Zolotarev Projection Methods

• Zolotarev projection is a family of methods that use optimal rational approximations to separate disjoint spectral regions or one-dimensional probability projections.
• Its formulation, via the third and fourth Zolotarev problems, underpins spectral filtering by approximating sign and indicator functions with sharp passband/stopband characteristics.
• Numerical techniques such as AAA, AAA-Lawson, and the Loewner framework are employed to construct Zolotarev filters for robust eigenvalue computations in complex geometries.

“Zolotarev projection” is not a single universally standardized term. In numerical linear algebra, it usually denotes a rational approximate spectral projector built from Zolotarev’s extremal approximation to sign or indicator functions on separated spectral sets. Closely related work formulates the third and fourth Zolotarev problems on disjoint sets $E$ and $F$, then converts sign approximants into separator or filter functions that emulate projectors. In probability theory, by contrast, projection refers to one-dimensional pushforwards $\mu_\theta$ entering a quantitative Cramér–Wold theorem for Zolotarev distances. These usages share the Zolotarev name but not a common operator-theoretic definition (Guettel et al., 2014, Trefethen et al., 2024, Bobkov et al., 21 Jun 2025).

1. Terminological scope

The phrase is best understood as a family of constructions rather than a named canonical operator. In the spectral-filtering literature, the central object is an approximate projector onto a target eigenspace, implemented by a rational function that is close to $1$ on one spectral region and close to $0$ on another. In the rational-approximation literature, the same idea appears as optimal separation of two disjoint sets by a rational function. In the probability-metric literature, “projection” instead means linear projection of a multivariate law onto one-dimensional marginals.

Usage	Core object	Representative source
Spectral projection/filtering	Rational approximate spectral projector	(Guettel et al., 2014)
Rational separation on $E,F$	Zolotarev sign and ratio problems	(Trefethen et al., 2024, Antoulas et al., 6 Nov 2025)
Probability-theoretic projection	Pushforward $\mu_\theta$ under $x\mapsto \langle x,\theta\rangle$	(Bobkov et al., 21 Jun 2025)

A recurring misconception is that the term denotes a literal projection operator defined by nearest-point or orthogonality principles. The cited literature does not support that reading. Mattner’s normal-approximation framework, for example, is explicitly relevant only in the broad sense of Zolotarev-type probability metrics and “does not define a literal projection operator onto the normal law” (Mattner, 2022). Likewise, the classification of twisted Zolotarev fractions is structural background on rational maps with four critical values, not a projection algorithm (Bogatyrev, 2015).

2. Rational sign and ratio formulations

A modern approximation-theoretic formulation begins with two disjoint closed sets $E,F\subseteq \mathbb C\cup\{\infty\}$ and rational functions of bounded degree. The third Zolotarev problem $\mathrm{Z3}$ seeks a rational separator $F$0 with

$F$1

that minimizes

$F$2

Equivalently, it minimizes the ratio

$F$3

The fourth Zolotarev problem $F$4 seeks a minimax rational approximation $F$5 to the two-valued sign function

$F$6

by minimizing

$F$7

These two problems are equivalent through explicit Möbius formulas, and in the small-error regime the relation simplifies to

$F$8

(Trefethen et al., 2024).

This equivalence is the conceptual core of Zolotarev projection in the spectral sense. A sign approximant is a rational spectral splitter; a ratio minimizer is a rational attenuation/amplification filter. This suggests that once a matrix $F$9 has spectrum separated between $\mu_\theta$0 and $\mu_\theta$1, a plausible implication is that

$\mu_\theta$2

act as approximate spectral projectors. That formula is not written explicitly in the cited paper, but it follows directly from the sign-approximation viewpoint emphasized there (Trefethen et al., 2024).

The numerical study of the third and fourth Zolotarev problems reinforces this projector interpretation. It treats the Zolotarev sign problem as approximation of a piecewise constant separator and the Zolotarev ratio problem as construction of a rational function that is small on one set and large on the other. The paper compares the Loewner framework, AAA, AAA-sign, and AAA-Lawson, and reports that the Loewner framework is fast and reliable, provides approximants with a high level of accuracy, is often more accurate than near-optimal AAA-Lawson at higher degree, and has running time that remains constant with degree (Antoulas et al., 6 Nov 2025).

3. Approximate spectral projectors in FEAST

The clearest operator-theoretic use of the term appears in the FEAST eigensolver literature. For the Hermitian definite generalized eigenvalue problem

$\mu_\theta$3

with $\mu_\theta$4 Hermitian and $\mu_\theta$5 Hermitian positive definite, FEAST is analyzed through

$\mu_\theta$6

The exact object of interest is the spectral projector onto eigenvalues inside a target contour $\mu_\theta$7,

$\mu_\theta$8

FEAST does not apply this projector exactly; it applies a rational filter $\mu_\theta$9, and the iteration is equivalent to subspace iteration with implicit orthogonalization (Guettel et al., 2014).

The convergence mechanism is fully spectral. If $1$0 is the $1$1-orthogonal projector onto the filtered subspace, then

$1$2

Hence performance is determined by passband/stopband separation. The paper formalizes the worst-case filter quality by

$1$3

A good Zolotarev projector is therefore a rational approximant to the indicator of $1$4 that is near $1$5 on the interval and near $1$6 outside it (Guettel et al., 2014).

The Zolotarev construction begins with the best uniform rational approximant $1$7 of type $1$8 to $1$9 on

$0$0

After the Möbius transformation

$0$1

the FEAST filter becomes

$0$2

This is the best uniform rational approximant of type $0$3 to the interval indicator on the passband $0$4 and the stopband $0$5. Its approximation error satisfies

$0$6

and the resulting worst-case FEAST convergence factor obeys

$0$7

(Guettel et al., 2014).

The practical significance is explicit. The paper attributes to the Zolotarev filter a smaller worst-case convergence factor, a steeper slope near interval boundaries, minimax worst-case optimality on the relevant passband and stopband, and more predictable convergence across search intervals. On the Caffeinep2 problem with $0$8, Gauss with $0$9 reaches only $E,F$0 after $E,F$1 iterations, whereas Zolotarev with $E,F$2 reaches $E,F$3 in $E,F$4 iterations. In a two-interval load-balancing experiment with $E,F$5, Gauss required $E,F$6 and $E,F$7 iterations on the two intervals, while Zolotarev required $E,F$8 and $E,F$9 (Guettel et al., 2014).

4. Numerical construction of Zolotarev filters

For general complex geometries, the central computational issue is how to obtain the rational separator itself. One approach computes the sign approximation first and then converts it into the ratio-optimal separator. The paper on computation of Zolotarev rational functions describes this as the first reliable numerical algorithm for general complex sets $\mu_\theta$0 and $\mu_\theta$1. Its workflow is: solve $\mu_\theta$2 by modified AAA and AAA-Lawson, then convert the resulting $\mu_\theta$3 to a $\mu_\theta$4 solution $\mu_\theta$5 and recover $\mu_\theta$6, poles, and zeros. The implementation uses barycentric rational form, a modified “blending of singular values” step in AAA for sign problems, and damped AAA-Lawson with

$\mu_\theta$7

to improve robustness (Trefethen et al., 2024).

The method is designed for broad geometric flexibility. The sets $\mu_\theta$8 and $\mu_\theta$9 may be intervals, unions of intervals, circles, ellipses, polygons, crosses, nested domains, disconnected unions of continua, or more general closed sets sampled on boundaries. This breadth matters for projection applications because explicit elliptic-function formulas are available only in special geometries, whereas spectral sets arising in practice are often nonclassical (Trefethen et al., 2024).

A complementary data-driven approach is the Loewner framework. It starts from sampled data

$x\mapsto \langle x,\theta\rangle$0

partitions them into left and right sets, constructs the Loewner and shifted Loewner matrices, and derives a reduced realization

$x\mapsto \langle x,\theta\rangle$1

Model order is selected by singular value decay, commonly using the threshold $x\mapsto \langle x,\theta\rangle$2, and in one spiral test $x\mapsto \langle x,\theta\rangle$3 (Antoulas et al., 6 Nov 2025).

The reported comparison is not merely about supremum error. The Loewner framework is described as fast and reliable, with results consistent across machines, and as often preserving real coefficients, symmetry, correct parity or alternation structure, cleaner pole-zero geometry, and lower effective complexity. AAA and AAA-Lawson frequently achieve strong Zolotarev numbers at low or moderate degree, but can introduce spurious poles, zeros, and complex artifacts. In the symmetric-circle benchmark with $x\mapsto \langle x,\theta\rangle$4, the exact optimal value is

$x\mapsto \langle x,\theta\rangle$5

while LF gives $x\mapsto \langle x,\theta\rangle$6, AAA-L200 gives $x\mapsto \langle x,\theta\rangle$7, and AAA-L1000 gives $x\mapsto \langle x,\theta\rangle$8. At much higher degree, for case $x\mapsto \langle x,\theta\rangle$9 with $E,F\subseteq \mathbb C\cup\{\infty\}$0, the optimal value is approximately $E,F\subseteq \mathbb C\cup\{\infty\}$1, while LF achieves approximately $E,F\subseteq \mathbb C\cup\{\infty\}$2, which the paper describes as very close to optimal (Antoulas et al., 6 Nov 2025).

5. Geometric and conformal foundations

The quantitative backbone of Zolotarev projection is the Zolotarev number

$E,F\subseteq \mathbb C\cup\{\infty\}$3

for disjoint sets $E,F\subseteq \mathbb C\cup\{\infty\}$4. It measures how sharply a type-$E,F\subseteq \mathbb C\cup\{\infty\}$5 rational function can separate $E,F\subseteq \mathbb C\cup\{\infty\}$6 from $E,F\subseteq \mathbb C\cup\{\infty\}$7. Because $E,F\subseteq \mathbb C\cup\{\infty\}$8 iff $E,F\subseteq \mathbb C\cup\{\infty\}$9, the number is symmetric: $\mathrm{Z3}$0 When the doubly connected complement is

$\mathrm{Z3}$1

its condenser capacity determines the annulus modulus

$\mathrm{Z3}$2

and the asymptotic rate satisfies

$\mathrm{Z3}$3

(Rubin et al., 2019).

A constructive realization of this asymptotic geometry uses Faber rational functions. For disjoint, simply connected, compact sets with rectifiable Jordan boundaries, the cited paper derives the explicit bound

$\mathrm{Z3}$4

where $\mathrm{Z3}$5 and $\mathrm{Z3}$6 are the total rotations of the boundaries. If $\mathrm{Z3}$7 and $\mathrm{Z3}$8 are convex, then $\mathrm{Z3}$9, so

$F$00

When the conformal map to the annulus is a Möbius transformation, the result is exact: $F$01 This is the ideal regime in which the rational filter is not merely near-optimal in rate but exactly optimal (Rubin et al., 2019).

These geometric bounds feed directly into algorithmic separation results. For the Sylvester equation

$F$02

ADI-type iterations satisfy

$F$03

Thus zeros and poles of near-optimal rational separators become nearly optimal ADI shifts (Rubin et al., 2019). This suggests that Zolotarev projection and optimal shift selection are two views of the same rational-separation problem.

The deeper function-theoretic background explains why elliptic parametrizations recur. Twisted Zolotarev fractions are rational functions on $F$04 whose critical points are all simple and whose critical values lie in a $F$05-element set. The paper classifies their Möbius equivalence classes by pairs of embedded rank-two lattices $F$06, modulo scaling, with two exceptions: $F$07 and $F$08. It also shows that classical Zolotarev fractions, elliptic rational functions, and Chebyshev–Blaschke products lie in the same Möbius-equivalence classes (Bogatyrev, 2015). For projection theory, this is structural rather than algorithmic, but it clarifies why the relevant rational filters form a rigid elliptic family.

6. Probability-theoretic projection and related non-operator uses

A distinct research line uses “projection” in the Cramér–Wold sense. For probability measures on $F$09, the projected law $F$10 is the pushforward under

$F$11

The quantitative Cramér–Wold theorem for Zolotarev distances asks whether closeness of all one-dimensional projections implies closeness of the full $F$12-dimensional laws. Under suitable moment assumptions, the answer is affirmative: $F$13 for $F$14 with $F$15, provided the moment-matching conditions needed for finiteness of $F$16 hold (Bobkov et al., 21 Jun 2025).

This is a genuine projection theorem, but not a spectral projector theorem. Its “projection” is the linear functional $F$17, and its core mechanism is quantitative recovery of a multivariate Zolotarev metric from one-dimensional projected metrics. The same paper proves the characteristic-function estimate

$F$18

which is the precise quantitative Cramér–Wold step (Bobkov et al., 21 Jun 2025).

Related probability-metric literature confirms the terminological split. Mattner’s Berry–Esseen refinement uses $F$19 and $F$20 to quantify distance to Gaussianity and derives

$F$21

with $F$22 admissible, but it explicitly does not define a projection onto the Gaussian family (Mattner, 2022). Likewise, sharp inequalities between the second-order Zolotarev distance and Wasserstein distance,

$F$23

provide dimension-free comparison tools for equal-barycenter measures, yet again do not introduce a projection operator (Bołbotowski et al., 31 Oct 2025).

The resulting picture is therefore layered rather than singular. In numerical linear algebra, Zolotarev projection denotes rational approximate spectral projectors and filters; in rational approximation, it is the extremal separation of two sets by solutions of the third and fourth Zolotarev problems; in probability, it refers to one-dimensional projections used to control multivariate Zolotarev metrics. The common denominator is not an abstract projection formalism, but the use of Zolotarev extremality to turn a difficult multivariate or operator-level separation problem into a tractable rational or one-dimensional surrogate.