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Let \( \omega \) be the arcsine distribution of a closed interval \( \Delta \) and \( \mu \) be any positive Borel measure there. Denote by \( P_n(z) \) the \( n \)-th monic orthogonal polynomials w.r.t. \( \mu \), i.e., \[ \int x^k P_n(x) d\mu(x) =0, \quad k\in\{0,1,\ldots,n-1\}, \] where \( P_n(x) = x^n + \cdots \). It is a classical result of Szegő that if \( v\in L^1(\omega) \), where \( d\mu = vd\omega + d\mu^s \) and \( \mu^s \) is singular to the Lebesgue measure, then \[ P_n(z) = \exp\left\{ n\int\log(z-x)d\omega(x)\right\} \frac{G_\mu(\infty)}{G_\mu(z)} \] locally uniformly in \( \overline{\mathbb C}\setminus\Delta \), where \( G_\mu(z) \) is the Szegő function of \( \mu \), i.e., it is an outer function in \( \overline{\mathbb C}\setminus\Delta \) such that \( G_\mu(\infty)>0 \) and \( |G_{\mu\pm}(x)|^2 = v(x) \) a.e. on \( \Delta \). In this talk an extension of this result to multiple orthogonal polynomials for Angelesco systems of measures will be discussed. More precisely, positive Borel measures \( \mu_1,\ldots,\mu_d \) form an Angelesco system if \( \Delta_1<\Delta_2<\cdots<\Delta_d \), where \( \Delta_i \) is a convex hull of \( \mathrm{supp}\,\mu_i \). For these systems, given a multi-index \( \vec n\in\mathbb N^d \) there exists a unique monic polynomial \( P_{\vec n}(z) \) of degree \( |\vec n| = n_1+\dots + n_d \) such that \[ \int x^k P_{\vec n}(x) d\mu_i(x) =0, \quad k\in\{0,1,\ldots,n_i-1\}, \quad i\in\{1,2,\ldots,d\}. \] Strong asymptotics of these polynomials is derived along ray sequences \( \mathcal N(\vec c) = \{\vec n: n_i/|\vec n|\to c_i\text{ for each }i\} \), where \( \vec c=(c_1,c_2,\ldots,c_d) \), \( c_i>0 \) and \( c_1+c_2 + \cdots + c_d=1 \). It turns out that in general Szegő condition \( v_i\in L^1(\omega_i) \), where \( d\mu_i = vd\omega_i + d\mu_i^s \) and \( \omega_i \) is the arcsine distribution of \( \Delta_i \), is not sufficient to get the desired asymptotics. The proof utilizes certain generalization of results of Totik on asymptotics of orthogonal polynomials with varying weighs and de la Calle Ysern and Lopez Lagomasino on asymptotics of orthogonal polynomials on the unit circle with reciprocal polynomial weights.

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The goal of this talk is to show how orthogonal polynomials are a common tool in studying such disparate questions as asymptotic behavior of special solutions of Painlevé equations, asymptotic behavior of partition functions from statistical mechanics, and map enumeration on compact Riemann surfaces.

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It is known that monic polynomials orthogonal with respect to a compactly supported non–trivial Borel measure on the real line satisfy three–term recurrence relations with coefficients that are uniformly bounded. The coefficients then can be used to define a bounded operator on the space of square–summable sequences. This operator can be symmetrized and the spectral measure of the symmetrized operator is in fact the measure of orthogonality of the polynomials themselves. One way of arriving at the subject of orthogonal polynomials is via Padé approximation (Padé approximants are rational interpolants of a given holomorphic function; when the function is a Cauchy transform of a Borel measure on the real line, the denominators of the approximants are the orthogonal polynomials). Padé approximants can be extended to the setting of a vector of holomorphic functions and a vector of rational interpolants (this construction was introduced by Hermite to prove transcendency of e). Vector rational interpolants naturally lead to multiple orthogonal polynomials. Spectral theory of multiple orthogonal polynomials is not yet fully developed. I shall describe some of the recent advancements in this area.

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The multipoint Padé approximants to Cauchy integrals of analytic non–vanishing densities will be discussed in the situation where the (symmetric) contour attracting the poles of the approximants does separate the plane.

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Ever since the work of Runge in the late 19th century, it is known that functions analytic in a neighborhood of a compact set can be approximated arbitrarily close by rational functions (later Vitushkin characterized the compacta on which such an approximation is possible). Early in the 20th century, Walsh has shown that \[ \limsup_{n\to\infty} \inf_{r\in\mathcal R_n} \|f-r\|_A \leq \inf_B \exp\left\{ - 1/\mathrm{cap}(A,B) \right\}, \] where \( f \) is holomorphic in a neighborhood of a continuum \( A \), \( \mathcal R_n \) is the set of rational functions of type \( (n,n) \), \( \mathrm{cap}(A,B) \) is the condenser capacity, and the infimum on the right is taken over all compact sets \( B \) such that \( f \) is holomorphic in the complement of \( B \) (the complement must be connected and necessarily contain \( A \)). In general this bound is sharp. Driven by evidence from certain classes of functions, Gonchar has conjectured that \[ \liminf_{n\to\infty} \inf_{r\in\mathcal R_n} \|f-r\|_A \leq \inf_B \exp\left\{ - 2/\mathrm{cap}(A,B) \right\}. \] This conjecture was shown to be true by Parfenov with the help of Adamyan–Arov–Krein approximants. Elaborating on the work of Stahl, Gonchar and Rakhmanov have shown that \[ \lim_{n\to\infty} \inf_{r\in\mathcal R_n} \|f-r\|_A = \inf_B \exp\left\{ - 2/\mathrm{cap}(A,B) \right\} \] if \( f \) is a multi–valued function meromorphic outside of a compact polar set. For a subclass of such functions, asymptotic distribution of poles of sequences of rational approximants \( \{ r_n \} \) such that \[ \lim_{n\to\infty} \|f-r_n\|_A = \inf_B \exp\left\{ - 2/\mathrm{cap}(A,B) \right\}, \] where \( A \) is a continuum, will be discussed.

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Jacobi matrices on trees whose coefficients are generated by multiple orthogonal polynomials will be introduced. The definition is such that the matrices are self-adjoint when multiple orthogonal polynomials come from Angelesco systems of measures (but is not restricted to this case). Full description of eigenvalues and eigenvectors is obtained for finite Jacobi matrices under very mild assumption that «consecutive» multiple orthogonal polynomials do not have common zeroes. For infinite Jacobi matrices of Angelesco systems Hilbert space decomposition into an orthogonal sum of cyclic subspaces is obtained. For each of these subspaces, generators and the generalized eigenfunctions are found and written in terms of the orthogonal polynomials. This leads to the identification of the spectrum and the spectral type of such matrices. It also shown that Jacobi matrices of Nikishin systems are neither self–adjoint nor bounded.

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The essence of Stahl–Gonchar–Rakhmanov theory of symmetric contours as applied to the multipoint Padeé 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 non–identically zero 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. I will discuss a construction of a class of pairs interpolation scheme/symmetric contour with the help of hyperelliptic Riemann surfaces (following the ideas of Nuttall and Singh) and then convergence of rational interpolants with free poles of Cauchy transforms of non–vanishing complex densities on such contours under mild smoothness assumptions on the density.

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In this talk certain algebraic and analytic properties of polynomials (multiple) orthogonal with respect to a vector of measures on the real line are discussed.

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In 1873, Hermite proved that the number e is transcendental. To do so he cleverly used a connection between diophantine and rational approximation. At the heart of his construction lay the problem of approximation of the vector of exponentials ( e^z, e^2z, … , e^mz ) by certain vectors of rational functions ( P₁/Q, P₂/Q, … , P_m/Q ) where P_k/Q is chosen to interpolate e^kz at the origin with prescribed order. Such a vector of rational approximants with common denominator is now called an Hermite–Padé approximant. Even though the description of the behavior of the Hermite–Padé approximants for entire functions is not complete, it is well understood. However, for more complicated classes of functions even the conjectural understanding of the convergence is not fully developed. In this talk, I will try to describe some aspects of the theory when the approximated functions are Markov functions.

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Padé approximants (PAs) are rational functions that interpolate a given holomorphic functions at one point with maximal order. In other words, PAs are a rational function analog of Taylor polynomials. PAs have free poles; that is, one does not prescribe the location of the poles but rather obtains the coefficients of the polynomials forming a PA as a solution of a linear system involving Fourier coefficients of the approximated functions. Hence, PAs are easy to construct but their convergence properties are not as transparent. While most of the poles exhibit structured behavior (converge to the minimal capacity contour corresponding to the approximated function as shown by Stahl) some of them behave erratically wandering all over the complex plane. The latter poles received the name wandering/spurious. In my talk I will concentrate on the nearly uniform convergence of PAs to algebraic functions and Cauchy integrals of smooth densities (an extension of Nuttall's theorem), explain the nature of the wandering poles for this class of functions and their impact on the convergence.

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Padé approximants (PAs) are rational functions that interpolate a given holomorphic functions at one point with maximal order. They were introduced by Hermite in connection with transcendental number theory and later studied by his student Padé on their own right. PAs are rational approximants with free poles; that is, one does not prescribe the location of the poles but rather obtains the coefficients of the polynomials forming Padé approximant as a solution of a linear system involving Fourier coefficients of the approximated functions. Hence, PAs are easy to construct but their convergence properties are not as transparent. While most of the poles exhibit structured behavior (converge to the minimal capacity contour corresponding to the approximated function as shown by Stahl) some of them behave erratically wandering all over the complex plane. The latter poles received the name spurious. In my talk I will discuss convergence properties of PAs to different classes of holomorphic functions and «explain» the presence of the spurious poles in Padé approximation of algebraic functions with finitely many branch points.

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In this talk we discuss convergence of rational interpolants, also known as Padé approximants, to algebraic functions and Cauchy integrals of non–vanishing densities. Starting from the overview of the essential ideas developed through the second half of the previous century, we more to more recent work. Namely, multipoint Padé approximation of Cauchy transforms of complex measures. It is shown that if the support of a measure is an analytic Jordan arc and the density of this measure is sufficiently smooth, then the diagonal multipoint Padé approximants associated with «admissible» interpolation schemes converge locally uniformly to the approximated Cauchy transform. The existence of such interpolation schemes is shown.

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The first example of a transcendental number was given by Liouville in 1844. In 1873 Hermite proved that \(e\) is transcendental. In order to do that he utilized the analogy between simultaneous Diophantine approximation of real numbers and simultaneous rational approximation of analytic functions. In this talk I will explain Hermite's proof and its later modification by Lindemann who showed that π is transcendental as well.

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Recent results on strong asymptotics of multiple orthogonal polynomials for Angelesco systems are discussed.

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

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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).

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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. In this talk, the asymptotics of the scaled kernel in a neighborhood of a point on the unit circle will be described.

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Let \(\widehat\sigma\) 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 \(\widehat\sigma\) is a rational function \(P/Q\), \(\deg(P)\leq m\), \(\deg(Q)\leq n\), such that the first \(m+n+1\) Fourier coefficients of the linear form \(Q\widehat\sigma-P\) vanish when the form is developed into a series with respect to the polynomials \(p_n\). Asymptotics of the Frobenius–Padé approximants to \(\widehat\sigma\) along ray sequences \(\frac n{n+m+1}\to c>0\), \(n-1\leq m\), is presented 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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An Angelesco system with complex weights is a vector of Cauchy transforms of complex measures compactly and disjointly supported on the real line. An Hermite–Padé approximant to such a system is a vector of rational functions all having the same denominator where each component of the approximant interpolates the corresponding component of the vector function at infinity with prescribed order. The results describe asymptotic properties of such approximants when the measures have smooth derivatives with respect to the Lebesgue measure.

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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 N poles if and only if the winding number of f + h on T is bigger or equal to −N for any h A such that f + h ≠ 0 on T.

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The generalized Mahler measure of a polynomial is a measure of complexity formed by integrating the logarithm of the absolute value of the polynomial against ω_K, the logarithmic equilibrium distribution of a set K (in the case of the classical Mahler measure, K is the unit disk). The roots of a complex degree N polynomial chosen uniformly from the set of polynomials of Mahler measure at most 1 yields a determinantal point process on the complex plane. According to the large deviation principle, when N is large, it is exponentially likely to find the roots near the outer boundary of K distributed as ω_K. Determinantal nature of this point process means that the local statistics of the roots can be expressed via determinants of a single kernel function. We investigate the asymptotics of this kernel in different regions of the complex plane.

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Let f be an algebraic function holomorphic at infinity with all its singularities contained in the unit disk, D. Let further {r_n} be a sequence of H²–best rational approximants to f on the unit circle. We show that {r_n} converges in capacity to f in C\K, the unique domain characterized by the property of minimal condenser capacity of the compact K relative to D among all compacts that make f single-valued, and that the counting measures of the poles of r_n weakly converge to the Green equilibrium distribution on K relative to D. En route to this result we show that for any Borel probability measure ν, supp(ν)⊆D, there exists the unique weighted extremal domain C\Γ_ν such that rational interpolants to f whose interpolation points are distributed asymptotically as ν^* converge to f in capacity in C\(Γ_ν ∪ supp(ν^*)), where ν^* is the reciprocal measure of ν.

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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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We consider multipoint Padé approximation to Cauchy transforms of complex measures. It is known 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–continuous 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. In this talk we show that this convergence holds also for measures whose Radon–Nikodym derivative is a Jacobi weight modified by a Hölder continuous function. The asymptotics behavior of Padé approximants is deduced from the analysis of underlying non–Hermitian orthogonal polynomials, for which we use Riemann–Hilbert–∂ method.

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Let \(G\) be a simply connected domain and \(E \subset G\) a regular compact. In this talk we describe the asymptotic behavior of Kolmogorov's \(k\)–width, \(k = k_n\), of the unit ball of \( H^\infty \cap P_n \) restricted to \(E\) in \(C(E)\), where \(H^\infty\) 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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In the talk we consider best rational approximants to Cauchy transforms in the \(L^2\)–norm on the unit circle. If the approximated function \(F\) is the Cauchy transform of a sufficiently smooth complex–valued measure supported on a hyperbolic geodesic of the unit disk, it is shown that these approximants converge to \(F\) uniformly in the domain of analyticity of \(F\). Moreover, the so–called strong asymptotics for the error of approximation is provided. Using the Index theorem and formulae of the strong asymptotics for the error in multipoint Padé interpolation, the asymptotic uniqueness of rational approximants to \(F\) is derived. Behind all the convergence results lies the analysis of the asymptotic behavior of non–Hermitian orthogonal polynomials that is presented at the end of the talk.

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In this talk we consider AAK approximants for Cauchy transforms of complex measures perturbed by a rational function. Results are combined into two groups depending on the assumptions on the measure. In the first setting measures may vanish on a significant portion of the convex hull of its support but only convergence in capacity is obtained. In the second setting measures are much more smooth but strong asymptotics for the error of approximation is derived. Some numerical experiment are adduced at the end of the talk.