Do orthogonal polynomials dream of symmetric curves?

Do orthogonal polynomials dream of symmetric curves?

A. Martínez-Finkelshtein Department of Mathematics, University of Almería, Almería, Spain andrei@ual.es  and  E. A. Rakhmanov Department of Mathematics, University of South Florida, Tampa (FL), USA rakhmano@mail.usf.edu
Abstract.

The complex or non-Hermitian orthogonal polynomials with analytic weights are ubiquitous in several areas such as approximation theory, random matrix models, theoretical physics and in numerical analysis, to mention a few. Due to the freedom in the choice of the integration contour for such polynomials, the location of their zeros is a priori not clear. Nevertheless, numerical experiments, such as those presented in this paper, show that the zeros not simply cluster somewhere on the plane, but persistently choose to align on certain curves, and in a very regular fashion.

The problem of the limit zero distribution for the non-Hermitian orthogonal polynomials is one of the central aspects of their theory. Several important results in this direction have been obtained, especially in the last 30 years, and describing them is one of the goals of the first parts of this paper. However, the general theory is far from being complete, and many natural questions remain unanswered or have only a partial explanation.

Thus, the second motivation of this paper is to discuss some “mysterious” configurations of zeros of polynomials, defined by an orthogonality condition with respect to a sum of exponential functions on the plane, that appeared as a results of our numerical experiments. In this apparently simple situation the zeros of these orthogonal polynomials may exhibit different behaviors: for some of them we state the rigorous results, while others are presented as conjectures (apparently, within a reach of modern techniques). Finally, there are cases for which it is not yet clear how to explain our numerical results, and where we cannot go beyond an empirical discussion.

Key words and phrases:
Non-hermitian orthogonality, zero asymptotics, logarithmic potential theory, extremal problems, equilibrium on the complex plane, critical measures, -property, quadratic differentials.
2010 Mathematics Subject Classification:
Primary: 42C05; Secondary: 26C10; 30C25; 31A15; 41A21

1. Introduction

One of the motivations of this paper is to discuss some “mysterious” configurations of zeros of polynomials, defined by an orthogonality condition with respect to a sum of exponential functions on the plane, that appeared as a results of our numerical experiments. It turned out that in this apparently simple situation the orthogonal polynomials may exhibit a behavior which existing theoretical models do not explain, or the explanation is not straightforward. In order to make our arguments self-contained, we present a brief outline of the fundamental concepts and known results and discuss their possible generalizations.

The so-called complex or non-Hermitian orthogonal polynomials with analytic weights appear in approximation theory as denominators of rational approximants to analytic functions [32, 50] and in the study of continued fractions. Recently, non-Hermitian orthogonality found applications in several new areas, for instance in the description of the rational solutions to Painlevé equations [18, 14], in theoretical physics [1, 2, 3, 19, 20] and in numerical analysis [22].

Observe that due to analyticity, there is a freedom in the choice of the integration contour for the non-Hermitian orthogonal polynomials, which means that the location of their zeros is a priori not clear. The problem of their limit zero distribution is one of the central aspects studied in the theory of orthogonal polynomials, especially in the last few decades. Several important general results in this direction have been obtained, and describing them is one of the goals of the first parts of this paper. However, the general theory is far from being complete, and many natural questions remain unanswered or have only a partial explanation, as some of the examples presented in the second part of this work will illustrate. We will deal with one of the simplest situation that is still posing many open questions.

Complex non-Hermitian orthogonal polynomials are denominators of the diagonal Padé approximants to functions with branch points and thus play a key role in the study of the asymptotic behavior of these approximants, in particular, in their convergence. Since the mid-twentieth century convergence problems for Padé approximants have been attracting wide interest, and consequently, complex orthogonal polynomials have become one of the central topics in analysis and approximation theory. There is a natural historical parallel of this situation with the one occurred in the middle of the ninetieth century, when Padé approximants (studied then as continued fractions) for Markov- and Stieltjes-type functions led to the introduction of general orthogonal polynomials on the real line. The original fundamental theorems by P. Chebyshev, A. Markov and T. Stieltjes on the subject gave birth to the theory of general orthogonal polynomials.

In 1986 Stahl [66, 68] proved a fundamental theorem explaining the geometry of configurations of zeros of non-Hermitian orthogonal polynomials and presented an analytic description of the curves “drawn” by the strings of zeros. Those curves are important particular cases of what we now call -curves. They may be defined by the symmetry property of their Green functions or as trajectories of some quadratic differential.

The fact that the denominators of the diagonal Padé approximants to an analytic function at infinity satisfy non-Hermitian orthogonality relations is straightforward and was definitely known in the nineteenth century. Just nobody believed that such an orthogonality could be used to study the properties Padé denominators. Stahl’s theorem showed that complex orthogonality relations may be effectively used for these purposes, at least for functions with a “small” set of singular points, some of them being branch points. This, without any doubt, was a beginning of a new theory of orthogonal polynomials.

Before the work of Stahl, asymptotics of these polynomials was studied for some subclasses of functions and by appealing to their additional properties. For instance, several important results were obtained by Gonchar [27, 28] and collaborators in 1970s and the beginning of the 1980s. The geometry of their zero distribution was conjectured (and in partially proved, e.g., for hyperelliptic functions) by J. Nuttall and collaborators [51, 54]. Later, in [53], the case when the logarithmic derivative of the approximated function is rational (the so-called semiclassical or Laguerre class) was analyzed. The associated orthogonal polynomials, known as the semiclassical or generalized Jacobi polynomials, satisfy a second-order differential equation, and the classical Liouville–Green (a.k.a WKB) method may be used to study their strong asymptotics, as it was done by Nuttall, see also a recent paper [46].

Stahl’s ideas were considerably extended by Gonchar and Rakhmanov [32] to cover the case when the orthogonality weight depends on the degree of the polynomial, which requires the inclusion of a non-trivial external field (or background potential) in the picture. The curves, describing the location of the strings of zeros of the orthogonal polynomials, feature a symmetry property (the -property, so we call them the -curves), and their geometry is much more involved. The resulting Gonchar–Rakhmanov–Stahl (or GRS) theory, founded by [32, 66, 68], allows to formulate statements about the asymptotics of the zeros of complex orthogonal polynomials conditional to the existence of the -curves, which is a non-trivial problem from the geometric function theory. Further contributions in this direction, worth mentioning here, are [17, 16, 37, 59].

The notion of the -property can be interpreted also in the light of the Deift-Zhou’s nonlinear steepest descent method for the Riemann–Hilbert problems [23]. One of the key steps in the asymptotic analysis is the deformation of the contours, the so-called lens opening, along the level sets of certain functions. It is precisely the -property of these sets which guarantees that the contribution on all non-relevant contours becomes asymptotically negligible.

An important further development was a systematic investigation of the critical measures, presented in [43]. Critical measures are a wider class that encompasses the equilibrium measures on -curves, see Sect. 4 for the precise definition and further details. One of the contributions of [43] was the description of their supports in terms of trajectories of certain quadratic differentials (this description for the equilibrium measures with the -property is originally due to Stahl [70]). In this way, the problem of existence of the appropriate -curves is reduced to the question about the global structure of such trajectories.

Let us finish by describing the content of this paper. Section 2 is a showcase of some zero configurations of polynomials of complex orthogonality, appearing in different settings. The presentation is mostly informal, it relies on some numerical experiments, and its goal is mainly to illustrate the situation and eventually to arouse the reader’s curiosity.

Sect. 3 contains a brief overview of some basic definitions from the logarithmic potential theory, necessary for the subsequent discussion, as well as some simple applications of these notions to polynomials. This section is essentially introductory, and a knowledgeable reader may skip it safely.

In Sect. 4 we present the known basic theorem on asymptotics of complex orthogonal polynomials. We simplify settings as much as possible without losing essential content. The definitions and results contained here constitute the core of what we call the GRS theory. Altogether, Sects. 24 are expository.

Finally, in Sections 5 and 6 we present some recent or totally new results. For instance, Section 5 is about the so-called vector critical measures, which find applications in the analysis of the Hermite–Padé approximants of the second kind for a couple of power series at infinity of a special form, as well as in the study of the problems tackled in Sect. 6. This last section of the paper deals with the orthogonality with respect to a sum of two (or more) analytic weights. In order to build some intuition, we present another set of curious numerical results in Sect. 6.2, and the title of this paper (partially borrowed from Philip K. Dick) is motivated by the amazing variety and beauty of possible configurations. As the analysis of these experiments shows even for the simplest model, corresponding to the sum of two exponential weights, in some domain in the parameter space of the problem the standard GRS theory still explains the observed behavior, while in other domains it needs to be modified or adapted, which leads to some new equilibrium problems. Finally, there are regions in the parameter space where it is not yet clear how to generalize the GRS theory to explain our numerical results, and we cannot go beyond an empirical discussion.

2. Zeros showcase

We start by presenting some motivating examples that should illustrate the choice of the title of this work.

2.1. Padé approximants

Padé approximants [6, 13] are the locally best rational approximants of a power series; in a broader sense, they are constructive rational approximants with free poles.

Let denote the set of algebraic polynomials with complex coefficients and degree , and let

(2.1)

be a (formal) power series. For any arbitrary non-negative integer there always exist polynomials  and , , satisfying the condition

(2.2)

This equation is again formal and means that (called the remainder) is a power series in descending powers of , starting at least at . In order to find polynomials  and we first use condition (2.2) to determine the coefficients of , after which is just the truncation of at the terms of nonnegative degree. It is easy to see that condition (2.2) does not determine the pair uniquely. Nevertheless, the corresponding rational function is unique, and it is known as the (diagonal) Padé approximant to at of degree .

Hence, the denominator is the central object in the construction of diagonal Padé approximants, and its poles constitute the main obstruction to convergence of in a domain of the complex plane .

With this definition we can associate a formal orthogonality verified by the denominators (see, e.g., a recent survey [75] and the references therein). However, the most interesting theory is developed when is an analytic germ at infinity. Indeed, if (2.1) converges for , then choosing a Jordan closed curve in and using the Cauchy theorem we conclude that

This condition is an example of a non-Hermitian orthogonality satisfied by the denominators .

In particular interesting is the case when corresponds to an algebraic (multivalued) function, being approximated by intrinsically single-valued rational functions . As an illustration we plot in Figure 1 the poles of for the analytic germs at infinity of two functions,

(2.3)

both normalized by . These functions belong to the so-called Laguerre class (or are also known as “semiclassical”): their logarithmic derivatives are rational functions.

\OVP@calc \OVP@calc
Figure 1. Poles of the Padé approximant for (left) and (right), see (2.3).

A quick examination of the pictures puts forward two phenomena:

  1. generally, poles of distribute on in a rather regular way. Our eye cannot avoid “drawing” curves along which the zeros align almost perfectly.

  2. there are some exceptions to this beautiful order: observe a clear outlier on Figure 1, right. These “outliers” are known as the spurious poles of the Padé approximants.

2.2. Jacobi polynomials

This is the “most classical” family of polynomials, which includes the Chebyshev polynomials as a particular case. They can be defined explicitly (see [55, 74]) ,

(2.4)

or, equivalently, by the well-known Rodrigues formula

(2.5)

Incidentally, Jacobi polynomials could have been considered in the previous section: for they are also denominators of the diagonal Padé approximants (at infinity) to the function

(2.6)

In fact, denominators of the diagonal Padé approximants to semiclassical functions as in (2.3) are known as generalized Jacobi polynomials, see [53, 46, 10].

Clearly, polynomials are entire functions of the complex parameters . When they are orthogonal on with respect to the positive weight ,

(2.7)

and in consequence their zeros are all real, simple, and belong to .

What happens if at least one of the parameters is “non-classical”? In Figure 2 we depicted the zeros of for , and . We can appreciate again the same feature: a very regular distribution of the zeros along certain imaginary lines on the plane.

\OVP@calc
Figure 2. Zeros of the polynomial for and .

2.3. Heine–Stieltjes polynomials

These are a natural generalization of the Jacobi polynomials. Given a set of pairwise distinct points fixed on the complex plane ,

(2.8)

(), and two polynomials,

(2.9)

we are interested in the polynomial solutions of the generalized Lamé differential equation (in algebraic form),

(2.10)

where ; if , then is monic. An alternative perspective on the same problem can be stated in terms of the second order differential operator

and the associated generalized spectral problem (or multiparameter eigenvalue problem, see [76]),

(2.11)

where is the “spectral polynomial”.

If we take in (2.10), we are back in the case of Jacobi polynomials (hypergeometric differential equation). For we get the Heun’s equation, which still attracts interest and poses open questions (see [61]). Moreover, denominators of Padé approximants for semiclassical functions are also Heine–Stieltjes polynomials, see e.g. [53, 46].

Heine [34] proved that for every there exist at most

(2.12)

different polynomials such that (2.10) (or (2.11)) admits a polynomial solution . These particular are called Van Vleck polynomials, and the corresponding polynomial solutions are known as Heine-Stieltjes (or simply Stieltjes) polynomials; see [65] for further details (Fig. 3).

\OVP@calc
Figure 3. Zeros of a Heine-Stieltjes polynomial of degree for . The fat dot is the zero of the corresponding Van Vleck polynomial (of degree ).

Stieltjes discovered an electrostatic interpretation of zeros of the polynomials discussed in [34], which attracted general attention to the problem. He studied the problem (2.10) in a particular setting, assuming that and that all residues in

(2.13)

are strictly positive (which is equivalent to the assumption that the zeros of alternate with those of and that the leading coefficient of is positive). He proved in [73] (see also [74, Theorem 6.8]) that in this case for each there are exactly different Van Vleck polynomials of degree and the same number of corresponding Heine-Stieltjes polynomials of degree , given by all possible ways how the zeros of can be distributed in the open intervals defined by . Obviously, this models applies also in the case , i.e. to the zeros of a Jacobi polynomial. For a more detailed discussion see [39], although we describe the electrostatic model of Stieltjes for in the next Section.

2.4. Hermite–Padé approximants

Let us return to the Padé approximants. The situation gets more involved if we consider now a pair of power series at infinity, say

(2.14)

fix a non-negative integer , and seek three polynomials, , and , with all and not all , such that

(2.15)

are the type I Hermite–Padé (HP) polynomials, corresponding to the pair (or more precisely, to the vector ), and function defined in (2.15) is again the remainder of the Hermite–Padé approximation to .

Hermite used in 1858 a construction slightly more general than (2.15), involving , in order to prove that the number is transcendental. HP polynomials play an important role in analysis and have significant applications in approximation theory, number theory, random matrices, mathematical physics, and other fields. For details and further references see [6, 7, 33, 52, 58, 69, 75].

\OVP@calc
Figure 4. Zeros of for , see (2.16) (picture courtesy of S. P. Suetin).

HP polynomials is another classical construction closely related to orthogonal polynomials, but essentially more complicated. An asymptotic theory for such polynomials is not available yet, even though there is a number of separate results.

As a single illustration of the sophisticated beauty and complexity of this situation, we plot in Figure 4 the zeros of for the analytic germs at infinity of two functions,

(2.16)

both normalized by .

As in (2.3), functions and are semiclassical, and it is known [40] that polynomials satisfy a differential equation. Observe the interesting and non-trivial configuration of the curves where their zeros lie.

At any rate, let us insist in a general conclusion we can extract from all previous examples: zeros of analyzed polynomials tend to distribute along certain curves on in a very regular way that clearly needs an explanation.

3. Cauchy transforms and the logarithmic potential theory

How can we count the zeros of a polynomial? A trivial observation is that if

is a monic polynomial of degree , then its logarithmic derivative can be written as

(3.1)

where

is the normalized zero-counting measure for . Observe that we count each zero in accordance with its multiplicity.

The integral in the right-hand side of (3.1) is related to the so-called Cauchy or Stieltjes transform of this measure. In general, given a finite Borel measure on , its Cauchy transform (in the sense of the principal value) is

(3.2)

Hence, our first identity is

(3.3)

which is valid as long as , or equivalently, for .

The second, apparently trivial observation is that

For a finite Borel measure on we define its logarithmic potential

Hence,

(3.4)

valid again for .

The apparently innocent identities (3.3)–(3.4) establish the connection between the analytic properties of polynomials and two very important areas of analysis: the harmonic analysis and singular integrals, and the logarithmic potential theory.

Let us start with a classical example: the zero distribution of the Jacobi polynomials, defined in (2.4). An already mentioned fact is that the Jacobi polynomial is solution of a second-order differential equation (hypergeometric equation)

(3.5)

where , see [74, Theorem 4.2.2]. For all zeros of are real, simple, and belong to . With and , consider the normalized zero-counting measures . Standard arguments using weak compactness of the sequence show that there exist and a unit measure on such that

(3.6)

Here we denote by the weak-* convergence of measures.

An expression for the Cauchy transform of can be obtained in an elementary way directly from (3.5). The derivation of the continued fraction for from the differential equation appears in the Perron’s monograph [56, §80], although the original ideas are contained already in the work of Euler. For more recent applications, see [63, 25, 41].

The differential equation (3.5) can be rewritten in terms of the function

well defined at least for . We get

(3.7)

Since zeros of and interlace on , functions are analytic and uniformly bounded in , and by our assumption,

uniformly on compact subsets of (a.k.a. locally uniformly in) . Thus, taking limits in (3.7), we obtain that is an algebraic function satisfying a very simple equation:

In other words,

(3.8)

where we take the branch of the square root satisfying for . In possession of the additional information that we can use the Stieltjes–Perron (or Sokhotsky–Plemelj) inversion formula to recover the measure : we conclude that is absolutely continuous on , , and

(3.9)

Due to the uniqueness of this expression, we conclude that the limit in (3.6) holds for the whole sequence .

Limit (3.6) with given by (3.9) holds actually for families of orthogonal polynomials with respect to a wide class of measures on . The explanation lies in the properties of the measure : it is the equilibrium measure of the interval . In order to define it properly, we need to introduce the concept of the logarithmic energy of a Borel measure :

(3.10)

Moreover, given a real-valued function (the external field), we consider also the weighted energy

(3.11)

The electrostatic model of Stieltjes for the zeros of the Jacobi polynomials says precisely that the normalized zero-counting measure minimizes , with

(3.12)

among all discrete measures of the form supported on . Notice that vanishes asymptotically as for , so that it is not surprising that the weak-* limit of , given by (3.9) on , minimizes the logarithmic energy among all probability Borel measures living on that interval:

In the terminology of potential theory, is the equilibrium measure of the interval , the value is its Robin constant, and

is its logarithmic capacity.

As it was mentioned, this asymptotic zero behavior is in a sense universal: it corresponds not only to , but to any sequence of orthogonal polynomials on with respect to a measure a.e. (see, e.g., [48, 49, 72]), or even with respect to complex measures with argument of bounded variation and polynomial decay at each point of its support [15]. We no longer have a differential equation, but we do have the minimal property of the orthogonal polynomial, which at the end suffices.

Notice that the arguments above extend easily to the case of parameters dependent on . For instance, when the limits

exist, the *-limit of the zero-counting measure of minimizes the weighted energy with a non-trivial external field

(see [36, 41, 42, 44] for a general treatment of this case). In this situation the sequence satisfies varying orthogonality conditions, when the weight in (2.7) depends on the degree of the polynomial. As it was shown by Gonchar and Rakhmanov [30], under mild assumptions same conclusions will be valid for general sequences of polynomials satisfying varying orthogonality conditions.

But let us go back to the zeros of an individual Jacobi polynomial . Under the assumption of , why do they belong to the real line? A standard explanation invokes orthogonality (2.7), but why are we integrating along the real line? The integrand in (2.7) is analytic, so any deformation of the integration path joining and leaves the integral unchanged. Why do the zeros still go to ?

We can modify Stieltjes’ electrostatic model to make it “-free” (see [39]): on a continuum joining and find a discrete measures of the form , supported on , minimizing , with as in (3.12); the minimizer is not necessarily unique. Denote this minimal value by . Now maximize among all possible continua joining and . The resulting value is , and the max–min configuration is on , given by the zeros of .

This max–min ansatz remains valid in the limit : among all measures supported on a continuum joining and , in (3.9) maximizes the minimal energy. Or equivalently, is the set of minimal logarithmic capacity among all continua joining and .

If we recall that the Jacobi polynomials are denominators of the diagonal Padé approximants (at infinity) to the function (2.6), using Markov’s theorem (see [50]) we can formulate our conclusion as follows: the diagonal Padé approximants to this converge (locally uniformly) in , where is the continuum of minimal capacity joining and . It is easy to see that is a multivalued analytic function with branch points at .

It turns out that this fact is much more general, and is one of the outcomes of the Gonchar–Rakhmanov–Stahl (or GRS) theory.

4. The GRS theory and critical measures

4.1. -curves

Let us denote by the class of functions holomorphic (i.e., analytic and single-valued) in a domain ,

(4.1)

and let be defined by the minimal capacity property

(4.2)

Stahl [66, 68] proved that the sequence converges to in capacity in the complement to ,

under the assumption that the set of singularities of has capacity . The convergence in capacity (instead of uniform convergence) is the strongest possible assertion for an arbitrary function due to the presence of the so-called spurious poles of the Padé approximants that can be everywhere dense, even for an entire , see [67, 70, 71], as well as Figure 1.

More precisely, Stahl established the existence of a unique of minimal capacity, comprised of a union of analytic arcs, such that the jump across each arc is , as well as the fact that for the denominator of Padé approximants we have , where is equilibrium measure for . Here and in what follows we denote by (resp., ) the left (resp., right) boundary values of a function on an oriented curve.

The original work of Stahl contained not only the proof of existence, but a very useful characterization of the extremal set : on each arc of this set

(4.3)

where are the normal vectors to pointing in the opposite directions. This relation is known as the -property of the compact .

Notice that Stahl’s assertion is not conditional, and the existence of such a compact set of minimal capacity is guaranteed. In the case of a finite set of singularities, the simplest instance of such a statement is the content of the so-called Chebotarev’s problem from the geometric function theory about existence and characterization of a continuum of minimal capacity containing a given finite set. It was solved independently by Grötzsch and Lavrentiev in the 1930s. A particular case of Stahl’s results, related to Chebotarev’s problem, states that given a finite set of distinct points in there exist a unique set

where is the class of continua with . The complex Green function for has the form

where and is a polynomial uniquely defined by . In particular, we have

and (4.3) is an immediate consequence of these expressions. Another consequence is that is a union of arcs of critical trajectories of the quadratic differential . This is also the zero level of the (real) Green function of the two-sheeted Riemann surface for .

In order to study the limit zero distribution of Padé denominators Stahl [68] created an original potential theoretic method based directly on the non-Hermitian orthogonality relations

satisfied by these polynomials; incidentally, he also showed for the first time how to deal with a non-Hermitian orthogonality. The method was further developed by Gonchar and Rakhmanov in [32] for the case of varying orthogonality (see also [16]). The underlying potential theoretic model in this case must be modified by including a non–trivial external field . If the set on the plane is comprised of a finite number of piece-wise analytic arcs, we say that it exhibits the -property in the external field if

(4.4)

where is now the minimizer of the weighted energy (3.11) among all probability measures supported on , and . In other words, is the equilibrium measure of in the external field , and can be characterized by the following variational (equilibrium) conditions: there exists a constant (the equilibrium constant) such that

(4.5)

Equation (4.5) uniquely defines both the probability measure on and the constant .

The pair of conditions (4.4)–(4.5) has a standard electrostatic interpretation, which turns useful for understanding the structure of the -configurations. Indeed, it follows from (4.5) that distribution of a positive charge presented by is in equilibrium on the fixed conductor S, while the -property of compact in (4.4) means that forces acting on the element of charge at from both sides of are equal. So, the distribution of an -curve will remain in equilibrium if we remove the condition (“scaffolding”) that the charge belongs to and make the whole plane a conductor (except for a few exceptional insulating points, such as the endpoints of some of arcs in the support of ). In other words, is a distribution of charges which is in an (unstable) equilibrium in a conducting domain.

Let be a domain in , a compact subset of of positive capacity, , and let the sequence . Assume that polynomials are defined by the non-Hermitian orthogonality relations

(4.6)

where

(4.7)

the integration goes along the boundary of (if such integral exists, otherwise integration goes over an equivalent cycle in ).

A slightly simplified version of one of the main results of Gonchar and Rakhmanov [32] is the following:

Theorem 4.1.

Assume that converge locally uniformly in (as ) to a function . If has the -property in and if the complement to the support of the equilibrium measure is connected, then .

Theorem 4.1 was proved in [32], where it was called “generalized Stahl’s theorem”. Observe that unlike the original Stahl’s theorem, its statement is conditional: if we are able to find a compact set with the -property (in a harmonic external field) and connected complement, then the weak-* convergence is assured. Under general assumptions on the class of integration paths and in the presence of a non–trivial external field, neither existence nor uniqueness of a set with the -property are guaranteed.

A general method of solving the existence problem is based on the maximization of the equilibrium energy, which is inspired by the minimum capacity characterization (in the absence of an external field), see (4.2). More exactly, consider the problem of finding with the property

(4.8)

where is the set of all probability Borel measures on . If a solution of this extremal problem exists then under “normal circumstances” is an -curve in the external field , see [59].

4.2. Critical measures

Critical measures are Borel measures on the plane with a vanishing local variation of their energy in the class of all smooth local variations with a prescribed set of fixed points. They are, therefore equilibrium distributions in a conducting plane with a number of insulating points of the external field.

The zeros of the Heine–Stieltjes polynomials (see Section 2.3) are (discrete) critical measures. This observation lead in [43] to a theorem on asymptotics of these polynomials in terms of continuous critical measures with a finite number of insulating points of the external field.

It turns out that the equilibrium measures of compact sets with the -property are critical; the reciprocal statement is that the critical measures may be interpreted as the equilibrium measures of -curves in piece-wise constant external fields. Both notions, however, are defined in a somewhat different geometric settings, and it is in many ways convenient to distinguish and use both concepts in the study of many particular situations.

The idea of studying critical (stationary) measures has its origins in [32]. Later it was used in [35, 60] in combination with the min–max ansatz, and systematically in [43], see also [45, 37]. The formal definition is as follows: let be a domain, be a finite set and a harmonic function in . Let and , satisfying . Then for any (signed) Borel measure the mapping (“-variation”) defines the pull-back measure , as well as the associated variation of the weighted energy,

(4.9)

where was introduced in (3.11). We say that is -critical if for any -variation, such that the limit above exists, we have

(4.10)

when , we write -critical instead of -critical measure.

The relation between the critical measures and the -property (4.4) is very tight: every equilibrium measures with an -property is critical, while the potential of any critical measure satisfies (4.4). However, in some occasion it is more convenient to analyze the larger set of critical measures.

As it was proved in [43], for any -critical measure we have

(4.11)

This formula implies the following description of : it consists of a finite number of critical or closed trajectories of the quadratic differential on (moreover, all trajectories of this quadratic differential are either closed or critical, see [43, Theorem 5.1] or [32, p. 333]). Together with (4.11) this yields the representation

(4.12)

Finally, the -property (4.4) on and the formula for the density on any open arc of follow directly from (4.12). In this way, function becomes the main parameter in the construction of a critical measure: if we know it (or can guess it correctly), then consider the problem solved.

Example 4.2.

Let us apply the GRS theory to the following simple example: we want to study zeros asymptotics of the polynomials defined by the orthogonality relation

(4.13)

where the varying (depending on ) weight function has the form

(4.14)

and the integration goes along a Jordan arc connecting and . Since

uniformly on every compact subset of , the application of the GRS theory is reduced to finding a Jordan arc , connecting and , such that the equilibrium measure has the -property in the external field .

For , such an is just the vertical straight segment connecting both endpoints. Using (4.11) and the results of [43] it is easy to show that for small values of (roughly speaking, for ),

valid a.e. (with respect to the plane Lebesgue measure) on . In this case, is connected, and is the critical trajectory of the quadratic differential

(4.15)

connecting . As an illustration of this case we depicted the zeros of for , see Figure 5, left111The procedure used to compute these zeros numerically is briefly explained in Section 6.2..

\OVP@calc \OVP@calc
Figure 5. Zeros of , defined by (4.13), for (left) and (right).

At a critical value of the double zero hits the critical trajectory and splits (for ) into two simple zeros, that can be computed, so that now

In this situation is made of two real-symmetric open arcs, each connecting with lying in the same half-plane, delimited by ; they are the critical trajectory of the quadratic differential

on . For illustration, see the zeros of for , see Figure 5, right.

Formula (4.12) and the fact that is comprised of arcs of trajectories of on show that is a constant on . However, this constant is not necessarily the same on each connected component of : this additional condition singles out the equilibrium measures with the -property within the class of critical measures. Equivalently, the equilibrium measure can be identified by the validity of the variational condition (4.5).

Recall that one of the motivations for the study of critical (instead of just equilibrium) measures is the characterization of the zero distributions of Heine–Stieltjes polynomials, see Section 2.3. One of the main results in [43] is the following. If we have a convergent sequence of Van Vleck polynomials , then for corresponding Heine–Stieltjes polynomials we have

where is an -critical measure. Moreover, any -critical measure may be obtained this way. In electrostatic terms, is a discrete critical measure and the result simply means that a weak limit of a sequence of discrete critical measures is a (continuous) critical measure. In the case of the Heine–Stieltjes polynomials, in (4.11) is a rational function of the form , with given by (2.9), and the polynomial determined by a system of nonlinear equations.

In the case of varying Jacobi polynomials, , with , satisfying

(4.16)

we can obtain the explicit expression for using the arguments of Section 3. By the GRS theory, the problem of the weak-* asymptotics of the zeros of such polynomials boils down to the proof of the existence of a critical trajectory of the corresponding quadratic differential, joining the two zeros of , and of the connectedness of its complement in , see [36, 41, 42, 44], as well as Figure 6.

\OVP@calc
Figure 6. Critical trajectories of , with and in (4.16), and the zeros of the polynomial from Figure 2 superimposed.

5. Vector critical measures

If we check the motivating examples from Section 2, we will realize that at this point we only lack tools to explain the asymptotics of the zeros of the Hermite–Padé polynomials (Section 2.4). For this, we need to extend the notion of critical measures (and in particular, of equilibrium measures on sets with an -property) to a vector case.

Assume we are given a vector of nonnegative measures , compactly supported on the plane, a symmetric and positive-semidefinite interaction matrix , and a vector of real-valued harmonic external fields , , . We consider the total (vector) energy functional of the form [31]

(5.1)

(compare with (3.11)), where

is the mutual logarithmic energy of two Borel measures and .

Typical cases of the matrix for are

corresponding to the so-called Angelesco and Nikishin systems, respectively, see [29, 9, 4].

As in the scalar situation, for and , denote by the push-forward measure of induced by the variation of the plane , . We say that is a critical vector measure (or a saddle point of the energy ) if

(5.2)

for every function .

Usually, critical vector measures a sought within a class specified by their possible support and by some constraints on the size of each component. The vector equilibrium problems deal with the minimizers of the energy functional (5.1) over such a family of measures .

For instance, we can be given families of analytic curves , so that , , and additionally some constraints on the size of each component of . In a classical setting this means that we fix the values , such that , and impose the condition

see e.g. [31, 29, 8, 7]. More recently new type of constraints have found applications in this type of problems, when the conditions are imposed on a linear combination of .

For instance, [47] considers the case of , with the interaction matrix

polynomial external fields, and conditions

(5.3)

where is a parameter; see also [5] and [58] for and its applications to Hermite-Padé approximants. From the electrostatic point of view it means that we no longer fix the total masses of each component , but charges (of either sign) can “flow” from one component to another.

There is a natural generalization of the scalar -property to the vector setting. Continuing with the results of the recent work [47], it was proved that under some additional natural assumptions on the critical vector measure , if is an open analytic arc in , for some , then there exists a constant for which both the variational equation (equilibrium condition)

(5.4)

and the -property

hold true a.e. on , where again are the unit normal vectors to , pointing in the opposite directions.

Clearly, the problem of existence of compact sets with the -property for the vector problem are even more difficult than in the scalar case. The possibility to reduce this question to the global structure of critical trajectories of some quadratic differentials is the most valuable tool we have nowadays. This is given by higher-dimension analogues of the equation (4.11) for critical vector measures. Again, in the situation of [47] it was proved that functions

(5.5)

are solutions of the cubic equation

(5.6)

where is a polynomial and a rational function with poles of order at most . This fact yields that measures are supported on a finite union of analytic arcs, that are trajectories of a quadratic differential living on the Riemann surface of (5.6), and which is explicitly given on each sheet of this Riemann surface as . The fact that the support of critical (or equilibrium) vector measures is described in terms of trajectories on a compact Riemann surface (and thus, what we actually see is their projection on the plane) explains the apparent geometric complexity of the limit set for the zeros of Hermite–Padé polynomials, see Figure 4.

6. Orthogonality with respect to a sum of weights

6.1. General considerations

Now we step into a new territory. It is well known that a multiplicative modification of an orthogonality measure is easier to handle (from the point of view of the asymptotic theory) than an additive one. However, this kind of problems appears very naturally. For instance, a large class of transfer functions corresponding to time-delay systems can be written as the ratio of two functions, each of them a polynomial combination of exponentials (see e.g. [38]). In the simplest case, let