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Weekly Seminar
 WS Season 2009/2010

# Seminar on Orthogonality, Approximation Theory and Applications

Upcoming seminars

There are no upcoming seminars. We wish everyone a good summer and hope to see you here for the next season!

Past seminars

1. Friday. July 2, 2010. 16:00. Room: 2.2.D.08

Title: Strong contracted asymptotics for Hermite and Mexiner polynomials and their zeros from the point of view of their difference equations.

Speaker: Jeffrey Geronimo (Georgia Institute of Technology, USA)

Abstract: Using potential theory with an external field and the difference equation satisfied by Hermite and Meixner polynomials we will obtain strong contracted asymptotics for these orthogonal polynomial systems and their zeros.

2. Thursday. July 1, 2010. 16:00. Room: 2.2.D.08.

Title: Theory and applications of Kapteyn series

Speaker: Diego Dominici (State University of New York, New Paltz)

Abstract: Kapteyn series have a long history, with results going back to Joseph Louis de Lagrange's groundbreaking paper "Sur le Problème de Képler" (1770). They were rediscovered independently by Friedrich Wilhelm Bessel and appeared in his paper "Untersuchung des Theils der planetarischen Störungen" (1824), where he introduced for the first time the functions that now bear his name. Both Lagrange and Bessel considered Kapteyn series as a solution of Kepler's Equation, formulated by Johannes Kepler in his masterpiece "Astronomia Nova" (1609). Besides their application in Astronomy and Astrophysics, Kapteyn series have also been applied by George Adolphus Schott and others to problems in Electromagnetic Radiation.  In this talk we will review the history and theory of Kapteyn series, present some old and new results, discuss their applications and their connection with the theory of Hypergeometric Functions.

3. Thursday. June 17, 2010. 16:00. Room: 2.2.D.08.

Title: Quasidiagonality, Folner sequences and spectral approximation

Abstract: Consider a sequence of operators T_n in a Hilbert space H that approximates a given operator T in a suitable sense. A fundamental question is how the spectrum of T_n relates with the spectrum of T when n grows. There are two important notions in operator theory that can be applied to the mentioned spectral approximation problem: quasidiagonality and the existence of a Folner sequence. In both cases a sequence of finite rank projections converging strongly to the identity and adapted to the operator has to be constructed. In this talk I will introduce these notions and mention several examples. Time permitting I will present a new result on the existence of Folner sequences for some representations of C*-dynamical systems (also called crossed products).

4. Tuesday. June 15, 2010. 16:00. Room: 2.2.D.08.

Title: Multiple orthogonal polynomials of mixed type: Gauss-Borel factorization and the multi-component 2D Toda hierarchy

Speaker:
Manuel Mañas. Departamento de Física Teórica II, UCM, Madrid

Abstract: Multiple orthogonality is considered in the realm of a Gauss--Borel factorization problem for a semi-infinite moment matrix. Perfect combinations of weights and a finite Borel measure are constructed in terms of Nikishin systems. These perfect combinations ensure that the problem of mixed multiple orthogonality has a unique
solution, that can be obtained from the solution to a Gauss--Borel factorization problem for a semi-infinite matrix, which plays the role of a moment matrix. This leads to sequences of multiple orthogonal polynomials, their duals and second kind functions. It also gives the corresponding linear forms that are bi-orthogonal to the dual linear forms. Expressions for these objects in terms of determinants from the moment matrix are given, recursion relations are found, which imply a multi-diagonal Jacobi type matrix with double staircase shape, and results like the ABC theorem or the Christoffel--Darboux formula are re-derived in this context (using the factorization problem and the
generalized Hankel symmetry of the moment matrix). The connection between this description of multiple orthogonality and the multi-component 2D Toda hierarchy, which can be also understood and studied through a Gauss--Borel factorization problem, is discussed. Deformations of the weights, natural for certain type of Nikishin systems, are considered and the correspondence with solutions to the integrable hierarchy, represented as a collection of Lax equations, is explored. Corresponding Lax and Zakharov--Shabat matrices as well as wave functions and their adjoints are determined. The construction of discrete flows is discussed in terms of Miwa transformations which involve Darboux transformations for the multiple orthogonality conditions. The bilinear equations is
derived and the $\tau$-function representation of the multiple orthogonality is given.

5. Thursday. June 10, 2010. 16:00. Room: 2.2.D.08.

Title: Recent results on classical orthogonal polynomials

Speaker: Roberto S. Costas Santos (U. California Santa Barbara)

Abstract: We present some of the latest algebraic and structural results on classical orthogonal polynomials. Among others, the characterization theorem for q-polynomials on the quadratic lattice, a degenerate version of the Favard's theorem, some ways to generalize classical orthogonal polynomials, etc; among others. Also an elementary way to prove the interlacing property of zeros of classical orthogonal polynomials is presented. This is joint work with J. F. Sánchez-Lara.

6. Thursday. May 20, 2010. 16:00. Room: 2.2.D.08.

Title: Zeros of Orthogonal Polynomials Generated by Canonical Perturbations of the Standard Measure.

Speaker: Edmundo J. Huertas. UC3M, Madrid

Abstract:  In the last years some attention has been paid to the so called canonical spectral transformations of measure supported on the real line. Our contribution is focused on the behaviour of zeros of MOPS associated with the Christoffel and Uvarov transformations of such measures.

The outline of the talk is the following.  In the first part we introduce the representation of the perturbed MOPS in terms of the initial ones and we analyze the behaviour of the zeros of the MOPS when an Uvarov transform is introduced. In particular, we obtain such a behavior when the mass N tends to infinity as well as we characterize the values of the mass N such the smallest (respectively, the largest) zero of these MOPS is located outside the support of the measure. The second part of the talk is devoted to the electrostatic interpretation of the zero distribution as equilibrium points in a logarithmic potential interaction under the action of an external field. We analyze such an equilibrium problem when the mass point is located either in the boundary or in the exterior of the support of the measure, respectively.

7. Thursday. May 6, 2010. 16:00. Room: 2.2.D.08.

Title: Some Markov-Bernstein type inequalities and certain class of Sobolev polynomials.

Speaker: Yamilet Quintana (Department of Mathematics, Universidad Simón Bolívar, Venezuela)

Abstract: Let $(\mu_{0},\mu_{1})$  be a vector of non-negative measures on the real line, with $\mu_{0}$ not identically zero, finite moments of all orders, compact or non compact supports, and at least one of them having an infinite number of points in its support. We show that for any linear operator $T$ on the space of polynomials with complex coefficients and any integer $n\geq 0$, there is a constant $\gamma_{n}(T)\geq 0$,  such that     $$\|Tp\|_{S}\leq \gamma_{n}(T)\|p\|_{S},$$ for any polynomial $p$ of degree $\leq n$, where $\gamma_{n}(T)$ is independent of $p$, and
$$\|p\|_{S}= \left\{ \int|p(x)|^{2}d\mu_{0}(x)+ \int|p^{\prime}(x)|^{2}d\mu_{1}(x)\right\}^{\frac{1}{2}}.$$

We find a formula for the best possible value $_{n}(T)$ of $\gamma_{n}(T)$ and inequalities for $_{n}(T)$. Also, we give some examples when $T$ is a differentiation operator and $(\mu_{0},\mu_{1})$ is a vector of orthogonalizing  measures for classical orthogonal polynomials.
This is joint work with Dilcia Pérez.

8. Thursday. April 29, 2010. 16:00. Room: 2.2.D.08.

Title: codimension-1 exceptional orthogonal polynomials

Speaker: Robert Milson (Dalhousie University)

Abstract: The orthogonal polynomials of Hermite, Laguerre, and Jacobi can be realized as the eigenfunctions of 2nd order, self-adjoint spectral problems.  About 100 years ago, Bochner proved that this property characterizes the classical orthogonal polynomials modulo certain technical assumptions.  One of these assumptions is that the sequence of orthogonal polynomials contains a polynomial of every degree n=0,1,2,...  However, if we relax this assumption, and allow sequences of orthogonal polynomials that begin with a degree $m>0$ polynomial, then we escape the boundaries of Bochner's theorem and arrive at novel families of orthogonal polynomials defined by 2nd order eigenvalue equations.

If we limit ourselves to the case of $m=1$, then a full classification is possible. The key result is the following theorem.  Let Pn denote the n+1 dimensional vector space of univariate polynomials of degree n or less.  Let Un be a codimension 1 subspace of Pn.  Let us call Un exceptional or X1 if there exists a 2nd order differential operator

T(y) = p(x) y'' + q(x) y' + r(x) y

such that U is an invariant subspace but Pn isn't.  Then, up to a fractional linear transformation

Un = span { x-1, x2, ... , x^n }

Novel orthogonal polynomials arise when we consider 2nd order differential operators T(y) that leave invariant an infinite flag of exceptional codimension 1 subspaces.

9. Thursday. April 22, 2010. 16:00. Room: 2.2.D.08.

Title: A canonical family of multiple orthogonal polynomials for Nikishin systems

Speaker: Ignacio Álvarez Rocha, UPM, Madrid

Abstract:  For any pair of non-intersecting compact intervals of the real line $\Delta_1, \Delta_2,$ we obtain two absolutely continuous measures $\mu_1,\tau_1$ supported on $\Delta_1$ and $\Delta_2$, respectively, such that the Nikishin system ${\mathcal{N}}(\mu_1,\tau_1)$ has orthogonal polynomials which satisfy a four term recurrence relation with constant coefficients of period 2. The measures are obtained from the functions which give the ratio asymptotics of orthogonal polynomials with respect to arbitrary Nikishin systems ${\mathcal{N}}(\sigma_1,\sigma_2)$ such that $\supp(\sigma_1) = \Delta_1, \sigma_2' > 0$ a.e. on $\Delta_1$ and $\supp(\sigma_2) = \Delta_2, \sigma_2' > 0$ a.e. on $\Delta_1$ whose orthogonal polynomials.

10. Thursday. April 8, 2010. 16:00. Room: 2.2.D.08.

Title: A fast method for the computation of Legendre expansions

Speaker: Arieh Iserles, University of Cambridge, UK

Abstract: In this talk we introduce an approach for O(n log n) computation of the first coefficients of an expansion into Legendre polynomials. This approach is based on a sequence of perhaps counter-intuitive steps, manipulating hypergeometric expansions in the complex plane. We describe the underlying mathematics, prove the O(n log n) cost, present few numerical results and, time allowing, discuss generalisations.

11. Thursday. March 25, 2010. 16:00. Room: 2.2.D.08.

Title: Asymptotic solvers for oscillatory systems of ODEs

Speaker: Alfredo Deaño Cabrera,  UC3M, Madrid

Abstract: Systems of ODEs subject to oscillatory forcing terms are important in several applications, but they are usually difficult to solve numerically using standard methods such as Runge-Kutta. Following recent advances in the theoretical and numerical understanding of these problems, we present and alternative method that combines asymptotic expansions of the solution in inverse powers of the oscillatory parameter with modulated Fourier series. Numerical examples are included to show the efficiency of this approach. This is joint work with M. Condon (Dublin) and A. Iserles (Cambridge).

12. Thursday. March 11, 2010. 16:00. Room: 2.2.D.08.

Title: How much indeterminacy may ﬁt in a moment problem. An example

Speaker: Franciszek Hugon Szafraniec, Jagiellonski University, Krakow, Poland

Abstract: Employing the possibility of extending a symmetric operator to a selfadjoint, which goes beyond the Hilbert space, we construct examples of indeterminate Hamburger moment sequences admitting a family of representing measures such that: the support of each of them is in arithmetic progression, the supports of all the measures together partition R, none of them is N-extremal, and all of them are of inﬁnite order. In fact our goal is twofold: to present an example and to propose a method which may be useful for obtaining other results of this kind.

13. Thursday. March 4, 2010. 16:00. Room: 2.2.D.08.

Title: Leave it to Smith: Preserving structure in matrix polynomials

Speaker: Steven Mackey, Department of Mathematics, Western Michigan University

Abstract: Polynomial eigenvalue problems arise in many applications, such as the vibration analysis of mechanical systems, optical waveguide design, molecular dynamics, and optimal control. A much-used computational approach to such problems starts with a linearization of the underlying matrix polynomial P, such as the companion form, and then applies a general purpose algorithm to the linearization. Often, however, the polynomial P has some additional algebraic structure, leading to physically significant spectral symmetries which are important for computational methods to respect. In this situation it can be advantageous to use a linearization with the same structure as P, if one can be found. It turns out that there are structures polynomials for which a linearization with the same structure does not exist. Using the Smith form as the central tool, we describe which matrix polynomials from the classes of alternating, palindromic, and skew symmetric polynomials allow a linearization with the same structure.

14. Thursday. February 25, 2010. 16:00. Room: 2.2.D.08.

Title: Nikishin systems and simultaneous quadrature

Speaker: Guillermo López Lagomasino, UC3M, Spain

Abstract: This talk is a continuation of the one given on November 19, 2009. We will construct simultaneous quadrature rules for Nikishin systems of measures, prove their convergence, and show how this problem is connected with an extension of Markov's theorem on the convergence of diagonal Padé approximants.

15. Thursday. February 10, 2010. 16:00. Room: 2.2.D.08.

Title: Twisted Green's (CMV-like) matrices and their factorizations, Laurent polynomials and Digital Filter Structures

Speaker: Pavel Zhlobich (Department of Mathematics, University of Connecticut)

Abstract: Several new classes of structured matrices have appeared recently in the scientific literature. Among them there are so-called CMV and Fiedler matrices which are found to be related to polynomials orthogonal on the unit circle and Horner polynomials, respectively. Both matrices are five diagonal and have a similar structure, although they have appeared under completely different circumstances.

In a recent paper by Bella, Olshevsky and Zhlobich, it was proposed a unified approach to the above mentioned matrices. Namely, it was shown that all of them belong to a wider class of twisted Green's matrices.

We will use this idea to show that the factorizability of CMV and Fiedler matrices into a product of planar rotations in the n-dimensional space is also inherited by twisted Green's matrices. Shortly, for a given Hessenberg Green's matrix of size n, the intarchange of factors in the factorization leads to 2^n different twisted Green's matrices.

Polynomials orthogonal on the unit circle (those related to CMV matrices) and Horner polynomials (related to Fidler matrices) satisfy different types of recurrence relations. We will show that the characteristic polynomials of twisted Green's matrices satisfy certain recurrence relations which are, therefore valid for both Szegö's and Horner polynomials.

CMV matrix appeared in the scientific literature in connection with Laurent polynomials orthogonal on the unit circle. Fiedler matrix was developed purely from its factorization. We will show that an infinite-dimensional twisted Green's matrix serve as the operator of multiplication by "z" in the linear space of complex Laurent polynomials. Our development doesn't use orthogonallity in any sense and is based on the factorization and recurrence relations only. In the case of finite dimensional matrices we are able to give an explicit form of an eigenvector and all the generalized eigenvectors for a given eigenvalue.

The final part of our talk will be devoted to Kimura's approach to CMV matrices, i.e. Signal Flow Graphs (SFG) approach. We will exploit the tool of SFG to visualize all the theoretical results for twisted Green's matrices as well as to show how they can be used in construction of new types of Digital Filters.

This is joint work with Vadim Olshevsky and Gilbert Strang.

16. Thursday. January 28, 2010. 16:00. Room: 2.2.D.08.

Title: Gromov hyperbolic equivalence of the hyperbolic and quasihyperbolic metrics in Denjoy domains.

Speaker: Eva Touris Lojo, UC3M, Spain.

Abstract: In this article we investigate the Gromov hyperbolicity of Denjoy domains equipped with the hyperbolic or the quasihyperbolic metric. We first prove the existence of suitable families of quasigeodesics. The main result shows that a Denjoy domain is Gromov hyperbolic with respect to the hyperbolic metric if and only it is Gromov hyperbolic with respect to the quasihyperbolic metric. Using these tools we give a characterization in terms of Euclidean distances of when the domains are Gromov hyperbolic. We also give several concrete examples of families of domains satisfying the criteria of the theorems.

17. Thursday. January 21, 2010. 16:00. Room: 2.2.D.08.

Title: Linearizations of singular matrix polynomials.

Speaker: Fernando de Terán Vergara, UC3M, Spain.

Abstract: In this talk we give an overview to the systematic study of linearizations of singular matrix polynomials initiated by the authors in the past few years. This work was motivated by the introduction, in 2004 and 2006, of new families of linearizations (for regular matrix polynomials) extending the classical companion forms and enjoying some interesting features for applications. We will show that these families can be extended to the singular case and we will review some relevant features of these linearizations.

18. Thursday. January 14, 2010. 16:00. Room: 2.2.D.08.

Title: Memory on birth-and-death processes.

Speaker: Ulises Fidalgo Prieto, UC3M, Spain.

Abstract: Following Karlin's and Mc Gregor's point of view on one dimensional Brownian motion corresponding to a single particle we study a kind of generalized birth and death processes with memory. In such processes the transition probability functions satisfy differential equations with operators that have n-diagonal matrix representations (n > 3). In the research we needed to analyze strong asymptotic behaviours on mixed type of multi-orthogonality of polynomials. To this end we used Riemann-Hilbert problems techniques. As well, new results on momentum problems associated to multi-orthogonality were very important. The works on this last field (momentum problems) is still in progress.

19. Thursday. December 17, 2009. 16:00. Room: 2.2.D.08

Title: Fast transforms for classical orthogonal polynomials and their associated functions (Colloquium).

Speaker: Jens Keiner, Universitaet zu Luebeck. Institut fuer Mathematik, Germany.

Abstract: Classical orthogonal polynomials and functions have lots of applications. They are, for example, important for several generalizations of the discrete Fourier transform. These involve classical orthogonal polynomials or their associated functions rather than the usual complex exponentials. Often, these transforms are motivated by a known theoretical framework from mathematical physics. Nowadays, there is a demand to enable efficient and, at the same time, stable numerical computation of these transforms. While the plain discrete Fourier transform and its fast cousin, the fast Fourier transform (FFT), have excellent numerical properties, the calculation of generalized Fourier transforms involving classical orthogonal polynomials or their associated functions turns out to be much more challenging. In recent years, substantial progress has been made towards more efficient and more stable numerical algorithms which are required for any serious real world application. My talk will focus on two techniques that offer compelling advantages over other options. The ﬁrst approach is based on a connection to semiseparable matrices. The upshot is that the connection problem for classical orthogonal polynomials and their associated functions can be reduced to a structured eigenproblem that can be handled efficiently This allows to obtain a building block that, with a few things added, yields the desired transformation. While being useful for numerical applications, this new line of thought is also interesting from a purely theoretical point of view. The second technique is based on applying the well-known fast multipole method (FMM) to the connection problem. New results establish the conditions under which this can be done. As an application, we will show how the obtained methods can be used to compute fast Fourier transforms on the sphere S2 and the rotation group SO(3).

20. Thursday. December 10, 2009. 16:00. Room: 2.2.D.08.

Speaker: Pablo Linares Briones, U.C.M., Spain.

Abstract: En esta charla haremos un recorrido por la teoría de polinomios ortogonalmente aditivos en retículos de Banach haciendo especial hincapié en el Teorema de Representación que relaciona este tipo de polinomios con aplicaciones lineales. Este teorema ha probado ser muy versátil teniendo aplicaciones en diversas áreas de la matemática. Nos centraremos aquí en un nuevo concepto de ortogonalidad, basado en la ortogonalidad clásica de polinomios con respecto a una aplicación lineal, pero definida ahora respecto a un polinomio ortogonalmente aditivo.Este trabajo se realiza conjuntamente con Alberto Ibort y José Luis G. Llavona.

21. Thursday. December 3, 2009. 14:30. Room: 2.2.D.08.

Title: Asymptotics for a generalization of Laguerre polynomials (Colloquium).

Speaker: Ana Peña Arenas, U. Zaragoza, Spain.

Abstract: We consider a generalization of the classical Laguerre polynomials by the addition of r terms involving derivatives in the inner product. This type of generalization has been studied in the literature from the point of view of the algebraic properties. Thus, our aim is to study the asymptotics of this sequence of nonstandard orthogonal polynomials. In fact, we obtain the relative asymptotics and Mehler--Heine type formulas for these polynomials and, as a consequence, we prove that there exists an acceleration of the convergence of the smallest positive zeros of these generalized Laguerre polynomials towards the origin.

22. Thursday. November 19, 2009. 15:30. Room: 2.2.D.08.

Title: Sistemas de Nikishin y cuadraturas simultaneas.

Speaker: Guillermo López Lagomasino, U. Carlos III de Madrid, Spain.

Abstract: Recientemente, U. Fidalgo y yo hemos probado que los sistemas de Nikihin son perfectos. Veremos como ello permite construir cuadraturas simultaneas interpolatorias de tipo Gauss-Jacobi para aproximar con un mismo sistema de nodos las integrales de una cierta funcion respecto a varias medidas.

23. Thursday. November 12, 2009. 16:00. Room: 2.2.D.08.

Title: Nuevas técnicas de cálculo de variaciones en el estudio del operador de multiplicación.

Speaker: José Manuel Rodríguez García, U. Carlos III de Madrid, Spain.

Abstract: La acotación del operador de multiplicación por la variable independiente tiene importantes consecuencias en el estudio de los polinomios ortogonales de Sobolev, puesto que permite garantizar la acotación uniforme de los ceros de dichos polinomios, y esto a su vez permite encontrar el comportamiento asintótico de dichos polinomios ortogonales. En el estudio de la acotación de dicho operador se han venido utilizado diversos métodos, pero hasta ahora no se habían usado para ello técnicas de cálculo de variaciones. Explicaremos cómo con estas técnicas se puede garantizar la acotación para una amplia clase de productos de Sobolev no diagonales. Este es un trabajo conjunto con Ana Portilla, Yamilet Quintana y Eva Tourís.

24. Thursday. November 5, 2009. 16:00. Room: 2.2.D.08.

Title: Integrable systems, spectral transformations, and orthogonal polynomials on the unit circle.

Speaker: Francisco Marcellán Español, U. Carlos III de Madrid, Spain.

Abstract: In this talk we will present a survey about properties of orthogonal polynomials on the unit circle (OPUC) related to one-parameter deformations of the corresponding nontrivial probability measure of orthogonality μ. First, we analyze the connection between Toda lattices and orthogonal polynomials in the real line. Second, we study the time dynamics of the Verblunsky parameters, i.e. the evaluation at z=0 of such orthogonal polynomials, focussing our attention in the Schur flow, which is characterized by a complex semidiscrete modified KdV equation and where a discrete analogue of the Miura transformation appears. Third, the Lax pair for the CMV and GGT matrices associated with such deformations is discussed. Finally, some open problems in the framework of spectral transformations of probability measures supported on the unit circle will be analyzed. The study of such perturbations of measures from the point of view of the relation between the corresponding sequences of orthogonal polynomials and, as a consequence, between their GGT matrices.

25. Thursday. October 29, 2009. 16:00. Room: 2.2.D.08.

Title: Orthogonality with respect to a Jacobi differential operator and a fluid dynamics model.

Speaker: Héctor Pijeira Cabrera, U. Carlos III de Madrid, Spain.

Abstract: We study algebraic and analytic properties of the polynomial solutions of the differential equation A(x) y'' + B(x) y'= λ Ln(x), where A(x):= 1-x2, B(x):=β- α -(α + β + 2)x, λ:= n(1+n+α +β), α , β > -1 and Ln the n-th Monic Orthogonal Polynomial with respect to a no negative, finite Borel measure μ with support on [-1,1] (paper joint with Jorge Borrego).

26. Thursday. October 22, 2009. 16:00. Room: 2.2.D.08.

Title: A new algorithm for computing the Geronimus transformation with large shifts.

Speaker: Alfredo Deaño Cabrera, U. Carlos III de Madrid, Spain.

Abstract: A monic Jacobi matrix is a tridiagonal matrix which contains the parameters of the three-term recurrence relation satisfied by the sequence of monic polynomials orthogonal with respect to a measure μ. The basic Geronimus transformation with shift α transforms the monic Jacobi matrix associated with a measure μ into the monic Jacobi matrix associated with the new measure d μ/(x-α)+C δ(x-α), for some constant C. We will examine the algorithms available to compute this transformation and will propose a more accurate method, estimate its forward errors, and prove that it is forward stable. In particular, we show that for C=0 the problem is very ill-conditioned. (Joint work with M. I. Bueno (UC Santa Barbara) and E. Tavernetti (UC Davis)).

27. Thursday. October 15, 2009. 16:00. Room: 2.2.D.08.

Title: Asymptotics of orthogonal polynomials for a weight with a jump: local behavior.

Speaker: Andrei Martínez Finkelshtein, U. de Almería, Spain.

Abstract: For orthogonal polynomials on [−1, 1] with respect to a generalized Jacobi weight modified by a step-like function we obtain strong uniform asymptotics in the whole plane. In this talk the main focus is on the local behavior at the jump. We study the asymptotics of the Christoffel-Darboux kernel and show that the zeros of the orthogonal polynomials no longer exhibit the clock behavior. The main tool is the steepest descendent method of Deift and Zhou applied to the non-commutative Riemann-Hilbert problems characterizing the orthogonal polynomials. The local analysis at the jump is carried out in terms of the confluent hypergeometric functions. Incidentally, we establish some properties of these functions that may have an independent interest.

Last Updated on Monday, 12 July 2010 18:54

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