Zolotarev Norms in Approximation & Probability

• Zolotarev Norms are extremal metrics used to measure minimal deviations in rational and polynomial approximations and to define distances between probability measures.
• They involve explicit constructions via elliptic, theta, and Faber rational functions, offering sharp bounds and algorithmic frameworks for numerical analysis.
• Their applications span quantitative limit theorems, operator theory, and optimal filter design by bridging approximation theory with duality in transport problems.

The term "Zolotarev norms" encompasses a collection of extremal functionals and distances, arising in rational approximation, probability metrics, and uniform minimax problems for polynomials and rational functions. There are distinct but related uses: (i) the Zolotarev uniform norm of extremal polynomials; (ii) the Zolotarev (ideal) metrics on probability measures defined via test functions of given smoothness or moment vanishing conditions; and (iii) the Zolotarev (ratio) norms associated with the third Zolotarev problem for rational functions on disjoint sets in the complex plane. These objects are central in modern approximation theory, probability, and numerical analysis, as they provide sharp constants and universality properties in their respective regimes.

1. Zolotarev Norms in Rational Polynomial and Rational Function Approximation

The classical Zolotarev norm arises in the study of uniform approximation of functions by constrained polynomials or rational functions. For polynomials, the Zolotarev norm of order $n$ is the minimal maximal deviation on $[-1,1]$ among all degree-$n$ polynomials with prescribed leading coefficients (Zolotarev's first problem). For rational functions, the norm is expressed as a minimax ratio problem over disjoint compact sets. These norms have explicit connections to elliptic and theta functions, and serve as extremal constants in best approximation scenarios.

For rational functions, the "Zolotarev norm" of degree $n$ for two disjoint compact sets $E, F \subseteq \widehat{\mathbb{C}}$ is: $$\sigma_n = \min_{\substack{r \in R_n \ \min_{z \in F}|r(z)| = 1}}\ \max_{z \in E}|r(z)| \ =\ \min_{\deg(r)\le n}\frac{\sup_{z \in E}|r(z)|}{\sup_{z \in F}|r(z)|},$$ with the minimizer called the optimal Zolotarev ratio function of degree $n$ (Trefethen et al., 2024, Antoulas et al., 6 Nov 2025).

2. Zolotarev (Ideal) Probability Metrics and Smooth Norms

For probability measures, the Zolotarev norm (or distance) of order $p$ ($p \in \mathbb{N}$, or more generally $s > 0$) quantifies the "distance" between two distributions via test functions with vanishing moments and bounded smoothness. For measures $[-1,1]$0 on $[-1,1]$1 with matched lower moments, the norm is

$[-1,1]$2

where the supremum is over $[-1,1]$3 with all mixed lower-order moments vanishing and all partial derivatives of order $[-1,1]$4 bounded by 1. For $[-1,1]$5, this reduces to the $[-1,1]$6 (Kantorovich–Wasserstein) distance; for $[-1,1]$7 and $[-1,1]$8, one requires barycenter and covariance equality, respectively (Bobkov et al., 21 Jun 2025, Bołbotowski et al., 31 Oct 2025).

In dimension one, dual representations exist in terms of integrated tails and variational functionals. These norms are interpolation metrics between total variation and moments, and they fully metrize weak convergence under moment control (Mattner, 2022).

3. Zolotarev Norms and Duality: Kantorovich–Rubinstein Structures

The Zolotarev $[-1,1]$9 norm admits a novel duality, extending the Kantorovich–Rubinstein duality for $n$0. For centered probability measures $n$1, the $n$2 distance is

$n$3

and has a dual optimal transport formulation involving three-marginal couplings with martingale-like constraints: $n$4 where $n$5 enforces appropriate marginal and martingale balance conditions (Bołbotowski et al., 31 Oct 2025).

4. Sharp Inequalities and Metric Comparisons

Zolotarev norms are tightly related to quadratic Wasserstein distances and play a role in optimal constants for inequalities, limit theorems, and error bounds.

• Zolotarev–Wasserstein Relations: For measures $n$6,

$n$7

with equality in the upper bound iff $n$8 is a dilation of $n$9, and in the lower bound only if $n$0 (Bołbotowski et al., 31 Oct 2025). These constants are optimal and describe the minimal and maximal possible ratio between $n$1 and $n$2 over measures with fixed variances.

• Moment Slices and Quantitative Reduction: The multivariate Zolotarev distance can be controlled by the supremum of the corresponding one-dimensional distances over all projections, with explicit dimension- and moment-dependent exponents:

$n$3

where $n$4 for $n$5 and $n$6 the $n$7th moment bound (Bobkov et al., 21 Jun 2025).

5. Algorithmic and Constructive Theory

Computing Zolotarev norms and the associated extremal objects is nontrivial. Recent advances cover both explicit and algorithmic approaches:

• Explicit Polynomial Parametrization: Proper Zolotarev polynomials of degree $n$8 can be constructed as normalized extremal polynomials on $n$9 with prescribed top coefficients. For $E, F \subseteq \widehat{\mathbb{C}}$0, rational parametrizations are available; for $E, F \subseteq \widehat{\mathbb{C}}$1, explicit radical or nested radical representations have been achieved, resolving century-old open cases (Rack et al., 2019, Rack et al., 2020). These polynomials equioscillate between $E, F \subseteq \widehat{\mathbb{C}}$2 on a fixed set of extremal points, and their minimal deviation yields the classical Zolotarev norm.
• Rational Function Construction: The third Zolotarev problem is equivalent to a best uniform approximation by rational functions (of type $E, F \subseteq \widehat{\mathbb{C}}$3) with prescribed behavior on two sets. Its minimax solutions can be characterized by equioscillation and by the structure of level curves in the complex plane. In practice, AAA and AAA-Lawson algorithms, as well as the Loewner framework, are deployed for numerical construction. The Loewner framework, in particular, achieves near-minimax accuracy with superior computational efficiency at high degrees (Trefethen et al., 2024, Antoulas et al., 6 Nov 2025).
• Faber Rational Functions and Bounds: Explicit upper and lower bounds for Zolotarev numbers (extremal ratios) are derivable via Faber rational functions built from conformal maps onto doubly-connected domains. Such bounds have immediate applications to the analysis of operator singular values and to the parameter selection in iterative matrix solvers (Rubin et al., 2019).

6. Applications in Probability, Approximation, and Numerical Analysis

Zolotarev-type norms (especially the $E, F \subseteq \widehat{\mathbb{C}}$4 and $E, F \subseteq \widehat{\mathbb{C}}$5 metrics) have become ubiquitous in quantitative limit theorems:

• Central Limit Theorem: Error bounds for the Kolmogorov or Wasserstein distances between normalized sums and the Gaussian distribution can be sharply expressed in terms of Zolotarev norms:

$E, F \subseteq \widehat{\mathbb{C}}$6

for some universal constant $E, F \subseteq \widehat{\mathbb{C}}$7; this strictly improves on classical Berry–Esseen and Katz bounds by accounting for "Zolotarev-closeness to normality" rather than just absolute moments (Jonas et al., 9 Mar 2025, Mattner, 2022).

• Operator Theory and Solvers: The decay rate of singular values of structured matrices (e.g., Cauchy, Vandermonde) can be estimated in terms of Zolotarev numbers for the spectrum-containing sets, with explicit connection to the rational minimax constants. The optimal (or near-optimal) choice of shift parameters in ADI-type methods for the Sylvester equation can be directly extracted from Faber rational functions associated with Zolotarev bounds (Rubin et al., 2019).
• Approximation Theory and Filtering: The explicit forms for Zolotarev polynomials and rational functions are crucial for the design of optimal filters, digital signal processing, and high-accuracy quadrature on irregular geometries.

7. Summary Table: Main Zolotarev Norms and Their Contexts

Norm/Quantity	Definition/Context	Domain
$E, F \subseteq \widehat{\mathbb{C}}$8	$E, F \subseteq \widehat{\mathbb{C}}$9 (polynomial dev.)	$$\sigma_n = \min_{\substack{r \in R_n \ \min_{z \in F}|r(z)| = 1}}\ \max_{z \in E}|r(z)| \ =\ \min_{\deg(r)\le n}\frac{\sup_{z \in E}|r(z)|}{\sup_{z \in F}|r(z)|},$$0 (polynomials)
$$\sigma_n = \min_{\substack{r \in R_n \ \min_{z \in F}|r(z)| = 1}}\ \max_{z \in E}|r(z)| \ =\ \min_{\deg(r)\le n}\frac{\sup_{z \in E}|r(z)|}{\sup_{z \in F}|r(z)|},$$1	$$\sigma_n = \min_{\substack{r \in R_n \ \min_{z \in F}|r(z)| = 1}}\ \max_{z \in E}|r(z)| \ =\ \min_{\deg(r)\le n}\frac{\sup_{z \in E}|r(z)|}{\sup_{z \in F}|r(z)|},$$2 (ratio)	$$\sigma_n = \min_{\substack{r \in R_n \ \min_{z \in F}|r(z)| = 1}}\ \max_{z \in E}|r(z)| \ =\ \min_{\deg(r)\le n}\frac{\sup_{z \in E}|r(z)|}{\sup_{z \in F}|r(z)|},$$3
$$\sigma_n = \min_{\substack{r \in R_n \ \min_{z \in F}|r(z)| = 1}}\ \max_{z \in E}|r(z)| \ =\ \min_{\deg(r)\le n}\frac{\sup_{z \in E}|r(z)|}{\sup_{z \in F}|r(z)|},$$4	$$\sigma_n = \min_{\substack{r \in R_n \ \min_{z \in F}|r(z)| = 1}}\ \max_{z \in E}|r(z)| \ =\ \min_{\deg(r)\le n}\frac{\sup_{z \in E}|r(z)|}{\sup_{z \in F}|r(z)|},$$5, $$\sigma_n = \min_{\substack{r \in R_n \ \min_{z \in F}|r(z)| = 1}}\ \max_{z \in E}|r(z)| \ =\ \min_{\deg(r)\le n}\frac{\sup_{z \in E}|r(z)|}{\sup_{z \in F}|r(z)|},$$6 $$\sigma_n = \min_{\substack{r \in R_n \ \min_{z \in F}|r(z)| = 1}}\ \max_{z \in E}|r(z)| \ =\ \min_{\deg(r)\le n}\frac{\sup_{z \in E}|r(z)|}{\sup_{z \in F}|r(z)|},$$7, Deriv. bounds	$$\sigma_n = \min_{\substack{r \in R_n \ \min_{z \in F}|r(z)| = 1}}\ \max_{z \in E}|r(z)| \ =\ \min_{\deg(r)\le n}\frac{\sup_{z \in E}|r(z)|}{\sup_{z \in F}|r(z)|},$$8 (probability)
$$\sigma_n = \min_{\substack{r \in R_n \ \min_{z \in F}|r(z)| = 1}}\ \max_{z \in E}|r(z)| \ =\ \min_{\deg(r)\le n}\frac{\sup_{z \in E}|r(z)|}{\sup_{z \in F}|r(z)|},$$9	$n$0	$n$1 (probability)

These definitions represent, respectively, the minimal uniform norm for polynomial deviation, the minimax rational ratio on sets, the ideal metric on measures via smooth-test functions, and the second-order Kantorovich-type Zolotarev distance.

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Zolotarev norms thus provide the sharpest possible metrics and extremal constants in a diverse set of settings ranging from best polynomial/rational approximation, to quantitative limit theorems and stability of numerical algorithms. Their theory synthesizes explicit analytic, algebraic, and geometric methods and continues to expand with new structural dualities and computational frameworks (Rack et al., 2019, Trefethen et al., 2024, Bołbotowski et al., 31 Oct 2025, Bobkov et al., 21 Jun 2025, Antoulas et al., 6 Nov 2025, Jonas et al., 9 Mar 2025, Mattner, 2022, Rubin et al., 2019, Rack et al., 2020).