## Invited Speakers and Plenary Conference Abstracts

## How big the orthonormal polynomial from the Steklov class can be?

**Alexander Aptekarev (Keldysh Institute of Applied Mathematics, Russia).**

The famous problem of Steklov is to find bounds for the polynomial sequences, which are orthonormal with respect to the strictly positive weight. In 1921 V. A. Steklov made a conjecture that such a sequence of polynomials is bounded on the support of the orthogonality measure. In 1979 E. A. Rakhmanov disproved this conjecture constructing a weight from the Steklov class, for which a subsequence of the polynomials demonstrates a logarithmic growth at one point of the support. Then a natural question have arisen: how fast this growth could be?

Let S_{δ} be the space of measures σ on the unit circle, such that
σ'(θ)>δ>0 at every Lebesgue point, and let φ_{n}(z) be the
orthonormal polynomials with respect to σ in S_{δ}. In our talk we
consider the following variational problem. Fix n in **N** and δ>0. Let

M_{n}=sup_{{σ in Sδ}} |φ_{n}|_{{L∞ (T)}}=sup_{{σ in Sδ}|φn(1)|.
}

Elementary considerations yield M_{n}~< n^{1/2}. Rakhmanov has proved in 1981, that

M_{n} >~n^{1/2}/ (ln(n))^{3/2}.

The main result of our joint work 1. with S.A. Denisov and D.N.
Tulyakov is M_{n} >~n^{1/2}. I.e. the elementary upper estimate is
sharp.

- A. Aptekarev, S. Denisov, D. Tulyakov, The sharp estimates on the orthogonal polynomials from the Steklov class, ArXiv: 1308.6614v1

## Algebraic properties of robust Padé approximants

**Bernhard Beckermann (Laboratoire Painlevé, UFR Mathématiques, Université Lille 1, France). Joint work with Ana C. Matos (Lille). **

It has been conjectured [3] that recently introduced so-called robust Padé approximants computed through SVD techniques do not have so-called spurious poles [3], that is, poles with a close-by zero or poles with small residuals. Such a result would have a major impact on the convergence theory of Padé approximants since it is known that convergence in capacity plus absence of poles in some domain D implies locally uniform convergence in D.

Following [2], we prove in the present talk the conjecture for the subclass of so-called well-conditioned Padé approximants, and discuss related questions. It turns out that it is not sufficient to discuss only linear algebra properties of the underlying rectangular Toeplitz matrix, since in our results other matrices like Sylvester matrices also occur. This type of matrices have been used before in numerical greatest common divisor computations.

- B. Beckermann and A.C. Matos, Algebraic properties of robust Padé approximants. Manuscript (2013).
- P. Gonnet, S. Güttel and L. N. Trefethen, Robust Padé approximation via SVD, SIAM Review, 55 (2013), pp. 101-117.
- H. Stahl, Spurious poles in Padé approximation, J. Comp. Appl. Math., 99 (1998), 511-527.

## Around Padé-type approximation and rational interpolation

**Claude Brezinski (University of Sciences and Technologies of Lille, France)**
**joint work with Michela Redivo-Zaglia (University of Padua, Italy)**

Three ideas will be presented in this talk

- In Padé-type approximants, the denominator can be arbitrarily chosen. We will show how to choose it so that, in addition, these approximants also interpolate the function to be approximated.
- In barycentric rational interpolation, the weights of the interpolants can be arbitrarily chosen. We will how how to choose them so that, in addition, they also satisfy a Padé-type approximation property.
- We will show how to write Padé approximants under a barycentric form.

## Pablo González-Vera, a quadrature of his work

**Adhemar Bultheel (Department of Computer Science, KU Leuven, Belgium.)**

In this talk I will try to estimate the breadth and width of Pablo's mathematical work. Measuring the influence he had, and still has, on the work of all the people who have known him professionally. That ranges from the two-point Padé approximation that he started with, but that quickly came to blossom in many papers on rational approximation with many more points of interpolation (countably many). His favored application of these was the design of numerical quadrature formulas.

This was mainly developed together with his seven PhD students in the group in La Laguna and colleagues from abroad. He was rarely the sole author of a paper showing his skill as a team player and an excellent team leader.

It is a difficult task to do this in just one lecture. So like quadrature is finding a square with the same area as a more amorphous region, I will only be constructing approximations that may be about exact for certain subsections but it will be largely an approximate recollection valid within rounding errors caused by observations done with finite precision and finite memory storage.

There are of course many other aspects of Pablo as an administrator, a sports enthousiast, a musician, a person, a husband, a father, a friend. It was impossible to collaborate with him and not instantly be charmed by his warm personality. His colleagues and students were friends by definition. However I will avoid this emotional quicksand and mainly stick to the mathematics in this lecture.

## Matrix methods for quadrature formulas on the unit circle

**María J. Cantero (Univ. Zaragoza, Spain)**

In this talk we present some results concerning the computation of quadrature formulas on the unite circle. The recurrence relation satisfied by the orthogonal Laurent polynomials with respect to a measure defined on the unit circle, gives rise to a certain unitary five-diagonal matrix. This matrix can be used to compute the nodes and the weights of Szegö's quadrature formulas on the unit circle. On the other hand, these quadrature formulas can be computed alternatively using Hesenberg matrices.

Orthogonal polynomials are a particular case of orthogonal rational functions with prescribed poles. Szegö's quadrature formulas can be generalized to orthogonal rational functions. A way to calculate the nodes and the weights to the rational Szegö's quadrature formulas is using matrix representations for orthogonal rational functions with prescribed poles.

In both cases we make a comparative study using different matrix representations to compute such quadrature formulas. We illustrate the preceding results with some numerical examples .

The results presented in this talk are the fruit of joint works with Pablo González-Vera and one of his closest collaborators, Adhemar Bultheel.

## There's something about approximation beyond extremality

**Bernardo De la Calle (Univ. Politécnica de Madrid, Spain)**

Classical results on approximation of analytic functions by Taylor series or row sequences of Padé approximants have been extended over the last decades to other type of approximations by means of potential theory, with equilibrium measures and extremal approximants playing a major role.

In this lecture we will show how some of the above results can be extended to general classes of interpolatory (non-extremal) approximants as long as the information given by the table of interpolation points can be properly plugged into the formulation of the problem. Namely, we will be concerned by the characterization of the region of analytic (or meromorphic) continuation of a function in terms of the geometric rate of convergence of its approximants on a given compact set and by the extension of the classical Jentzsch-Szegö theorem on zeros of Taylor polynomials.

## Phase transitions and equilibrium measures in random matrix models

**Andrei Martínez Finkelshtein (Univ. Almería, Spain)**

We are interested in the so-called phase transitions in the Hermitian random matrix models with a polynomial potential. Or, in a language more familiar to approximators, we study families of equilibrium measures on the real line in a polynomial external field. The total mass of the measure is considered as the main parameter, which may be interpreted also either as temperature or time. By phase transitions we understand the loss of analyticity of the equilibrium energy.

Our main tools are differentiation formulas with respect to the parameters of the problem, and a representation of the equilibrium potential in terms of a hyperelliptic integral. This allows to find a dynamical system that describes the evolution of families of equilibrium measures. On this basis we are able to systematically derive results on phase transitions, such as the local behavior of the system at all kinds of phase transitions. We discuss in depth the case of the quartic external field.

This is a joint work with R. Orive, and E. A. Rakhmanov.

## A saga of canary approximators: The legacy of Pablo

**Francisco Perdomo Pío (Univ. La Laguna, Spain)**

Along his very fruitful career, Pablo González Vera was building a nice and quite large family of canary approximators and numerical integrators: his academic "sons". In the first part of this talk, a brief overview of the progressive growth of this family will be made. In the second part, some of the contributions of the youngest son of Pablo (that is, myself) will be revised

This is a Joint work with Matías Camacho Machín, Ramón A. Orive Rodríguez, Mateo M. Jiménez Paiz, Juan C. Santos León, Carlos J. Díaz Mendoza, Leyla Daruis Luis and Ruymán Cruz Barroso

## Zero distribution of Hermite - Pade polynomials

**Evguenii A. Rakhmanov (Univ. South Florida in Tampa, USA)**

The lecture will be devoted to a review of some old and new conjectures and results related to zero distribution (weak asymptotics) of Hermite-Padé polynomials.

## Projective joint spectra, commuting operators and optimal recovery of analytic functions

**Michael Stessin (Univ, Albany, NY, USA)**

A connection between projective joint spectra of Toeplitz operators and optimal algorithms in optimal recovery of analytic functions was discovered by P.González-Vera and M.Stessin in 2012. Pablo and I planned to further work on the geometry of joint spectra. In the talk we will discuss the latest developments in this area.

## Orthogonal polynomials for Minkowski's question mark function

**Walter Van Assche (KU Leuven, Belgium)**

Hermann Minkowski introduced a function in 1904 which maps quadratic irrational numbers to rational numbers and this function is now known as Minkowski's question mark function since Minkowski used the notation ?(x). This function turns out to be a monotone increasing and continuous function on [0,1] with ?(0)=0 and ?(1)=1 which is singular. Hence it defines a singular continuous measure q on [0,1] and one can show that the support of this measure is [0,1]. The question mark function is also known as the slippery devil's staircase. There are several ways to define the question mark function or the corresponding measure: one can use the continued fraction expansion of real numbers in [0,1], it is the asymptotic distribution of numbers in the nth Farey sequence as n tends to infinity and it can be given as the fixed point of an iterated function system consisting of two rational functions.

Our interest is in the (monic) orthogonal polynomials (P_n)_{n} for the Minkowski measure q and in particular in the behavior of the recurrence coefficients in their three term recurrence relation

xP_{n}(x) = P_{n+1}(x) + b_{n} P_{n}(x) + a_{n}^{2} P_{n-1}(x).

The symmetry of the question mark function gives b_{n}=1/2 for all n >= 0. The behavior of a_{n}^{2} is more complicated and is still an open problem. We will show some numerical experiments using the Stieltjes-Gautschi method with a discrete measure supported on the Farey sequence. We also explain how one can compute the moments of the measure q, from which one can also compute the recurrence coefficients. This is however a badly conditioned problem and does note allow the computation of sufficiently many a_{n}^{2} to draw some conclusions about their asymptotic behavior.