[x]
This manuscript reviews the study of the asymptotic behavior of meromorphic approximants to classes of functions holomorphic at infinity. The asymptotic theory of meromorphic approximation is primarily concerned with establishing the types of convergence, describing the domains where this convergence takes place, and identifying its exact rates. As the first question is classical, it is the latter two topics that this survey is mostly focused on with the greater emphasis on the exact rates. Three groups of approximants are introduced: meromorphic (AAK–type) approximants, \(L^2\)–best rational approximants, and rational interpolants with free poles. Despite the groups being distinctively different, they share one common feature: much of the information on their asymptotic behavior is encoded in the non–Hermitian orthogonality relations satisfied by the polynomials vanishing at the poles of the approximants with the weight of orthogonality coming from the approximated function. The main goal of the study is extracting the generic asymptotic behavior of the zeros of these polynomials from the orthogonality relations and tracking down those zeros that do not conform to the general pattern (wandering poles of the approximants).

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Given function \(f\) holomorphic at infinity, the \(n\)–th diagonal Padé approximant to \(f\), say \([n/n]_f\), is a rational function of type \((n,n)\) that has the highest order of contact with \(f\) at infinity. Equivalently, \([n/n]_f\) is the \(n\)–th convergent of the continued fraction representing \(f\) at infinity. Bernstein–Szegő theorem provides an explicit non–asymptotic formula for \([n/n]_f\) and all \(n\) large enough in the case where \(f\) is the Cauchy integral of the reciprocal of a polynomial with respect to the arcsine distribution on \([-1,1]\). In this note, Bernstein–Szegő theorem is extended to Cauchy integrals on the so–called algebraic S–contours.

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This article examines the Gonchar–Chudnovskies conjecture about the limited size of blocks of diagonal Padé approximants of algebraic functions. The statement of this conjecture is a functional analogue of the famous Thue–Siegel–Roth theorem. For algebraic functions with branch points in general position, we will show the validity of this conjecture as a consequence of recent results on the uniform convergence of the continued fraction for an analytic function with branch points. We will also discuss related problems on estimating the number of spurious (wandering) poles for rational approximations (Stahl's conjecture), and on the appearance and disappearance of defects (Froissart doublets).

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We consider a class of discrete Schröodinger operators on an infinite homogeneous rooted tree. Potentials for these operators are given by the coefficients of recurrence relations satisfied on a multidimensional lattice by multiple orthogonal polynomials. For operators on a binary tree with potentials generated by multiple orthogonal polynomials with respect to systems of measures supported on disjoint intervals (Angelesco systems) and for compact perturbations of such operators, we show that the essential spectrum is equal to the union of the intervals supporting the orthogonality measures.

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The notion of a symmetric contour introduced by Stahl and further generalized by Gonchar and Rakhmanov in connection with theory of rational interpolants with free poles is recalled. Refinement of this notion proposed by Baratchart and the author is discussed.

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Let \( \mu_1 \) and \( \mu_2 \) be two, in general complex–valued, Borel measures on the real line such that \( \mathrm{supp} \,\mu_1 =[\alpha_1,\beta_1] < \mathrm{supp}\,\mu_2 =[\alpha_2,\beta_2] \) and \( d\mu_i(x) = -\rho_i(x)dx/2\pi\mathrm{i} \), where \( \rho_i(x) \) is a restriction to \( [\alpha_i,\beta_i] \) of a function non–vanishing and holomorphic in some neighborhood of \( [\alpha_i,\beta_i] \). Strong asymptotics of multiple orthogonal polynomials is considered as their multi–indices \( (n_1,n_2) \) tend to infinity in both coordinates. The main goal of this work is to show that the error terms in the asymptotic formulae are uniform with respect to \( \min\{n_1,n_2\} \).

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Let \( D \) be a bounded Jordan domain and \( A \) be its complement on the Riemann sphere. We investigate the asymptotic behavior in \( D \) of the best rational approximants in the uniform norm on \( A \) of functions holomorphic on \( A \) that admit a multi-valued continuation to quasi every point of \( D \) with finitely many possible branches. More precisely, we study weak\(^*\) convergence of the normalized counting measures of the poles of such approximants as well as their convergence in capacity. We place best rational approximants into a larger class of \( n \)–th root optimal meromorphic approximants whose behavior we investigate using potential–theory on certain compact bordered Riemann surfaces.

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In this work, we establish strong asymptotics of multiple orthogonal polynomials of the second type for Angelesco systems with measures that satisfy Szegő conditions. We consider multi–indices that converge to infinity in the non–marginal directions.

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We describe the pole–free regions of the one–parameter family of special solutions of P\(_\mathrm{II}\), the second Painlevé equation, constructed from the Airy functions. This is achieved by exploiting the connection between these solutions and the recurrence coefficients of orthogonal polynomials that appear in the analysis of the ensemble of random matrices corresponding to the cubic potential.

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In this note we prove that \[ j!\,2^N \, \binom{N+j-1}{j} \, {}_2F_1\left(\begin{matrix}-j,-2j \\ -N-j+1 \end{matrix};-1\right) = \sum_{l=0}^N \binom{N}{l}\prod_{i=0}^{j-1}2(2i+1+l), \] where \( N \) and \( j \) are positive integers, which resolves a question posed by Ercolani, Lega, and Tippings.

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We show that the one–parameter family of special solutions of P\(_\mathrm{II}\), the second Painlevé equation, constructed from the Airy functions, as well as associated solutions of P\(_\mathrm{XXXIV}\) and S\(_\mathrm{II}\), can be expressed via the recurrence coefficients of orthogonal polynomials that appear in the analysis of the Hermitian random matrix ensemble with a cubic potential. Exploiting this connection we show that solutions of P\(_\mathrm{II}\) that depend only on the first Airy function \( \mathrm{Ai} \) (but not on \( \mathrm{Bi} \)) possess a scaling limit in the pole free region, which includes a disk around the origin whose radius grows with the parameter. We then use the scaling limit to show that these solutions are monotone in the parameter on the negative real axis.

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We investigate asymptotic behavior of polynomials \( Q_n(z) \) satisfying non–Hermitian orthogonality relations \[ \int_\Delta s^kQ_n(s)\rho(s)\mathrm ds =0, \quad k\in\{0,\ldots,n-1\}, \] where \( \Delta \) is a Chebotarëv (minimal capacity) contour connecting three non–collinear points and \( \rho(s) \) is a Jacobi–type weight including a possible power–type singularity at the Chebotarëv center of \( \Delta \).

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\( \bar\partial \)–extension of the matrix Riemann–Hilbert method is used to study asymptotics of the polynomials \( P_n(z) \) satisfying orthogonality relations \[ \int_{-1}^1 x^lP_n(x)\frac{\rho(x)dx}{\sqrt{1-x^2}}=0, \quad l\in\{0,\ldots,n-1\}, \] where \( \rho(x) \) is a positive \( m \) times continuously differentiable function on \( [-1,1] \), \( m\geq3 \).

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We investigate the phase diagram of the complex cubic unitary ensemble of random matrices with the potential \( V(M)=-\frac{1}{3}M^3+tM \) where \( t \) is a complex parameter. As proven in our previous paper, the whole phase space of the model, \( t\in\mathbb C \), is partitioned into two phase regions, \( O_{\mathsf{one-cut}} \) and \( O_{\mathsf{two-cut}} \), such that in \( O_{\mathsf{one-cut}} \) the equilibrium measure is supported by one Jordan arc (cut) and in \( O_{\mathsf{two-cut}} \) by two cuts. The regions \( O_{\mathsf{one-cut}} \) and \( O_{\mathsf{two-cut}} \) are separated by critical curves, which can be calculated in terms of critical trajectories of an auxiliary quadratic differential. In our previous work the one–cut phase region was investigated in detail. In the present paper we investigate the two–cut region. We prove that in the two–cut region the endpoints of the cuts are analytic functions of the real and imaginary parts of the parameter \( t \), but not of the parameter \( t \) itself. We also obtain the semiclassical asymptotics of the orthogonal polynomials associated with the ensemble of random matrices and their recurrence coefficients. The proofs are based on the Riemann–Hilbert approach to semiclassical asymptotics of the orthogonal polynomials and the theory of \( S \)–curves and quadratic differentials.

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Let \( f(z)=\int(z-x)^{-1}{\mathrm d}\mu(x) \), where \( \mu \) is a Borel measure supported on several subintervals of \( (-1,1) \) with smooth Radon–Nikodym derivative. We study strong asymptotic behavior of the error of approximation \( (f-r_n)(z) \), where \( r_n(z) \) is the \( L_{\mathbb R}^2 \)—best rational approximant to \( f(z) \) on the unit circle with \( n \) poles inside the unit disk.

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We study Jacobi matrices on trees whose coefficients are generated by multiple orthogonal polynomials. Hilbert space decomposition into an orthogonal sum of cyclic subspaces is obtained. For each subspace, we find generators and the generalized eigenfunctions written in terms of the orthogonal polynomials. The spectrum and its spectral type are studied for large classes of orthogonality measures.

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In this note we consider asymptotics of multipoint Padé approximants to Cauchy integrals of analytic non–vanishing densities defined on a Jordan arc connecting \( -1 \) and \( 1 \). We allow for the situation where the (symmetric) contour attracting the poles of the approximants does separate the plane.

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We continue studying the connection between Jacobi matrices defined on a tree and multiple orthogonal polynomials (MOPs) that was discovered previously by the authors. In this paper, we consider Angelesco systems formed by two analytic weights and obtain asymptotics of the recurrence coefficients and strong asymptotics of MOPs along all directions (including the marginal ones). These results are then applied to show that the essential spectrum of the related Jacobi matrix is the union of intervals of orthogonality.

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Let \( \{\varphi_i(z;\alpha)\}_{i=0}^\infty \), corresponding to \( \alpha\in(-1,1) \), be orthonormal Geronimus polynomials. We study asymptotic behavior of the expected number of real zeros, say \( \mathbb E_n(\alpha) \), of random polynomials \[ P_n(z) := \sum_{i=0}^n\eta_i\varphi_i(z;\alpha), \] where \( \eta_0,\dots,\eta_n \) are i.i.d. standard Gaussian random variables. When \( \alpha=0 \), \( \varphi_i(z;0)=z^i \) and \( P_n(z) \) are called Kac polynomials. In this case it was shown by Wilkins that \( \mathbb E_n(0) \) admits an asymptotic expansion of the form \[ \mathbb E_n(0) \sim \frac2\pi\log(n+1) + \sum_{p=0}^\infty A_p(n+1)^{-p} \] (Kac himself obtained the leading term of this expansion). In this work we obtain a similar expansion of \( \mathbb E(\alpha) \) for \( \alpha\neq 0 \). As it turns out, the leading term of the asymptotics in this case is \( (1/\pi)\log(n+1) \).

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Let \( f_0 \) and \( f_\infty \) be formal power series at the origin and infinity, and \( P_n/Q_n \) be a rational function such that \( \deg(P_n),\deg(Q_n)\leq n \) and \[ \left\{ \begin{array}{ll} (Q_nf_0-P_n)(z) = \mathcal{O}(z^n), & z\to 0,\\ (Q_nf_\infty-P_n)(z) = \mathcal{O}(z^{-1}), & z\to\infty. \end{array} \right. \] That is, \( P_n/Q_n \) simultaneously interpolates \( f_0 \) at the origin with order \( n \) and \( f_\infty \) at infinity with order \( n+1 \). When germs \( f_0,f_\infty \) represent multi–valued functions with finitely many branch points, it was shown by Buslaev that there exists a unique compact set \( F \) in the complement of which approximants converge in capacity to the approximated functions. The set \( F \) might or might not separate the plane. We study uniform convergence of the approximants for the geometrically simplest sets \( F \) that do separate the plane.

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We investigate asymptotic behavior of polynomials \( Q_n(z) \) satisfying non-Hermitian orthogonality relations \[ \int_\Delta s^kQ_n(s)\rho(s)\mathrm d s =0, \quad k\in\{0,\ldots,n-1\}, \] where \( \Delta := [-a,a]\cup [-\mathrm i b,\mathrm i b] \), \( a,b>0 \), and \( \rho(s) \) is a Jacobi–type weight. The primary motivation for this work is study of the convergence properties of the Padé approximants to functions of the form \[ f(z) = (z-a)^{\alpha_1}(z-\mathrm i b)^{\alpha_2}(z+a)^{\alpha_3}(z+\mathrm i b)^{\alpha_4}, \] where the exponents \( \alpha_i\not\in\mathbb Z \) add up to an integer.

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We consider a set of measures on the real line and the corresponding system of multiple orthogonal polynomials (MOPs) of the first and second type. Under some very mild assumptions, which are satisfied by Angelesco systems, we define self–adjoint Jacobi matrices on certain rooted trees. We express their Green's functions and the matrix elements in terms of MOPs. This provides a generalization of the well–known connection between the theory of polynomials orthogonal on the real line and Jacobi matrices on \(\mathbb{Z}_+\) to higher dimension. We illustrate importance of this connection by proving ratio asymptotics for MOPs using methods of operator theory.

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Let \( \{\varphi_i\}_{i=0}^\infty \) be a sequence of orthonormal polynomials on the unit circle with respect to a positive Borel measure \( \mu \) that is symmetric with respect to conjugation. We study asymptotic behavior of the expected number of real zeros, say \( \mathbb E_n(\mu) \), of random polynomials \[ P_n(z) := \sum_{i=0}^n\eta_i\varphi_i(z), \] where \( \eta_0,\dots,\eta_n \) are i.i.d. standard Gaussian random variables. When \( \mu \) is the acrlength measure on the unit circle such polynomials are called Kac polynomials and it was shown by Wilkins that \( \mathbb E_n(|d\xi|) \) admits an asymptotic expansion of the form \[ \mathbb E_n(|d\xi|) \sim \frac2\pi\log(n+1) + \sum_{p=0}^\infty A_p(n+1)^{-p} \] (Kac himself obtained the leading term of this expansion). In this work we generalize the result of Wilkins to the case where \( \mu \) is absolutely continuous with respect to arclength measure and its Radon–Nikodym derivative extends to a holomorphic non–vanishing function in some neighborhood of the unit circle. In this case \( \mathbb E_n(\mu) \) admits an analogous expansion with the coefficients \( A_p \) depending on the measure \( \mu \) for \( p\geq 1 \) (the leading order term and \( A_0 \) remain the same).

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We consider the roots of uniformly chosen complex and real reciprocal polynomials of degree \( N \) whose Mahler measure is bounded by a constant. After a change of variables this reduces to a generalization of Ginibre's complex and real ensembles of random matrices where the weight functions (on the eigenvalues of the matrices) is replaced by the exponentiated equilibrium potential of the interval \( [-2,2] \) on the real axis in the complex plane. In the complex (real) case the random roots form a determinantal (Pfaffian) point process, and in both cases the empirical measure on roots converges weakly to the arcsine distribution supported on \( [-2,2] \). Outside this region the kernels converge without scaling, implying among other things that there is a positive expected number of outliers away from \( [-2,2] \). These kernels, as well as the scaling limits for the kernels in the bulk \( (-2,2) \) and at the endpoints \( \{-2,2\} \) are presented. These kernels appear to be new, and we compare their behavior with related kernels which arise from the (non–reciprocal) Mahler measure ensemble of random polynomials as well as the classical Sine and Bessel kernels.

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Let \( \{\varphi_i\}_{i=0}^\infty \) be a sequence of orthonormal polynomials on the unit circle with respect to a probability measure \( \mu \). We study zero distribution of random linear combinations of the form \[P_n(z)=\sum_{i=0}^{n-1}\eta_i\varphi_i(z),\] where \( \eta_0,\dots,\eta_{n-1} \) are i.i.d. standard Gaussian variables. We use the Christoffel–Darboux formula to simplify the density functions provided by Vanderbei for the expected number real and complex of zeros of \( P_n \). From these expressions, under the assumption that \( \mu \) is in the Nevai class, we deduce the limiting value of these density functions away from the unit circle. Under mere assumption that \( \mu \) is doubling on subarcs of \( \mathbb T \) centered at \( 1 \) and \( -1 \), we show that the expected number of real zeros of \( P_n \) is at most \[ (2/\pi) \log n +O(1),\] and that the asymptotic equality holds when the corresponding recurrence coefficients decay no slower than \( n^{-(3+\epsilon)/2} \), \( \epsilon>0 \). We conclude with providing results that estimate the expected number of complex zeros of \( P_n \) in shrinking neighborhoods of compact subsets of \( \mathbb T \).

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The essence of Stahl–Gonchar–Rakhmanov theory of symmetric contours as applied to the multipoint Padé approximants is the fact that given a germ of an algebraic function and a sequence of rational interpolants with free poles of the germ, if there exists a contour that is “symmetric” with respect to the interpolation scheme, does not separate the plane, and in the complement of which the germ has a single–valued continuation with almost everywhere non–vanishing jump across the contour, then the interpolants converge to that continuation in logarithmic capacity in the complement of the contour. The existence of such a contour is not guaranteed. In this work we do construct a class of pairs interpolation scheme/symmetric contour with the help of hyperelliptic Riemann surfaces (following the ideas of Nuttall & Singh and Baratchart & the author). We consider rational interpolants with free poles of Cauchy transforms of non–vanishing complex densities on such contours under mild smoothness assumptions on the density. We utilize \( \bar\partial \)–extension of the Riemann–Hilbert technique to obtain formulae of strong asymptotics for the error of interpolation.

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We prove the topological expansion for the cubic log–gas partition function \[ Z_N(t)= \int_\Gamma\cdots\int_\Gamma\prod_{1\leq j \lt k\leq N}(z_j-z_k)^2 \prod_{k=1}^Ne^{-N\left(-\frac{z^3}{3}+tz\right)}\mathrm dz_1\cdots \mathrm dz_N, \] where \(t\) is a complex parameter and \(\Gamma\) is an unbounded contour on the complex plane extending from \(e^{\pi \mathrm i}\infty\) to \(e^{\pi \mathrm i/3}\infty\). The complex cubic log–gas model exhibits two phase regions on the complex \(t\)–plane, with one cut and two cuts, separated by analytic critical arcs of the two types of phase transition: split of a cut and birth of a cut. The common point of the critical arcs is a tricritical point of the Painlevé I type. In the present paper we prove the topological expansion for \(\log Z_N(t)\) in the one–cut phase region. The proof is based on the Riemann–Hilbert approach to semiclassical asymptotic expansions for the associated orthogonal polynomials and the theory of symmetric curves and quadratic differentials.

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Hermite–Padé approximants of type II are vectors of rational functions with common denominator that interpolate a given vector of power series at infinity with maximal order. We are interested in the situation when the approximated vector is given by a pair of Cauchy transforms of smooth complex measures supported on the real line. The convergence properties of the approximants are rather well understood when the supports consist of two disjoint intervals (Angelesco systems) or two intervals that coincide under the condition that the ratio of the measures is a restriction of the Cauchy transform of a third measure (Nikishin systems). In this work we consider the case where the supports form two overlapping intervals (in a symmetric way) and the ratio of the measures extends to a holomorphic function in a region that depends on the size of the overlap. We derive Szegő–type formulae for the asymptotics of the approximants, identify the convergence and divergence domains (the divergence domains appear for Angelesco systems but are not present for Nikishin systems), and show the presence of overinterpolation (a feature peculiar for Nikishin systems but not for Angelesco systems). Our analysis is based on a Riemann–Hilbert problem for multiple orthogonal polynomials (the common denominator).

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Let \(f_\sigma\) be a Cauchy transform of a possibly complex–valued Borel measure \(\sigma\) and \(\{p_n\}\) be a system of orthonormal polynomials with respect to a measure \(\mu\), \(\mathrm{supp}(\mu)\cap\mathrm{supp}(\sigma)=\varnothing\). An \((m,n)\)–th Frobenius–Padé approximant to \(f_\sigma\) is a rational function \(P/Q\), \(\mathrm{deg}(P) \leq m\), \(\mathrm{deg}(Q) \leq n\), such that the first \(m+n+1\) Fourier coefficients of the linear form \(Qf_\sigma-P\) vanish when the form is developed into a series with respect to the polynomials \(p_n\). We investigate the convergence of the Frobenius–Padé approximants to \(f_\sigma\) along ray sequences \(n/(n+m+1) \to c>0\), \(n-1 \leq m\), when \(\mu\) and \(\sigma\) are supported on intervals on the real line and their Radon–Nikodym derivatives with respect to the arcsine distribution of the respective interval are holomorphic functions.

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In this work type II Hermite–Padé approximants for a vector of Cauchy transforms of smooth Jacobi–type densities are considered. It is assumed that densities are supported on mutually disjoint intervals (an Angelesco system with complex weights). The formulae of strong asymptotics are derived for any ray sequence of multi–indices.

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Let f be a germ of an analytic function at infinity that can be analytically continued along any path in the complex plane deprived of a finite set of points, f ∈A(C \ A), #A ‹ ∞. J. Nuttall has put forward the important relation between the maximal domain of f where the function has a single–valued branch and the domain of convergence of the diagonal Padé approximants for f. The Padé approximants, which are rational functions and thus single–valued, approximate a holomorphic branch of f in the domain of their convergence. At the same time most of their poles tend to the boundary of the domain of convergence and the support of their limiting distribution models the system of cuts that makes the function f single–valued. Nuttall has conjectured (and proved for many important special cases) that this system of cuts has minimal logarithmic capacity among all other systems converting the function f to a single–valued branch. Thus the domain of convergence corresponds to the maximal (in the sense of minimal boundary) domain of single–valued holomorphy for the analytic function f ∈A(C \ A). The complete proof of Nuttall's conjecture (even in a more general setting where the set A has logarithmic capacity 0) was obtained by H. Stahl. In this work, we derive strong asymptotics for the denominators of the diagonal Padé approximants for this problem in a rather general setting. We assume that A is a finite set of branch points of f which have the algebro logarithmic character and which are placed in a generic position. The last restriction means that we exclude from our consideration some degenerated “constellations” of the branch points.

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A parametrization of a certain algebraic curve of genus 2, given by a cubic equation, is obtained. This curve appears in the study of Hermite–Padé approximants for a pair of functions with overlapping branch points on the real line. The suggested method of parametrization can be applied to other cubic curves as well.

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Given a function f holomorphic at infinity, the n–th diagonal Padé approximant to f, denoted by [n/n]_f, is a rational function of type (n,n) that has the highest order of contact with f at infinity. Nuttall's theorem provides an asymptotic formula for the error of approximation f-[n/n]_f in the case where f is the Cauchy integral of a smooth density with respect to the arcsine distribution on [-1,1]. In this note, Nuttall's theorem is extended to Cauchy integrals of analytic densities on the so–called algebraic S–contours (in the sense of Nuttall and Stahl).

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The Mahler measure of a polynomial is a measure of complexity formed by taking the modulus of the leading coefficient times the modulus of the product of its roots outside the unit circle. The roots of a real degree N polynomial chosen uniformly from the set of polynomials of Mahler measure at most 1 yields a Pfaffian point process on the complex plane. When N is large, with probability tending to 1, the roots tend to the unit circle, and we investigate the asymptotics of the scaled kernel in a neighborhood of a point on the unit circle. When this point is away from the real axis (on which there is a positive probability of finding a root) the scaled process degenerates to a determinantal point process with the same local statistics (i.e. scalar kernel) as the limiting process formed from the roots of complex polynomials chosen uniformly from the set of polynomials of Mahler measure at most 1. Three new matrix kernels appear in a neighborhood of ±1 which encode information about the correlations between real roots, between complex roots and between real and complex roots. Away from the unit circle, the kernels converge to new limiting kernels, which imply among other things that the expected number of roots in any open subset of C disjoint from the unit circle converges to a positive number. We also give ensembles with identical statistics drawn from two–dimensional electrostatics with potential theoretic weights, and normal matrices chosen with regard to their topological entropy as actions on Euclidean space.

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We investigate the question of uniform convergence of Padé approximants to elliptic functions that can be represented as Cauchy integrals of Dini–continuous non–vanishing densities given on 3–point Chebotarëv continua.

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A two–dimensional statistical model of N charged particles interacting via logarithmic repulsion in the presence of an oppositely charged regular closed region K whose charge density is determined by its equilibrium potential at an inverse temperature β is investigated. When the charge on the region, s, is greater than N, the particles accumulate in a neighborhood of the boundary of K, and form a point process in the complex plane. We describe the weak^* limits of the joint intensities of this point process and show that it is exponentially likely to find the process in a neighborhood of the equilibrium measure for K.

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We investigate a two–dimensional statistical model of N charged particles interacting via logarithmic repulsion in the presence of an oppositely charged compact region K whose charge density is determined by its equilibrium potential at an inverse temperature corresponding to β = 2. When the charge on the region, s, is greater than N, the particles accumulate in a neighborhood of the boundary of K, and form a determinantal point process on the complex plane. We investigate the scaling limit, as N → ∞, of the associated kernel in the neighborhood of a point on the boundary under the assumption that the boundary is sufficiently smooth. We find that the limiting kernel depends on the limiting value of N/s, and prove universality for these kernels. That is, we show that, the scaled kernel in a neighborhood of a point ζ ∈ ∂K can be succinctly expressed in terms of the scaled kernel for the closed unit disk, and the exterior conformal map which carries the complement K to the complement of the closed unit disk. When N / s → 0 we recover the universal kernel discovered by Lubinsky.

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Let f be holomorphically continuable over the complex plane except for finitely many branch points contained in the unit disk. We prove that best rational approximants to f of degree n, in the L²–sense on the unit circle, have poles that asymptotically distribute according to the equilibrium measure on the compact set outside of which f is single–valued and which has minimal Green capacity in the disk among all such sets. This provides us with n–th root asymptotics of the approximation error. By conformal mapping, we deduce further estimates in approximation by rational or meromorphic functions to f in the L²–sense on more general Jordan curves encompassing the branch points. The key to these approximation–theoretic results is a characterization of extremal domains of holomorphy for f in the sense of a weighted logarithmic potential, which is the technical core of the paper.

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Let T be the unit circle, f be an α-Hölder continuous function on T, α>1/2, and A be the algebra of continuous function in the closed unit disk D that are holomorphic in D. Then f extends to a meromorphic function in D with at most m poles if and only if the winding number of f+h on T is bigger or equal to -m for any h∈A such that f+h≠0 on T.

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We design convergent multipoint Padé interpolation schemes to Cauchy transforms of non–vanishing complex densities with respect to Jacobi–type weights on analytic arcs, under mild smoothness assumptions on the density. We rely on our earlier work for the choice of the interpolation points, and dwell on the Riemann–Hilbert approach to asymptotics of orthogonal polynomials introduced by Kuijlaars, McLaughlin, Van Assche, and Vanlessen in the case of a segment. We also elaborate on the ∂–extension of the Riemann–Hilbert technique, initiated by McLaughlin and Miller on the line to relax analyticity assumptions. This yields strong asymptotics for the denominator polynomials of the multipoint Padé interpolants, from which convergence follows.

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For all n large enough, we show uniqueness of a critical point in best rational approximation of degree n, in the L²–sense on the unit circle, to functions of the form with r a rational function and ⋅μ a complex–valued Dini–continuous function on [a,b]⊂(-1,1) which does not vanish, and whose argument is of bounded variation. Here ω_[a,b] indicates the normalized arcsine distribution on [a,b].

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Let [c,d] be an interval on the real line and μ be a measure of the form dμ=hh_zdω where h_z(t) = (t-c)^α(d-t)^β, α,β∈[0,1/2), h is a Dini–continuous non–vanishing function on [c,d] with an argument of bounded variation, and ω is the normalized arcsine distribution on [c,d]. Further, let p and q be two polynomials such that deg(p) ≤ deg(q) and [c,d] ∩ z(q) = ∅, where z(q) is the set of the zeros of q. We show that AAK–type meromorphic as well as diagonal multipoint Padé approximants to f(z):=∫(z-t)^-1dμ(t)+(p/q)(z) converge locally uniformly to f in D_f ∩ D and D_f, respectively, where D_f is the domain of analyticity of f and D is the unit disk. In the case of Padé approximants we need to assume that the interpolation scheme is “nearly” conjugate–symmetric. A noteworthy feature of this case is that we also allow the density dμ/dω to vanish on (c,d), even though in a strictly controlled manner.

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We consider multipoint Padé approximation to Cauchy transforms of complex measures. We show that if the support of a measure is an analytic Jordan arc and if the measure itself is absolutely continuous with respect to the equilibrium distribution of that arc with Dini–smooth non–vanishing density, then the diagonal multipoint Padé approximants associated with appropriate interpolation schemes converge locally uniformly to the approximated Cauchy transform in the complement of the arc. This asymptotic behavior of Padé approximants is deduced from the analysis of underlying non–Hermitian orthogonal polynomials, for which we use classical properties of Hankel and Toeplitz operators on smooth curves. A construction of the appropriate interpolation schemes is explicit granted the parameterization of the arc.

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Let G be a simply connected domain and E⊂G a regular compact with connected complement. In this paper we describe the asymptotic behavior of Kolmogorov's k–width, k = k_n, of the unit ball of H^∞∩P_n restricted to E in C(E), where H^∞ is the Hardy space of bounded analytic functions on G, P_n is the space of algebraic polynomials of degree at most n, and C(E) is the space of continuous functions on E.

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We study AAK–type meromorphic approximants to functions of the form where R is a rational function and λ is a complex measure with compact regular support included in (-1,1), whose argument has bounded variation on the support. The approximation is understood in L^p–norm of the unit circle, p≥2. We dwell on the fact that the denominators of such approximants satisfy certain non–Hermitian orthogonal relations with varying weights. They resemble the orthogonality relations that arise in the study of multipoint Padé approximants. However, the varying part of the weight implicitly depends on the orthogonal polynomials themselves, which constitutes the main novelty and the main difficulty of the undertaken analysis. We obtain that the counting measures of poles of the approximants converge to the Green equilibrium distribution on the support of λ relative to the unit disk, that the approximants themselves converge in capacity to F, and that the poles of R attract at least as many poles of the approximants as their multiplicity and not much more.

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We study diagonal multipoint Padé approximants to functions of the form where R is a rational function and λ is a complex measure with compact regular support included in R, whose argument has bounded variation on the support. Assuming that interpolation sets are such that their normalized counting measures converge sufficiently fast in the weak^* sense to some conjugate–symmetric distribution σ, we show that the counting measures of poles of the approximants converge to the balayage of σ onto the support of λ, in the weak^* sense, that the approximants themselves converge in capacity to F outside the support of λ, and that the poles of R attract at least as many poles of the approximants as their multiplicity and not much more.

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We derive a result on the boundedness of the multiplicity of the singular values for Hankel operator, whose symbol is of the form where λ is a complex measure with infinitely many points in its support which is contained in the interval (-1,1), and whose argument has bounded variation there, while R is a rational function with all its poles inside of the unit disk.

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Remez–type inequalities provide upper bounds for the uniform norms of polynomials p on given compact sets K, provided that |p(x)|≤1 for every x∈K\E, where E is a subset of K of small measure. In this note we obtain an asymptotically sharp Remez–type inequality for homogeneous polynomials on the unit sphere in R^d.

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Remez–type inequalities provide upper bounds for the uniform norms of polynomials p on given compact sets K, provided that |p(x)|≤1 for every x∈K\E, where E is a subset of K of small measure. In this paper we prove sharp Remez–type inequalities for homogeneous polynomials on star–like surfaces in R^d. In particular, this covers the case of spherical polynomials (when d=2 we deduce a result of T. Erdélyi for univariate trigonometric polynomials).