Ortogonalidad y Aproximación. Teoría y Aplicaciones Físicas y Técnicas. Print
Written by Guillermo López Lagomasino   
Sunday, 05 April 2009 00:00

Title: Orthogonality and Approximation. Theory and Physical and Technical Applications.

Code: MTM2006-13000-C03-02.

Center: Ministerio de Ciencia e Innovación de España.

Period: October, 2006- September, 2009.

Head: G. López Lagomasino

Number of participants: 20


Summary 


The aim of this project is, on one hand, to investigate the analytic properties of orthogonal polynomials with respect to several models of orthogonality and, on the other, to explore scientific, technological and medical (medical diagnosis by image) application of this study. More specifically, we will study the orthogonality: (a) of matrix type: with respect to a positive definite matrix of measures on the real line; furthermore, we explore the presence of matrix orthogonal polynomials that verify second order differential equatons in modeling quantum relativistic systems  (Dirac equation) with coulombian potential, as well as the corresponding time and limiting problems presumably associated with problems of limited angles in tensor tomography; (b) of Sobolev type: where the derivatives of the polynomials appear; Sobolev orthogonal polynomials present advantages for the numerical treatment, by means of spectral methods, of boundary value problems of (ordinary and partial) differential equations, as well as in approximation problems in Fourier-Sobolev series; (c) with respect to varying measures and its application to the study of certain (infinite dimensional SIMO) dynamical systems; (d) q-polynomials and other special functions and its application to the modeling of different discrete oscillatory quantum systems, and other physical and biological systems such as macromolecules and molecular motors. We will also consider other closely related fields: moment problems, rational approximation (mainly Pade approximation and its generalizations, with applications to the study of the stability of time delay dynamical systems, as well as computational methods for special functions relevant in physical and mathematical models), Fourier series and operator theory. The techniques used are, mainly, of matrix analysis, potential theory, Fourier analysis, operator theory, interpolation and classical complex analysis.


List of Publications

(you can download a copu of a given paper visiting the home page of a listed team member or collaborator)


  1. L. D. Abreu, F. Marcellán, S. Yakubovich. Hardy type theorem for orthogonal functions with respect to their zeros. The Jacobi weight case.  J. Math. Anal. Appl. 341 (2008), 803-812.
  2. M. Alfaro, F. Marcellán, A. Peña y M. L. Rezola. When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?. J. Comput.  Appl. Math. 233 (2010) 1446-1452.
  3. J.  Arvesú. On some properties of q-Hahn multiple orthogonal polynomials.  J.   Comput.   Appl. Math. (2009), 1462-1469.
  4. D. Barrios Rolanía, A. Branquinho, A. Foulquié Moreno. Dynamics and interpretation of some integrable systems via multiple orthogonal polynomials. J.  Math. Anal. Appl. 361 (2010), 358-370
  5. D. Barrios Rolanía, A. Branquinho, Complex high order Toda lattices, J. Difference Equations and Applications. 15 (2009), 197-213
  6. D. Barrios Rolanía, J.R. Gascón Márquez.  Spectrum and Generation of Solutions of the Toda Lattice, Discrete Dynamics in Nature and Society 2009, Article ID 23748 doi:10.1155/2009/237487 (2009)
  7. E. Berriochoa, A. Cachafeiro, F. Marcellan. A new quadrature formule on the unit circle. Numer. Algor. 44 (2007), 391-401.
  8. E. Berriochoa, A. Cachafeiro, J. M. García-Amor, F. Marcellán. New quadrature rules for  Bernstein measures on the interval [-1,1]. Electr. Trans.  Numer. Anal. 30 (2008), 278-290.
  9. A. Branquinho, U. Fidalgo,  A. Foulquié Moreno. Riemann-Hilbert problem associated with Angelesco systems. J. Comp. Appl. Math. 233 (2009), 643-651.
  10. A. Branquinho, A. Foulquié, F. Marcellán, M. N. Rebocho. Coherent pairs of linear functionals on the unit circle. J. Approx. Theory 153 (2008), 122-137.
  11. B. de la Calle Ysern, P. González Vera. Rational quadrature formulae on the unit circle with arbitrary poles. Numer. Math. 107 (2007), 559-587.
  12. B. de la Calle Ysern, F. Peherstorfer,  Ultraspherical Stieltjes polynomials and Gauss-Kronrod quadrature behave nicely for \lambda<0. SIAM J. Numer. Anal. 45 (2007), 770-786.
  13. B. de la Calle Ysern, G. López Lagomasino, L. Reichel. Stieltjes-type polynomials on the unit circle. Math. Comp. 266 (2009), 969-997.
  14. K. Castillo, L. Garza, F. Marcellan. Laurent Polynomials Perturbations of Linear Functionals. An  inverse problem. Electronic Transactions in Numerical Analysis 36 2010, 83-98.
  15. R. S. Costas-Santos. On the elementary symmetric functions of a sum of matrices. J. Algebra Number Theory, Adv. Appl.   1  (2009), 99-112.
  16. R. S. Costas-Santos,  C. R. Johnson, B. Tadchiev. Matrices Totally Positive Relative to a Tree. Electronic Linear Algebra  18  (2009),   211-221.
  17. R. S. Costas-Santos, F. Marcellan. Q-classical orthogonal polynomials. A general approach. Acta Appl. Math. (2010). Doi: 10.1007/ s10440-009-9536-z
  18. R. S. Costas-Santos, J. F. Sánchez-Lara Extensions of discrete classical orthogonal polynomials beyond the orthogonality. J. Comput. Applied Math. 225 (2009), 440-451.
  19. M.A. Delgado, Jeffrey S. Geronimo, Plamen Iliev, Yuan Xu, On a Two-Variable Class of Bernstein–Szego" Measures. Constructive Approx. 30 (2009), 71-91,
  20. J.I. de Vicente, Lower bounds on concurrence and separability conditions. Physical Review A 75, 052320 (2007)
  21. J.I. de Vicente.  Further results on entanglement detection and quantification from the correlation matrix criterion. Journal of Physics A:  41, 065309 (2008).
  22. J.I. de Vicente, S. Gandy, J. Sánchez-Ruiz, Information entropy of Gegenbauer polynomials of integer parameter. Journal of Physics A  40, 8345 (2007).
  23. J.I. de Vicente, J. Sánchez-Ruiz, Improved bounds on entropic uncertainty relations. Physical Review A  77, 042110 (2008).
  24. H. Dueñas, F. Marcellan. Laguerre type orthogonal polynomials. Electrostatic interpretation. Int. J. Pure Appl. Math. 38 (2007), 345-358.
  25. H. Dueñas, F. Marcellán. Jacobi-Type orthogonal polynomials: holonomic equation and electrostatic interpretation. Commun. in the Analytic Theory of Cont. Fractions 15 (2007), 4-19.
  26. H. Dueñas, F. Marcellán. Perturbations  of Laguerre-Hahn functional. Modification by the derivative  of a Dirac delta. Integral   Transf. and Special Funct. 20 (2009), 59-77.
  27. H. Dueñas, F. Marcellan. The Laguerre Sobolev orthogonal polynomials. J. Approx. Theory 162 (2010), 421-440.
  28. H. Dueñas, F. Marcellan. The holonomic equation of the Laguerre-Sobolev type orthogonal polynomials: A non diagonal case. J. Difference Equations Appl. (2010) doi: 10/1080/10236190903456063
  29. H. Dueñas, F. Marcellan. Asymptotic behaviour of Laguerre-Sobolev type orthogonal polynomials. A non diagonal case. J. Comput. Appl. Math. (2010). Doi: 10/1016/j. cam. 2009.07.055.
  30. B. Xh. Fejzullahu, F. Marcellán. On convergence and divergence of Fourier expansions with respect to some Gegenbauer-Sobolev type inner product. Commun. in the Analytic Theory of Continued Fractions 16 (2009), 1-11.
  31. B. Xh. Fejzullahu, F. Marcellán. A Cohen type inequality for Laguerre Sobolev expansions, J. Math. Anal. Appl. 352 (2009), 880-889.
  32. B. Xh Fejzullahu, F. Marcellan. Asymptotic properties of orthogonal polynomials with respect to a non discrete Jacobi-Sobolev inner product. Acta Appl. Math. (2010). Doi: 10.1007/ s10440-009-9511-8.
  33. U. Fidalgo Prieto, J. Illán, G. López Lagomasino. Convergence and computation of simultaneous rational quadrature rules. Numerische Mathematik 106 (2007), 99-128.
  34. U. Fidalgo Prieto, G. López Lagomasino. Generalized Hermite-Padé approximation for Nikishin systems of three functions. J.  Comput. Appl. Math. 233 (2010), 1525-1533.
  35. U. Fidalgo Prieto, A. López García, G. López Lagomasino, V. N. Sorokin. Mixed type multiple orthogonal polynomials for two Nikishin systems. Constructive Approximation   DOI:10.1007/s00365-009-9077-8.
  36. L. Garza, J. Hernandez, F. Marcellán. Orthogonal polynomials and measures on the unit cicle. The Geronimus  Transformations. J. Comput.  Appl. Math. 233 (2010), 1220-1231.
  37. L. Garza, F. Marcellán. Spectral transformations of measures supported on the unit circle and the Szegö transformation. Numer. Algor. 49 (2008), 169-185.
  38. L. Garza, F. Marcellán.  Szegö transforms and Nth order associated polynomials on the unit circle. Computers  and  Math. Appl. 57 (2009), 1659-1671.
  39. L. Garza, F. Marcellán, Linear spectral transformations and Laurent polynomials. Med.  J.  Math. 6 (2009), 273-278.
  40. L. Garza, F. Marcellán. Verblunsky parameters and linear spectral transformations. Meth. Appl. Analysis 16 (2009), 69-86.
  41. L. Garza, F. Marcellán. Szegö transformations and rational spectral transformations for associated polynomials. J. Comput. Appl. Math. 233 (2009), 730-738.
  42. J. Hernandez, F. Marcellán. Geronimus spectral transforms and measures on the complex plane. J. Comput. Appl. Math. 219 (2008), 441-456.
  43. R. H. Heredero, D. Levi, M. Petrera, C. Scimiterna, Multiscale  expansion on the lattice and integrability of partial difference  equations. Journal of Physics A 41 (2008) 1-12.
  44. R. H. Heredero, D. Levi, M. Petrera, C. Scimiterna, Multiscale  expansion and integrability properties of the lattice potential KdV  equation. J. of Nonlinear Math. Phys.15 (2008) 313-323.
  45. R. H. Heredero, E. Reyes. Nonlocal symmetries and a Darboux  transformation for the Camassa–Holm equation. Journal of Physics A, 42 (18) (2009) FTC 182002.
  46. A. López, G. López Lagomasino. Ratio asymptotics of Hermite-Padé orthogonal polynomials mials for Nikishin systems: II. Advances in Mathematics 218 (2008), 1081-1106.
  47. A. López, G. López Lagomasino. Relative asymptotics of multiple orthogonal polynomials of Nikishin systems. J. of Approx. Theory 158 (2009), 214-241
  48. G. López Lagomasino, A. Martínez, I. Pérez, H. Pijeira. Strong asymptotics for Sobolev orthogonal polynomials in the complex plane. J. Math. Anal. Appl. 340 (2008), 521–535.
  49. G. López Lagomasino, J. Mínguez. Fourier-Padé approximants for Nikishin systems. Constructive Approximation 30 (2009), 53-69.
  50. G. López Lagomasino, D. Pestana, J. M. Rodríguez, D. Yakubovich. Computation of conformal representations of compact Riemann surfaces. Math. of Comp. 79 (2010), 365-382.
  51. G. López Lagomasino, L. Reichel, L. Wunderlich. Matrices, moments and rational quadrature. Linear Algebra and its Appl. 429 (2008), 2540-2554.
  52. A. Martínez Finkelshtein,  A.M. Delgado, G.M. Castro, A. Zarzo, J.L. Alió. Comparative análisis of some modal reconstructionmethods of the shape of the cornea from corneal elevation data. IOVS 50 (2009), 5639-5645.
  53. F. Marcellán, R. Sfaxi. Inverse finite-type relations between sequences of polynomials. Rev. Acad. Colombiana de Ciencias Exactas, Fisicas y Naturales 32 (123) (2008), 245-255.
  54. H. Pijeira, C. Díaz, R. Orive. Zeros and logarithmic asymptotics of contracted Sobolev orthogonal polynomials for exponential weights. J. Comp. Appl. Math. 233 (2009),  691-698.
  55. H. Pijeira, C. Díaz, R. Orive. Asymptotically extremal polynomials with respect to varying weights and application to Sobolev orthogonality. J. Math. Anal. Appl. 346 (2008) 480-488.
  56. A. Portilla, Y. Quintana, J. M. Rodríguez, E. Tourís. Weighted Weierstrass' Theorem with first derivatives. J. Math. Anal. Appl. 334 (2007), 1167-1198
  57. A. Portilla, Y. Quintana, J. M. Rodríguez,  E. Tourís.  Weierstrass' Theorem in weighted Sobolev spaces with k derivatives. Rocky Mountain J. of Mathematics 37 (2007), 1989-2024.
  58. A. Portilla, J. M. Rodríguez, E. Tourís, Stability of Gromov hyperbolicity. Journal of Advanced Mathematical Studies 2 (2009), 1-20.
  59. A. Portilla, J. M. Rodríguez, E. Tourís, The multiplication operator, zero location and asymptotic for non-diagonal Sobolev norms, Acta Appl. Math. DOI 10.1007/s10440-009-9541-2.
  60. A. Portilla, E.Tourís. A new characterization of Gromov hyperbolicity of surfaces with negative variable curvature. Publications Matematiques, 53 (2009), 83-10.
  61. J. M. Rodríguez, A simple characterization of weighted Sobolev spaces with bounded multiplication operator, J. Approx. Theory 153 (2008), 53-72.
  62. J. M. Rodríguez, J. M. Sigarreta, Sobolev spaces with respect to measures in curves and zeros of Sobolev orthogonal polynomials, Acta Appl. Math. 104 (2008), 325-353.

Last Updated on Monday, 15 February 2010 19:53