## Orthogonality, Approximation Theory, and their applications in Science and Technology

**Title: Orthogonality, Approximation Theory, and their applications in Science and Technology****.**

**Head:** Francisco Marcellán Español (Universidad Carlos III de Madrid, Spain)**:**

**Funding Agency:** Ministerio de Ciencia e Innovación de España.

**Code:** MTM2009.12740-C03

**Institutions:** Universidad Carlos III de Madrid, Universidad de Sevilla, Universidad de La Rioja, Universidad de Zaragoza, University of Copenhagen, University of New South Wales.

**Participans:Section 1. Universidad Carlos III de Madrid.**

1) Francisco Marcellán Español (coordinator, U. Carlos III de Madrid, Spain)

2) Guillermo López Lagomasino (U. Carlos III de Madrid, Spain)

3) Elena Romera Colmenarejo (U. Carlos III de Madrid, Spain)

4) Jorge Sánchez Ruiz (U. Carlos III de Madrid, Spain)

5) Jorge Arvesu Carballo (U. Carlos III de Madrid, Spain)

6) Hector Pijeira Cabrera (U. Carlos III de Madrid, Spain)

7) Eva Touris Lojo (U. Carlos III de Madrid, Spain)

8) Ulises Fidalgo Prieto (U. Carlos III de Madrid, Spain)

9) Fernando Lledó Macau (U. Carlos III de Madrid, Spain)

10) Jorge Borrego Morell (U. Carlos III de Madrid, Spain)

11) Kenier Castillo Rodriguez (U. Carlos III de Madrid, Spain)

12) Edmundo Huertas Cejudo (U. Carlos III de Madrid, Spain)

13) Junot Cacoq (U. Carlos III de Madrid, Spain)

14) Ignacio Alvarez Rocha (U. Carlos III de Madrid, Spain)

15) Roberto Costas Santos (U. Carlos III de Madrid, Spain)

16) Ana Portilla Ferreira (U. Carlos III de Madrid, Spain)

**Section 2. Universidad de Sevilla**.

17) Antonio Durán Guardeño (coordinator, U. Sevilla, Spain)

18) Renato Álvarez Nodarse (U. Sevilla, Spain)

19) Guillermo Curbera Costello (U. Sevilla, Spain)

20) Pedro López Rodríguez (U. Sevilla, Spain)

21) Mirta Castro Smirnova (U. Sevilla, Spain)

22) Manuel Domínguez de la Iglesia (U. Sevilla, Spain)

23) Olvido Delgado Garrido (U. Sevilla, Spain)

24) Christian Berg (University of Copenhagen, Denmark)

25) Werner Ricker (University of New South Wales, Australia )

26) Francisco Javier Carrillo Alanís (U. Sevilla, Spain)

**Section 2. Universidad de La Rioja - Universidad de Zaragoza.**27) Oscar Ciaurri Ramírez (coordinator, U. La Rioja, Spain)

28) Juan Luis Varona Malumbres (U. La Rioja, Spain)

28) Javier Pérez Lázaro (U. La Rioja, Spain)

30) Judit Mínguez Ceniceros (U. La Rioja, Spain)

31) Francisco José Ruiz Blasco (U. La Rioja, Spain)

32) Mario Pérez Riera (U. Zaragoza, Spain)

33) Ana Peña Arenas (U. Zaragoza, Spain)

34) Jesus Munárriz Aldaz (U. Zaragoza, Spain)

35) Manuel Alfaro García (U. Zaragoza, Spain)

36) Maria Luisa Rezola Solaun (U. Zaragoza, Spain)

37) Hermann Render (U. La Rioja, Spain)

38) Jose Carlos Soares Petronilho (Universidad de Coimbra, Portugal)

**Abstract:** In this project we are dealing with the analytic properties of families of orthogonal polynomials with respect to several models of inner products and, on the other hand, we explore their scientific and technological applications (the modellisation of several discrete systems of quantum oscillators and other physical and biological systems like macromolecules and molecular motors among other illustrative examples). More precisely, we will focus our attention on three cases of orthogonality where the teams involved in the project have a recognized experience and worldwide leadership.

(a) Matrix orthogonality with respect to a positive definite matrix of measures supported on the realk line. Here we will deal with the spectral study of second order linear differential operators whose coefficients are matrix polynomials and their eigenfunctions are matrix orthogonal polynomials. As applications, we will consider the modellisation of relativistic quantum systems (Dirac equation) with Coulombian potential, discrete Markov chains when the interactions are not reduced to the closest neighbors and other problems with potential impact on the diagnosis by medical imaging using tensor tomography.

(b) Sobolev orthogonality where the derivatives of polynomials are involved in the weighted inner product. These orthogonal polynomials present some advantages with respect to the standard ones when spectral methods are considered in the numerical analysis of boundary value problems both for differential and partial differential equations as well as they improve the standard techniques in Approximation Theory when Fourier-Sobolev expansions are considered.

(c) Orthogonality with respect to vector measures and their applications in the study of some dynamical systems (infinite dimensional SIMO systems).

We will also deal with other related fields: Moment problem theory, rational approximation (mainly Padé approximants and their extensions, with applications in the study of the stability of time delay dynamical systems) as well as computational methods for Special Functions of relevance in physical-mathematical models, Number Theory, numerical quadrature, Fourier series, and Operator Theory. The techniques that we will use are Matrix Analysis, Potential Theory, Fourier Analysis, Operator Theory, Interpolation , and classical Complex Analysis.

Guillermo López Lagomasino

Elena Romera Colmenarejo

Jorge Sánchez Ruiz

Jorge Arvesu Carballo

Hector Pijeira cabrera

Eva Touris Lojo

Ulises Fidalgo Prieto

Fernando Lledó Macau

Jorge Borrego Morell

Kenier Castillo Rodriguez

Edmundo Huertas Cejudo

Junot Cacoq

Ignacio Alvarez Rocha

Roberto Costas Santos

Ana Portilla Ferreira

## Approximation and orthogonality: from theory to applications (AOTA)

**Title: Approximation and orthogonality: from theory to applications (AOTA).**

**Head:** Andrei Martínez Finkelshtein (Universidad de Almería, Spain)**:**

**Funding Agency:** Ministerio de Ciencia e Innovación de España.

**Code:** MTM2011-28952-C02-01

**Institutions:** Universidad de Almería, Universidad de Zaragoza, Universidad Politécnica de Madrid, Universidad de Granada, University of Oregon Eugene and Katholieke Universiteit Leuven.

**Participans:Section 1. Universidad de Almería.**

1) Andrei Martínez Finkelshtein (coordinator, U. Almería, Spain)

2) Juan J. Moreno Balcázar (U. Almeria, Spain)

3) Pedro Martínez González(U. Almeria, Spain)

4) Darío Ramos López (U. Almeria, Spain)

5) Luis Velázquez Campoy (U. Zaragoza, Spain)

6) María José Cantero Medina (U. Zaragoza, Spain)

7) Leandro Moral Ledesma (U. Zaragoza, Spain)

8) Alejandro Zarzo Altarejos (U. Pilitécnica de madrid, Spain)

9) Arnoldus B. J. Kuijlaars (KULeuven, Belgium)

**Section 2. Universidad de Granada**.

10) Miguel A. Piñar González (coordinador, U. Granada, Spain)

11) Teresa E. Pérez Fernández (U. Granada, Spain)

12) Lidia Fernández Rodríguez (U. Granada, Spain)

13) Antonia M. Delgado Amaro (U. Granada, Spain)

14) Yuan Xu (U. Oregon Eugene, USA)

**Abstract:** This project combines both ambitious goals in fundamental research on orthogonal polynomials, of one and several variables, special functions and their analytic and structural properties, with the applications of this knowledge in other branches of mathematics (stochastic processes, combinatorics, numerical analy- sis), physics (statistical physics, integrable systems, quantum mechanics, quantum computation), and technology (signal processing and diagnostic tools in ophthalmology, with applications in clinical practice). Some of the problems to be tackled are:

- Further development of new techniques, in particular, from complex analysis, operator theory, and the Riemann-Hilbert characterization, to the study of asymptotics and other properties of or- thogonal polynomials, both of one and several variables.

- Study of both classical stochastic processes (new random matrix models, non-intersecting diffu- sion processes, etc.) and quantum random walks by means of orthogonal polynomials approach, based on the results and methods obtained in the previous item.

- Asymptotic analysis of rational approximants and of polynomial solutions of ODE.

- Development of fast numerical algorithms related to orthogonal polynomials.

- Study of information measures of families of special functions, with applications in quantum me- chanics and medical imaging.

- Development of methods of surface reconstruction and medical imaging with clinical applications in ophthalmology and human vision.

## Orthogonal Polynomials, Gaussian and Rational Quadratures

**Title: Orthogonal Polynomials, Gaussian and Rational Quadratures.**

**Head:** M. Alicia Cachafeiro López (Universidad de Vigo, Spain)

**Funding Agency:** Ministerio de Ciencia e Innovación de España.

**Code:** MTM2008-00341

**Institutions:** Universidad de Vigo.

**Participans:**1) M. Alicia Cachafeiro López (U. Vigo, Spain)

2) Elías Berriochoa Esnaola (U. Vigo, Spain)

3) José Manuel García Amor (U. Vigo, Spain)

4) Jesús Illán González (U. Vigo, Spain)

5) Eduardo Martinez Brey (U. Vigo, Spain)

6) Carlos Pérez Iglesias (U. Vigo, Spain)

**Abstract:** The present project is inserted into the Numerical integration theory, Interpolation and Orthogonal and

Multiorthogonal Polynomial theory, all of which constitute a very active research area, with uncountable applications in Mathematics as well as in Physics and Engineering .

Among other purposes of this project we expect to obtain convergence results related to Lagrange, Hermite and generalized Hermite interpolation on the unit circle T. This study considers a class of nodal systems more general than the usual ones, under assumptions weak enough. Moreover, in the Hermite case when the derivatives of the interpolated function do not vanish, we try to find conditions which guarantee convergence when the interpolation procedure is applied to continuous functions. Further, we expect that the convergence behavior will be optimal in the sense that it can not be improved. In the generalized Hermite case, in which appear successive derivatives from 0 to n, the novelty consists in studying problems of convergence on T and in the exterior, with the precision of the speed of convergence.

Another objectives are the study of the convergence and the computation of single and

simultaneous quadrature formulas with a wide variety of modified weight functions. The main objective of this part is the evaluation of definite integrals whose mass is highly concentrated near some points and its numerical implementation.

## Numerical and Asymptotic Methods for the Computation of Mathematical Functions and Associated Numerical Software.

**Title: Numerical and Asymptotic Methods for the Computation of Mathematical Functions and Associated Numerical Software.**

**Head:** Javier Segura Sala (Universidad de Cantabria, Spain)**:**

**Funding Agency:** Ministerio de Ciencia e Innovación de España.

**Code:** MTM2009-11686

**Institutions:** Universidad de Cantabria, Universidad Carlos III de Madrid and Centrum Wiskunde & Informatica (*CWI*).

**Participans: **1) Javier Segura Sala (U. Cantabria, Spain)

2) Amparo Gil Gómez (U. Cantabria Spain)

3) Alfredo Deaño Cabrera (U. Carlos III de Madrid, Spain)

4) Nico M. Temme (CWI Amsterdam, Netherlands)

**Abstract:** The main goal of the project is the development of numerical and asymptotic methods for the computation of mathematical functions of relevance in many scientific fields (for example in engineering, astrophysics, financial mathematics, econometrics...). The intrinsic complexity of the problem of evaluating functions depending on several parameters requires a rigorous analysis of the several possible methods in order to build reliable and efficient computational schemes. Several tools are considered, among them convergent and divergent (asymptotic) expansions, quadrature methods, recurrences and continued fractions. The selection of one method or another, depending on the range of parameters, depends on efficiency and stability issues. The development of software for computing functions is a practical outcome of the project. The goal is to build verified software and enlarge the set of functions available in public numerical libraries. This project, which is a continuation of previous efforts (projects MTM2004-01357 and MTM2006-09050), is strongly related to an international effort for improving and enlarging the open libraries with verified numerical software for the computation of mathematical functions.

## Initial and boundary value problems, analytical techniques and advanced numerical methods.

**Title: Initial and boundary value problems, analytical techniques and advanced numerical methods.**

**Head:** José Luis López (Universidad Pública de Navarra, Spain)

**Funding Agency:** Ministerio de Ciencia e Innovación de España.

**Code:** MTM2010-21037

**Institutions:** Universidad Pública de Navarra and Universidad de Zaragoza.

**Participans (section of Special Functions): **1) José Luis López (U. Pública de Navarra, Spain)

2) Pedro Pagola (U. Pública de Navarra, Spain)

3) Ester Pérez (U. de Zaragoza, Spain)

4) Chelo Ferreira (U. de Zaragoza, Spain)

**Abstract:** Design new Liouville-Neumann methods for the approximation of solutions of initial value problems with regular singular points. From these new methos, obtain new analytic approximations of special functions and formulate a Picard-Lindelov´s theorem for singular initial value problems.

Design algorithms that approximate the solutions of one-dimensional boundary value problems based on the multi-point Taylor approximation. From these algorithms, formulate an existence and uniqueness theorem for one-dimensional boundary value problems.

Build a quasi-factorization method for linear differential equations of arbitrary order from which we can obtain a new family of equivalent integral equations. From these integral equations, design new Liouville-Neumann expansions and apply them to the approximation of special functions.