5èmes Journées Approximation International conference on constructive complex approximation Friday May 20, 2016 Salle de Réunion, Bâtiment M2 Laboratoire Paul Painlevé UMR 8524 Université de Lille 1, FRANCE

SCOPE

The aim of this meeting, the fifth one after similar meetings in 2000, 2004, 2008, and 2012, is to bring together people interested in complex approximation theory mainly from France and Belgium, but also from other european countries.

Subjects covered by the 2016 meeting include : orthogonal polynomials, rational approximation, numerical aspects of approximation, approximation by radial basis functions, low rank Tensor approximation in high dimensions, asymptotic analysis and Riemann-Hilbert problems, random matrices, quadrature formulas,....

PROGRAM

(*) Posters presented by: Paraskevi Roupa (U. Athens), Matteo Briani (U. Anvers), Olivier Sète (U. Oxford), Konstantin Usevich (U. Grenoble), Benjamin Fahs (UC Louvain), Thomas Hélart (U. Lille).

TITLES AND ABSTRACTS

• Annie Cuyt (U. Antwerp, Belgium): Sparse Interpolation
  Abstract.
• Sylvain Chevillard (INRIA Sophia Antipolis): An inverse magnetization problem in geosciences
  Abstract. When rocks are heated (typically when they are formed, or after subsequent alteration), they can become magnetized by the ambient magnetic field. This remanent magnetization is used to study important processes in Earth sciences, since it provides records of past variations of the geodynamo. It has been used, e.g., to study motion of tectonic plates and geomagnetic reversals. This magnetization itself produces a weak magnetic field. SQUID microscopes are sensitive instruments, able to measure the field produced by thin slabs of magnetized rocks. More precisely, it can measure the vertical component of the magnetic field on a plane slightly above the sample, with a good spatial resolution. The problem of recovering the magnetization distribution from the measurements provided by a SQUID microscope is severely ill-posed. However, an interesting quantity, namely the total net moment of the magnetization can, in theory, be recovered. This is already an important information for geoscientists, and not easy to measure with usual magnetometers. We will present some properties of the magnetization-to-field map and of the corresponding adjoint operator. We formulate a bounded extremal problem whose solution is a function phi such that the integral of phi against the measured field should provide one with a good approximation of the desired net moment. If time permits, we will also present an alternate method based on asymptotic formulas, which is accurate when the measurements are made on a fairly large square.
• Tom Claeys (UC Louvain la Neuve, Belgium): Thinned random matrix eigenvalues and conditional probabilities
  Abstract. Randomly incomplete, or thinned, spectra of random matrices were introduced by Bohigas and Pato. On one hand, one can study statistical properties of the incomplete spectrum. On the other hand, it is natural to investigate what the incomplete spectrum tells us about the complete spectrum. For instance, conditioning on a gap in the incomplete spectrum, what can we say about the complete spectrum? I will present identities in terms of orthogonal polynomials on the unit circle and Toeplitz determinants, and I will discuss asymptotic results as the size gets large. The talk will be based on joint work with Christophe Charlier.
• Albert Cohen (U. Paris VI): Algorithms for high dimensional interpolation
  Abstract. There exists many classical methods for interpolating a function of one or several variables Practically all of these methods however face difficulties when considering functions of a large number of variable. We shall discuss and compare two approaches that can handle high dimensional data, the first based on Gaussian processes and the second based on sparse polynomial expansions. Both approach give rise to adaptive greedy algorithms in which the interpolation points are chosen in a sequencial manner, and which are still not well understood from a theoretical point of view.
• Stefano de Marchi (U. Padova, Italy): Polynomial approximation on Lissajous curves on the d-cube
  Abstract. We present our recent studies on Lissajous curves in the d-cube that generate algebraic cubature formulas on a special family of rank-1 Chebyshev lattices. These formulas are used to construct hyper-interpolation polynomials via a single 1-dimensional Fast Chebyshev Transform (computed by the Chebfun package). In the case d = 2, 3 we are able to compute discrete extremal sets of Fekete and Leja type for polynomial interpolation as well. Applications arise in the frame-work of Lissajous sampling for MPI (Magnetic Particle Imaging). Joint work with L. Bos and M. Vianello.
• Alfredo Deano (U. Kent, England) Semiclassical Laguerre polynomials: asymptotics and applications
  Abstract. Semiclassical Laguerre polynomials, orthogonal with respect to the weight function $x^{\lambda} \exp(-x^2+tx)$ on $[0,\infty)$, appear in connection with solutions of the Painlevé IV differential equation, and also in random matrix theory, as a deformation of the LUE ensemble. In this talk we will consider their asymptotic behaviour as the degree tends to infinity, uniformly with respect to the other parameters, $t$ and $\lambda$. This analysis complements previous results on the structure of the recurrence coefficients and on their large $t$ asymptotics given by Clarkson and Jordaan (2013) and Clarkson, Jordaan and Kelil (2015). This is joint work with Nicholas Simm (Mathematics Institute, University of Warwick, UK).
• Antonio Duran (U. Sevilla, Spain): Beyond the classics: Krall and exceptional orthogonal polynomials
  Abstract. Krall and exceptional polynomials are two of the more important extensions of the classical families of Hermite, Laguerre and Jacobi. On the one hand, Krall or bispectral polynomials are orthogonal polynomials which are also eigenfunctions of a differential operator of order bigger than two. The first examples were introduced by H. Krall in 1940, and since the eighties a lot of effort has been devoted to this issue. If we consider difference operators, the first examples of Krall discrete polynomials were introduced by the author in 2012 using certain Christoffel transforms of the classical discrete weights. The first part of this talk will be devoted to show how these Krall discrete polynomials can be constructed. On the other hand, exceptional polynomials are orthogonal polynomials which are also eigenfunctions of a second order differential operator, but they differ from the classical polynomials in that their degree sequence contains a finite number of gaps. In mathematical physics, these functions allow to write exact solutions to rational extensions of classical quantum potentials. Taking into account the above definitions, it is scarcely surprising that no connection has been yet found between bispectral and exceptional polynomials. However, at the discrete level (difference operators), something very exciting happens: Duality interchanges Krall discrete and exceptional discrete polynomials. The second part of this talk will be devoted to explore some of the consequences of this unexpected connection.
• Dries Stivigny (KU Leuven, Belgium): Smallest eigenvalue distribution for products of random matrices.
  Abstract. We study the distribution of the smallest eigenvalues for certain classes of positive-definite Hermitian random matrices, in the limit where the size of the matrices becomes large. The limiting distributions that we will study can be expressed as Fredholm determinants of certain integral operators, and generalize in a natural way the extensively studied hard edge Bessel kernel determinant. We will express the logarithmic derivative of those Fredholm determinants identically in terms of a 2x2 Riemann-Hilbert problem, and use this representation to obtain the so-called large gap asymptotics. This is joint work with Tom Claeys and Manuela Girotti.

LOCATION OF THE WORKSHOP

• Maps of the campus, the Lille metro system, the highways around the campus. See it on Google Maps.
  

REGISTRATION AND ACCOMODATION

There is no inscription fee, but registration before April 30 is mandatory, please use the following form. The update of the list of participants will be done manually on a regular basis. There will be a poster session. Please let us know as soon as possible and before April 30 if you want to present a poster.

Most of the speakers will be at Hotel CALM in downtown Lille. There is also a hotel on the campus near the department of Mathematics called ASCOTEL. See this link for a complete list of hotels in Lille.

SOCIAL EVENTS

On Thursday May 19 in the evening we intend to go to some restaurant for those participants who arrive on time in Lille. Please tell us if you are interested to join us.

Some other events for those staying more time in Lille:

• Official website of the Lille tourism and convention bureau.
• Roubaix - La Piscine - Musée d'Art et d'Industrie de Roubaix André Diligent: an art-deco museum in a former swimming pool. Special exibition "Braïtou-Sala (1885-1972), l’élégance d’un monde en péril."
• Palais des Beaux Arts de Lille: biggest french general art museum in the north of Paris, situated downtown Lille at the place de la république (metro république). Special exposition ZEP (bande dessinée). Special exposition "Les Belles du Nord".
• Villeneuve d'Ascq - Modern Art Museum, situated in the north-west of Villeneuve d'Ascq. Special exposition Amedeo Modigliani.
• Opera de Lille: L'ORFEO.

SUPPORT FOR THE WORKSHOP

Laboratoire Paul Painlevé UMR 8524 (Université Lille 1 - CNRS)
Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville (Université Littoral - CNRS)
Fédération de Recherche Mathématique du Nord Pas de Calais
LABEX CEMPI
SMAI-SIGMA - interest group of the french mathematical society of applied mathematics

ORGANIZERS

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