logarithmic n-dependent error term.
Appendix A. Marcinkiewicz–Zygmund inequalities
This appendix explains how the Chui–Zhong Marcinkiewicz–Zygmund result is extended from one arc to a general analytic arc, including weights whose A2-characteristic may degenerate with n.
- General analytic arcs: The proof overview adapts Chui–Zhong’s result from a particular arc to a general analytic arc.The appendix reproduces key proof steps to explain the mechanism and identify where the generalization requires additional care.
- Degenerate weights: The argument allows the weight ωn(z) = |En+1(z)|2 to have an A2-characteristic that degenerates with n.This degeneration is the stated technical issue requiring extra care in the proof.
A.1. Control on Green curves.
This section establishes control on Green curves through an assumption on the distribution of chosen points and a comparison result for polynomials. The comparison is formulated with positive constants for any polynomial of degree n.
- A.1. Control on Green curves.: The key assumption concerns the distribution of the chosen points z_j.The supplied passage introduces this as the section’s key assumption, but does not include its full mathematical statement.
- A.1. Control on Green curves.: A useful comparison result is taken from [20, Lemma 10].The passage identifies this result as a comparison tool used in the section.
- A.1. Control on Green curves.: Lemma A.1 asserts the existence of positive constants c and C for any polynomial P of degree n.The supplied excerpts do not include the inequality relating c, C, P, and n.
- A.1. Control on Green curves.: The corresponding interval-and-exponential-weight statement is attributed to Levin–Lubinsky, with a proof adapting verbatim here.The cited prior statement concerns the interval [−1, 1] and an exponential weight.
A.2. A sampling theorem for Hardy spaces.
This section establishes a Hardy-space interpolation lemma for H2(Dn), using uniformly separated conformal images of interpolation points. Carleson’s theorem and the Shapiro–Shields interpolation estimate provide the required bound with constants uniform in n.
- A key input is an interpolation lemma for the Hardy space H2(Dn), where Dn is the domain enclosed by γn.
- The conformal images wj,n of the points zj must be uniformly separated, with a separation constant independent of n.
- Carleson’s interpolation theorem identifies the required condition with weak separation in the pseudohyperbolic metric and a Carleson measure estimate.
- The interpolation estimate follows from uniform geometric estimates, the H2-interpolation theorem of Shapiro–Shields, and composition with an appropriate conformal map.The relevant estimates are established with all constants uniform in n.
A.3. Uniform control of Cauchy transforms.
This section defines boundedness of the Cauchy transform on weighted L2 spaces and explains how uniform estimates are obtained for the varying curves γn. The argument combines explicit A2-weight dependence with uniform bounds for the unweighted transforms on γn.
- Operator framework: The Cauchy transform is treated as an operator on L2(Γ, ω |dz|), with norm defined by the best constant in its upper bound.The definition is stated for smooth functions f ∈ C∞(Γ).
- Operator framework: The general boundedness criterion requires Γ to be a Carleson curve and the weight to satisfy the relevant Muckenhoupt condition.Carleson curves are characterized by an upper arc-length bound |Γ ∩ B(x, ε)| ≤ CΓε.
- Uniform estimates: A2,γn ≍ log n, so the general boundedness theorem does not directly provide the explicit uniform norm control needed for the changing curves γn.The section therefore uses more precise estimates that make the dependence of the Cauchy-integral norm on the weight explicit.
- Uniform estimates: The weighted estimate is reduced to proving uniform bounds for the unweighted Cauchy transforms on γn, with constants independent of n.The unweighted norm on γn is denoted γn,1.
A.4. Proof of the Marcinkiewicz–Zygmund inequality. · Appendix B. Geometric lemmas
The Marcinkiewicz–Zygmund estimate is proved by constructing f in H2(Dn) from the polynomial samples, then controlling f − P through Cauchy-transform bounds and interpolation. The supplied passages describe this proof only; no geometric lemmas from Appendix B are included.
- A.4. Proof of the Marcinkiewicz–Zygmund inequality.: Applying Lemma A.2 with aj = P(zj) produces f ∈ H2(Dn) satisfying the desired H2-norm bound.This supplies the analytic function used to compare the sampled polynomial P with an H2 object.
- A.4. Proof of the Marcinkiewicz–Zygmund inequality.: The triangle inequality reduces the argument to estimating the norm of f − P.The proof explicitly identifies this difference as the remaining quantity to control.
- A.4. Proof of the Marcinkiewicz–Zygmund inequality.: Because (f − P)/E is holomorphic on Dn, Cauchy’s theorem yields an integral representation for controlling f − P.The decay P(w)E(w)(w−z) = O(1/w2) at infinity justifies the final contour step.
- A.4. Proof of the Marcinkiewicz–Zygmund inequality.: On γn, the resulting expression is identified using the principal-value Cauchy transform and Plemelj’s formula (A.1).The passages introduce Ch(z) as the Cauchy transform along γn and attribute the factor 1/2 to Plemelj’s formula.
- A.4. Proof of the Marcinkiewicz–Zygmund inequality.: Lemma A.3 bounds the Cauchy transform on L2(γn, ωn |dz|), allowing the norm of f − P to be controlled.The same estimate is also applied to 1/2I + C.
- A.4. Proof of the Marcinkiewicz–Zygmund inequality.: The resulting bound has an implicit constant depending only on the fixed arc γ and, combined with (A.3), yields the needed estimate.The final passage applies the interpolation bound (A.2) for f to complete the proof.
B.1. Growth of A2-characteristics.
The A2-characteristic of the weight |E|^2 is reduced to the elementary weight |z^2−1| on γ_n via the Faber approximation E0. Endpoint subarcs determine the growth, yielding logarithmic order in n.
- Reduction to an elementary weight: The reduction is based on the uniform comparison |E0(z)|^2 ≍ |z^2−1| for z ∈ γ_n.This comparison follows from the dual Faber weight, modulus matching, and dominance of the first term in the relevant expansion.
- Conformal parametrization: The conformal change of variables φ(σ) = (σ + σ^-1)/2 transfers subarc estimates from γ_n to Γ_n uniformly in n.The restriction of φ maps Γ_n homeomorphically onto γ_n, while the comparison remains uniform because Γ_n → Γ and Γ stays away from the origin.
- Endpoint analysis: The endpoint lower bound reaches log n by choosing a subarc with σ^-1_n(J) = [0,c].For this choice, the corresponding normalized endpoint parameter is cn, so the ratio is comparable to log n.
- Conclusion: Combining endpoint and non-endpoint subarc estimates gives an upper bound O(log n), and the reverse inequality follows from the endpoint lemma.Thus the A2-characteristic has logarithmic growth in n.
B.2. Separation of extreme points. · Benedikt Buchecker KU Leuven Leuven, Belgium benedikt.buchecker@kuleuven.be
This section proves separation estimates for the extremal points z_0,…,z_n by analyzing their angular parametrization and transferring the resulting spacing to the arc γ. The proof also controls the distance from each extremal point to the discrete set γ_n.
- B.2. Separation of extreme points.: The lemma establishes asymptotic relations governing the extremal points z_j and reduces the conclusion to two intermediate claims.The proof explicitly combines an asymptotic equality with a second estimate to obtain the lemma.
- B.2. Separation of extreme points.: Because (nA+B)′ ≍ n and A′ stays between positive constants, consecutive angular parameters s_j=θ(z_j)=A(t_j) are separated on the scale 1/n.The argument uses the parametrizations θ, A, and B to derive the spacing of the s_j.
- B.2. Separation of extreme points.: The endpoint normalization s_0=0 and s_n=2π yields corresponding estimates for the distances of s_j from both endpoints.Summing the consecutive-spacing estimate gives control near 0 and 2π.
- B.2. Separation of extreme points.: The inverse equilibrium parametrization identifies z_j with z_e(cos(s_j/2)), allowing angular separation to be translated into separation of the associated points x_j=cos(s_j/2).Regularity of z_e supplies the parametrization needed for this transfer.
- B.2. Separation of extreme points.: Elementary trigonometry together with the angular estimates proves the desired spacing estimate for the points {x_j: 0≤j≤n}.The proof expresses differences of the cosine-parametrized points through trigonometric factors involving consecutive s_j.
- B.2. Separation of extreme points.: A comparison estimate for points on the unit circle, combined with conformality of Φ^−1 near that circle, yields the distance estimate d(z_j,γ_n).The proof first establishes the unit-circle comparison, then applies the conformal map and combines the resulting estimates.