Orthogonality and Approximation. Theory and applications in Science and Technology 
Written by Guillermo López Lagomasino 
Wednesday, 18 September 2013 15:54 
Title: Orthogonality and Approximation. Theory and iApplications in Science and Technology. Code: MTM201236732C0301. Head: F. Marcellán Español Number of participants: 17 Summary The aim of this project is to investigate the analytic properties of orthogonal polynomials with respect to several models of inner products (models in which the teams involved in the project have a wide and credited experience): (a) Matrix orthogonality: with respect to a positive definite matrix of measures supported on the real line. We will focus our attention in the spectral study of second order differential and difference operators whose coefficients are matrix polynomials and whose eigenfunctions are matrix orthogonal polynomials; (b) Sobolev orthogonality: where the derivatives of polynomials are involved in the weighted inner product. For these orthogonal polynomials asymptotic properties in the case of measures with unbounded support as well as approximation problems using FourierSobolev series expansions will be analyzed; (c) Orthogonality with respect to measures supported on the unit circle and their applications in integrable systems as well as the matrix analysis of the multiplication operator and several factorization models related to spectral transformations; (d) Orthogonality with respect to vector measures and their applications in the implementation of simultaneous quadrature formulas and PadéHermite convergence. We will also deal with other related fields: bispectral problems for differential operators, difference operators and qdifferences, Rational Approximation (mainly Padé approximants and their extensions as well as computational methods for Special Functions of relevance in physicalmathematical models), Number Theory, Fourier series, and Operator Theory. The techniques that will be used are mainly Matrix Analysis, Potential Theory, Fourier Analysis, Operator Theory, Interpolation, and classical Complex Analysis. It is also a central objective of the project to explore scientific and technological applications, such as the modellization of quantum relativistic systems (Dirac equation), discrete systems of quantum oscillators and other physical and biological systems like macromolecules and molecular motors, as well as image filtering, random matrices and integrable systems, discrete Markov chains where the interactions are not confined to the closest neighbors.
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Last Updated on Thursday, 19 September 2013 18:02 